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Generalized Locally Toeplitz Sequences:Theory and Applications

Carlo Garoni and Stefano Serra-Capizzano

August 10, 2016

Contents

Preface 4

1 Introduction 51.1 Applications of the theory of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Summary of the theory of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Mathematical background 82.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Multi-index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Matrix-sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Multilevel diagonal sampling matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Separable functions and multivariate trigonometric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Preliminaries on measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Riemann-integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Preliminaries on Linear Algebra and Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.1 Tensor products and direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Singular value and eigenvalue distribution of a matrix-sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7.1 Clustering and attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7.2 Zero-distributed sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Spectral distribution of sequences of perturbed Hermitian matrices 283.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Approximating classes of sequences (a.c.s.) 334.1 The a.c.s. topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 The a.c.s. machinery as a tool for computing singular value and eigenvalue distributions . . . . . . . . . . . . . . . . . . 374.3 The a.c.s. algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Some criterions to identify a.c.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 An extension of the concept of a.c.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Multilevel Toeplitz matrices 495.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Properties of multilevel Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 Schatten p-norms of multilevel Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Multilevel circulant matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5 Spectral and singular value distribution of multilevel Toeplitz matrices: an a.c.s.-based proof . . . . . . . . . . . . . . . 575.6 Extreme eigenvalues of Hermitian multilevel Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 LT and sLT sequences 616.1 The Locally Toeplitz operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1.1 Properties of the Locally Toeplitz operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Definition of LT and sLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3 Zero-distributed sequences, sequences of multilevel diagonal sampling matrices

and sequences of multilevel Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.4 Properties and characterizations of LT and sLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2

7 GLT sequences 787.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Singular value and eigenvalue distribution of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.3 Approximation results for GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.3.1 Topological closure of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.4 Characterizations of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.5 The GLT algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.6 Algebraic-topological definition of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.7 Summary of the theory of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8 Applications 918.1 The algebra generated by Toeplitz sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.2 Variable-coefficient Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.2.1 Consequences of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.2.2 Possible extensions of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.3 Geometric means of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.4 PDE discretizations: the 1-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.4.1 FD discretization of diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.4.2 FD discretization of convection-diffusion-reaction equations (part I) . . . . . . . . . . . . . . . . . . . . . . . . 1018.4.3 FD discretization of convection-diffusion-reaction equations (part II) . . . . . . . . . . . . . . . . . . . . . . . 1038.4.4 FD discretization of convection-diffusion-reaction equations (part III) . . . . . . . . . . . . . . . . . . . . . . . 1058.4.5 FD discretization of higher-order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.4.6 Non-uniform FD discretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.4.7 FE approximation of convection-diffusion-reaction equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.4.8 Schur complements of matrices arising from the FE approximation of a system of PDEs . . . . . . . . . . . . . 1138.4.9 B-spline IgA collocation approximation of convection-diffusion-reaction equations . . . . . . . . . . . . . . . . 1168.4.10 Galerkin B-spline IgA approximation of convection-diffusion-reaction equations . . . . . . . . . . . . . . . . . 1258.4.11 Galerkin B-spline IgA approximation of second-order eigenvalue problems . . . . . . . . . . . . . . . . . . . . 130

8.5 PDE discretizations: the d-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.5.1 FD discretization of convection-diffusion-reaction equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.5.2 B-spline IgA collocation approximation of convection-diffusion-reaction equations . . . . . . . . . . . . . . . . 1388.5.3 Galerkin B-spline IgA approximation of convection-diffusion-reaction equations . . . . . . . . . . . . . . . . . 145

Conclusions and perspectives 152

Bibliography 153

Solutions to some of the exercises 158

3

Preface

Sequences of matrices with increasing size naturally arise in the discretization of continuous problems, such as Integral Equations (IEs)and Partial Differential Equations (PDEs). The theory of Generalized Locally Toeplitz (GLT) sequences was developed to analyze theasymptotic spectrum of these sequences of matrices. In this book, we present the theory of GLT sequences together with some of its mainapplications. We will also refer the reader to the available literature for further applications not included herein. Since the attention ofthis book is focused on a single subject (GLT sequences), we may classify it as a Monograph.

The book is conceptually divided into two parts. The first part (Chapters 1–7) covers the theory of GLT sequences. The second part(Chapter 8) is devoted to the applications, corroborated by several numerical illustrations. Some exercises are scattered in the text andtheir solutions are collected at the end. Each exercise is placed at a particular spot with the idea that the reader possesses at that stage allthe elements to solve it.

The book is intended for a use as a text for graduate courses. It should also be useful as a reference for researchers in the fields ofLinear Algebra, Numerical Analysis and Matrix Analysis. Given its analytic spirit, it could also be of interest for analysts, primarily thoseworking in the fields of Functional Analysis and Operator Theory. If we were to classify this book in some category, we would choosethe Series of Graduate Texts in Mathematics. This however does not prevent a conscious instructor from using the book in a one-semesterundergraduate course. Such a course can be constructed, for example, by presenting to the students the content of Chapters 1–8 in the1-level (1-dimensional) perspective only, thus avoiding tensors, multi-indices, multi-level matrices, and related technicalities.

The reader is expected to be familiar with basic Linear Algebra and Matrix Analysis. Any standard university course on LinearAlgebra covers all that is needed in this book. Concerning Matrix Analysis, an adequate preparation is provided by, e.g., [14] or [58]; inaddition, the reader who knows Chapters 1–4 of Bhatia’s book [10] will certainly take advantage of this. A basic knowledge of Real andComplex Analysis is necessary. For the purposes of this book, Rudin’s book [81] is more than enough. Finally, some familiarity withFourier Analysis and matrix functions will be of help. For Fourier Analysis, any standard university course contains everything is neededhere. For matrix functions, Chapter 1 of Higham’s book [60] is sufficient.

Assuming the reader possesses the above prerequisites, there exists a way of reading this book that allows one to omit all technicalitieswithout losing the core. This is probably ‘the best way of reading’ for those who love practice more than theory, but it is also advisable fortheorists, who can recover the missing details afterwards. It consists in reading carefully the summary in Sections 1.2 and 7.7, collectingeverything is needed to fully understand it, and passing to the applications of Chapter 8. The precise reading scheme is the following:

• Chapter 1;

• Sections 2.1, 2.2, 2.7 (Section 2.7.1 and the proofs of Theorems 2.9–2.10 can be omitted);

• beginning of Chapter 4, before Section 4.2;

• Section 5.1 (Lemma 5.1 can be omitted);

• Section 7.7;

• Chapter 8, except Method 1 of Section 8.4.1.

We conclude by expressing our gratitude to Bruno Iannazzo, Carla Manni and Hendrik Speleers, who awakened the interest in thetheory of GLT sequences and ultimately inspired the writing of this book. We also wish to thank all the colleagues that worked in the fieldof « Toeplitz matrices and spectral distributions », and contributed with their work to lay the foundations of the theory of GLT sequences.We mention in particular Albrecht Böttcher, Leonid Golinskii, Sergei Grudsky, Debora Sesana, Bernd Silbermann, Paolo Tilli, EugeneTyrtyshnikov and Nickolai Zamarashkin.

Carlo Garoni and Stefano Serra-Capizzano

4

Chapter 1

Introduction

The subject of this book is the theory of Generalized Locally Toeplitz (GLT) sequences, which stems from Tilli’s work on Locally Toeplitz(LT) sequences [101] and from the theory of classical Toeplitz operators [4, 19, 20, 24, 25, 59, 76, 100, 103, 104, 108], and was developedby the authors in [50, 51, 52, 89, 90]. Some applications are also presented in Chapter 8. We precisely begin with a discussion of the mainapplications in Section 1.1. Then, in Section 1.2, we summarize the main features of the theory. After reading the overview presented inthis introductory chapter, one is ready to enter into the mathematical details with an already-enough-precise idea of what we are going todeal with.

1.1 Applications of the theory of GLT sequencesAs already proved in several contexts, the theory of GLT sequences is a powerful apparatus for computing/analyzing the asymptoticspectral distribution of the discretization matrices arising from the numerical approximation of continuous problems, such as IntegralEquations (IEs) and, especially, Partial Differential Equations (PDEs). Let us explain this point in more detail. When discretizing alinear PDE by means of a linear numerical method, the actual computation of the numerical solution reduces to solving a linear systemAnun = bn. The size dn of this linear system increases when the discretization parameter n tends to infinity. Hence, what we actuallyhave is not just a single linear system, but an whole sequence of linear systems with increasing size; and what is often observed inpractice is that the sequence of discretization matrices Ann enjoys an asymptotic spectral distribution, which is somehow related tothe spectrum of the differential operator associated with the considered PDE. More precisely, it often happens that, for a large set of testfunctions F (usually, for all continuous functions F with bounded support), the following limit relation holds:

limn→∞

1

dn

dn∑j=1

F (λj(An)) =1

µk(D)

∫D

F (f(x))dx, (1.1)

where λj(An), j = 1, . . . , dn, are the eigenvalues of An, µk is the Lebesgue measure in Rk, and f : D ⊂ Rk → C is a measurablefunction. In this situation, f is referred to as the spectral symbol of the sequence Ann. The spectral symbol provides a ‘compact’and accurate description of the asymptotic spectral distribution of the discretization matrices An. Indeed, the informal (but important)meaning behind (1.1) can be summarized as follows: assuming that f is at least Riemann-integrable and n is large enough, a suitableordering of the eigenvalues λj(An), j = 1, . . . , dn, assigned in correspondence with a uniform grid on D, reconstructs approximatelythe surface x→ f(x) (the graph of f ). In other words, the spectrum of An ‘behaves’ (asymptotically) like a uniform sampling of f overD. The theory of GLT sequences, in combination with the results of [54, 57] concerning the asymptotic spectral distribution of perturbedsequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f . Indeed, the sequence ofdiscretization matrices Ann turns out to be a GLT sequence for many classes of PDEs and numerical methods, especially when thenumerical method belongs to the class of the so-called ‘local methods’. Local methods are, for example, Finite Difference (FD) methods,Finite Element (FE) methods with ‘locally supported’ basis functions, and collocation methods; in short, all standard numerical methodsfor the approximation of PDEs. We refer the reader to [89, 90, 94] for applications of the theory of GLT sequences in the context of FDdiscretizations of PDEs; to [8, 39, 90] for the FE and collocation settings; and to [44, 35, 45, 47, 48, 49, 79] for recent applications to thecase of Isogeometric Analysis (IgA) approximations of PDEs, both in the collocation and Galerkin frameworks.1 We also refer the readerto [1, 82] for a look at the GLT approach to deal with sequences of matrices coming from the approximation of IEs.

At this point, it is worth emphasizing that the discretization matrices An arising from the numerical approximation of PDEs are oftenill-conditioned for large n. In fact, their condition number diverges when n → ∞. The knowledge of the spectral symbol f , which can

1IgA is a modern and successful paradigm introduced in [28, 63] for analyzing problems governed by PDEs. Its goal is to improve the connection between numericalsimulation and Computer-Aided Design (CAD) systems. The main idea in IgA is to use directly the geometry provided by CAD systems and to approximate the solutionsof PDEs by the same type of functions (usually, B-splines or NURBS). In this way, it is possible to save about 80% of the CPU time, which is normally employed in thetranslation between two different languages (e.g., between FEs and CAD or between FDs and CAD).

5

be attained through the theory of GLT sequences, is not only interesting in itself, but could also be exploited in two different ways: (a)to analyze/predict the convergence rate of known iterative methods, such as preconditioned Krylov and multigrid methods, when theyare applied to the ill-conditioned linear systems with coefficient matrix An; (b) to design effective preconditioners and iterative solversfor these linear systems. The reason is clear: the convergence properties of general-purpose iterative methods depend on the spectralfeatures of the matrix to which they are applied. Hence, the spectral information provided by f can be conveniently used for designingfast iterative solvers and/or analyzing their convergence properties. In this respect, we recall that recent estimates on the superlinearconvergence of the CG (Conjugate Gradient) method are strictly related to the asymptotic spectral distribution of the matrices to whichthe method is applied; see [7]. We also refer the reader to [32, 33, 34] for recent developments in the IgA framework, where the spectralsymbol was exploited to design ad hoc iterative solvers for IgA discretization matrices.

1.2 Summary of the theory of GLT sequencesInformally speaking, a GLT sequence Ann is a sequence of matrices with increasing size, equipped with a Lebesgue-measurable(complex-valued) function κ. This function is referred to as the symbol (or kernel) of Ann and it is defined over a domain D of theform [0, 1]d × [−π, π]d, d ≥ 1. Due to the experience coming from the applications and to a detected analogy between the theory ofGLT sequences and the Fourier Analysis,2 a point of D = [0, 1]d × [−π, π]d is usually denoted by (x,θ), where x = (x1, . . . , xd) arethe so-called ‘physical variables’, while θ = (θ1, . . . , θd) are the ‘Fourier variables’.

The main theoretical properties of GLT sequences are summarized in items GLT 1 – GLT 8, which will be made more precise inSection 7.7, once the theory of GLT sequences has been developed. In the following, we write Ann ∼GLT κ to indicate that Ann isa GLT sequence with symbol κ.

GLT 1. The symbol κ : [0, 1]d × [−π, π]d → C of a GLT sequence Ann characterizes the asymptotic singular value distribution ofAnn. This means that, for all continuous functions F with bounded support, we have

limn→∞

1

dn

dn∑j=1

F (σj(An)) =1

(2π)d

∫[0,1]d×[−π,π]d

F (|κ(x,θ)|)dxdθ, (1.2)

where dn is the size ofAn and σj(An), j = 1, . . . , dn, are the singular values ofAn. If moreover the matricesAn are Hermitian,then the symbol κ also characterizes the asymptotic spectral distribution of Ann. This means that, for all continuous functionsF with bounded support, we have

limn→∞

1

dn

dn∑j=1

F (λj(An)) =1

(2π)d

∫[0,1]d×[−π,π]d

F (κ(x,θ))dxdθ, (1.3)

where λj(An), j = 1, . . . , dn, are the eigenvalues of An.

GLT 2. If Ann ∼GLT κ and the matrices An are ‘small perturbations’ of certain Hermitian matrices Xn (uniformly bounded inspectral norm), then (1.3) still holds. More precisely, if An = Xn + Yn, where:

• every Xn is Hermitian;

• the spectral norms of Xn and Yn are uniformly bounded with respect to n;

• the trace-norm of Yn divided by the matrix size dn converges to 0;

then (1.3) holds.

GLT 3. Here we list four important examples of GLT sequences.

• Any sequence of (multilevel) Toeplitz matrices Tn(f)n generated by a function f in L1([−π, π]d) is a GLT sequencewith symbol κ(x,θ) = f(θ).

• Any sequence of (multilevel) diagonal sampling matrices Dn(a)n containing the evaluations over a uniform grid of analmost everywhere continuous function a : [0, 1]d → C is a GLT sequence with symbol κ(x,θ) = a(x).

• Any zero-distributed sequence Znn, i.e., any sequence of matrices possessing an asymptotic singular value distributioncharacterized by the identically zero function, in the sense of eq. (1.2), is a GLT sequence with symbol κ(x,θ) = 0(identically).

• Any sequence of the form Dn(a) Tn(f)n is a GLT sequence with symbol κ(x,θ) = a(x)f(θ). Here, a : [0, 1]d → Cis a continuous function, f is a d-variate trigonometric polynomial, Dn(a) is the matrix obtained from Dn(a) by setting(Dn(a))ij = (Dn(a))min(i,j),min(i,j) for all i, j, and is the componentwise (or Hadamard) product of matrices.

2It was noted in [90] that the theory of GLT sequences can be seen as a generalized Fourier Analysis.

6

GLT 4. If Ann ∼GLT κ then A∗nn ∼GLT κ, where A∗n is the Hermitian transpose of An.

GLT 5. If An =∑ri=1 αi

∏qij=1A

(i,j)n , where r, q1, . . . , qr ∈ N, α1, . . . , αr ∈ C and A(i,j)

n n ∼GLT κij , then Ann ∼GLT κ =∑ri=1 αi

∏qij=1 κij .

GLT 6. If Ann ∼GLT κ and κ 6= 0 a.e., then A†nn ∼GLT κ−1, where A†n is the (Moore–Penrose) pseudoinverse of An.

GLT 7. If Ann ∼GLT κ and each An is Hermitian, then f(An)n ∼GLT f(κ) for all continuous functions f : R→ C.

GLT 8. Ann ∼GLT κ if and only if there exist GLT sequences Bn,mn ∼GLT κm such that κm converges to κ in measure andBn,mn ‘converges’ to Ann.

At this stage, we cannot be more precise about the ‘convergence’ of Bn,mn to Ann. We only anticipate that, by saying « Bn,mn‘converges’ to Ann », we mean that « Bn,mnm is an approximating class of sequences for Ann (as m → ∞) ». Things willbecome more clear in Chapter 4, where the notion of approximating classes of sequences is introduced. We note that items GLT 4 – GLT 6can be summarized by saying that the set of GLT sequences is a *-algebra. Roughly speaking, we may rephrase them as follows: supposethat A(1)

n n, . . . , A(r)n n are GLT sequences with symbols κ1, . . . , κr, and let An = ops(A(1)

n , . . . , A(r)n ) be the matrix obtained from

A(1)n , . . . , A

(r)n by means of certain algebraic operations ‘ops’, such as linear combinations, products, (pseudo)inversions and Hermitian

transpositions; then, Ann is a GLT sequence with symbol κ = ops(κ1, . . . , κr).

7

Chapter 2

Mathematical background

2.1 Notation and terminology• Rm×n (resp. Cm×n) is the space of real (resp. complex) m× n matrices.

• Om and Im denote, respectively, the m ×m zero matrix and the m ×m identity matrix. Sometimes, when the dimension m can beinferred from the context, O and I are used instead of Om and Im.

• If x is a vector and X is a matrix, then xT and x∗ (resp. XT and X∗) are the transpose and the transpose conjugate of x (resp. X).

• We use the abbreviations HPD, HPSD, SPD, SPSD for ‘Hermitian positive definite’, ‘Hermitian positive semidefinite’, ‘symmetricpositive definite’, ‘symmetric positive semidefinite’, respectively.

• If X,Y ∈ Cm×m, the notation X ≥ Y (resp. X > Y ) means that X,Y are Hermitian and X − Y is HPSD (resp. HPD).

• If X,Y ∈ Cm×m, we denote by X Y the componentwise (or Hadamard) product of X and Y : (X Y )ij = xijyij , i, j = 1, . . . ,m.

• If X ∈ Cm×m, we denote by X† the (Moore–Penrose) pseudoinverse of X .

• Given X ∈ Cm×m, Λ(X) is the spectrum of X and ρ(X) is the spectral radius of X , i.e., ρ(X) = maxλ∈Λ(X) |λ|. The eigenvaluesof X are denoted by λj(X), j = 1, . . . ,m; if they are real, the maximum and minimum eigenvalues are aso denoted by λmax(X) andλmin(X).

• If X ∈ Cm×m, we denote by σj(X), j = 1, . . . ,m, the singular values of X . The maximum and minimum singular values are alsodenoted by σmax(X) and σmin(X).

• If 1 ≤ p ≤ ∞, the symbol | · |p denotes both the p-norm of vectors and the associated operator norm for matrices:

|x|p =

(∑mi=1 |xi|p)

1/p if 1 ≤ p <∞,maxi=1,...,m |xi| if p =∞,

x ∈ Cm,

|X|p = maxx∈Cmx6=0

|Xx|p|x|p

, X ∈ Cm×m.

The 2-norm | · |2 coincides with the spectral norm and it will be preferably denoted by ‖ · ‖.

• Given X ∈ Cm×m and 1 ≤ p ≤ ∞, ‖X‖p denotes the Schatten p-norm of X , which is defined as the p-norm of the vector(σ1(X), . . . , σm(X)) formed by the singular values of X; see [10]. The Schatten 1-norm is also called the trace-norm. The Schatten∞-norm ‖X‖∞ = σmax(X) coincides with the spectral norm ‖X‖. The Schatten 2-norm ‖X‖2 = (

∑mi=1 σi(X)2)1/2 coincides with

the Frobenius norm ‖X‖ = (∑mi,j=1 |xij |2)1/2. Note that the Schatten norms are unitarily invariant, i.e., ‖UXV ‖p = ‖X‖p for all

p ∈ [1,∞], all X ∈ Cm×m and all unitary matrices U, V ∈ Cm×m. This follows from the fact that X and UXV have the samesingular values.

• <(X) and =(X) are, respectively, the real and the imaginary part of the (square) matrix X:

<(X) =X +X∗

2, =(X) =

X −X∗

2i,

where i is the imaginary unit (i2 = −1). It is clear that X = <(X) + i=(X) for all square matrices X .

8

• If z ∈ C and ε > 0, we denote by D(z, ε) the disk centered at z and with radius ε, i.e., D(z, ε) = w ∈ C : |w − z| < ε. If S ⊆ Cand ε > 0, we denote by D(S, ε) the ε-expansion of S, which is defined as D(S, ε) =

⋃z∈S D(z, ε).

• The symbol ‘ something t→t0−→ something else ’ means that ‘something’ tends to ‘something else’ when t→ t0.

• Given two sequences ζn and ξn, with ζn ≥ 0 and ξn > 0 for all n, the notation ζn = O(ξn) means that there exists a constant C,independent of n, such that ζn ≤ Cξn for all n; and the notation ζn = o(ξn) means that ζn/ξn → 0 as n→∞.

• Cc(C) (resp. Cc(R)) is the space of complex-valued continuous functions defined on C (resp. R) and with bounded support. Moreover,C1c (R) = Cc(R) ∩ C1(R), where C1(R) is the space of complex-valued functions F defined on R whose real and imaginary parts<(F ), =(F ) are of class C1 over R in the classical sense.

• A ‘functional’ φ is any function defined on some vector space (such as, for example, Cc(C) or Cc(R)) and taking values in C.

• If wi : Di → C, i = 1, . . . , d, are arbitrary functions, w1 ⊗ · · · ⊗ wd : D1 × · · · ×Dd → C denotes the tensor-product function

(w1 ⊗ · · · ⊗ wd)(ξ1, . . . , ξd) = w1(ξ1) · · ·wd(ξd), (ξ1, . . . , ξd) ∈ D1 × · · · ×Dd.

• If g : D → C, we set ‖g‖∞ = supξ∈D |g(ξ)|. If we need/want to specify the domain D of g, we write ‖g‖∞,D instead of ‖g‖∞.Clearly, ‖g‖∞ <∞ if and only if g is bounded over D.

• If h : D → C is continuous over D, with D ⊆ Ck for some k, we denote by ωh(·) the modulus of continuity of h,

ωh(δ) = supx,y∈D‖x−y‖≤δ

|h(x)− h(y)|.

If we need/want to specify D, we will say that ωh(·) is the modulus of continuity of h over D.

• χE is the characteristic (or indicator) function of the set E,

χE(ξ) =

1, if ξ ∈ E,0, otherwise.

• µk denotes the Lebesgue measure in Rk. Throughout this work, all the terminology coming from measure theory (such as ‘measure’,‘measurable’, ‘a.e.’, ‘in Lp’, etc.) is always referred to the Lebesgue measure.

• If f : D ⊆ Rk → C is in Lp(D) and the domain D is clear from the context, we write ‖f‖Lp instead of ‖f‖Lp(D) to indicate theLp-norm of f . Recall that ‖f‖Lp = (

∫D|f |p)1/p for 1 ≤ p <∞, and ‖f‖L∞ = ess supD |f | for p =∞.

• We use a notation borrowed from probability theory to indicate sets. For example, if f, g : D ⊆ Rk → C, then f 6= 1 = x ∈ D :f(x) 6= 1, f ∈ D(z, ε) = x ∈ D : f(x) ∈ D(z, ε), 0 ≤ f ≤ 1, g > 2 = x ∈ D : 0 ≤ f(x) ≤ 1, g(x) > 2, µkf >0, g < 0 is the measure of the set x ∈ D : f(x) > 0, g(x) < 0, χf=0 is the characteristic function of the set where f vanishes,and so on.

2.1.1 Multi-index notationThroughout this work, we will systematically use the multi-index notation. A multi-index i ∈ Zd, also called a d-index, is simply a vectorin Zd; its components are denoted by i1, . . . , id.

• 0, 1, 2, . . . are the vectors of all zeros, all ones, all twos, . . . (their size will be clear from the context).

• For any d-indexm, N(m) =∏dj=1mj andm→∞ means that min(m) = minj=1,...,dmj →∞.

• If h,k are d-indices, h ≤ k means that hr ≤ kr for all r = 1, . . . , d, while h 6≤ k means that hr > kr for at least one r ∈ 1, . . . , d.

• If h,k are d-indices such that h ≤ k, the multi-index range h, . . . ,k is the set j ∈ Zd : h ≤ j ≤ k. We assume for the multi-indexrange h, . . . ,k the standard lexicographic ordering:[

. . .[

[ (j1, . . . , jd) ]jd=hd,...,kd

]jd−1=hd−1,...,kd−1

. . .

]j1=h1,...,k1

. (2.1)

For instance, in the case d = 2 the ordering is

(h1, h2), (h1, h2 + 1), . . . , (h1, k2), (h1 + 1, h2), (h1 + 1, h2 + 1), . . . , (h1 + 1, k2),

. . . . . . , (k1, h2), (k1, h2 + 1), . . . , (k1, k2).

9

• When a d-index j varies over a multi-index range h, . . . ,k (this is sometimes written as j = h, . . . ,k), it is understood that j variesfrom h to k following the specific ordering (2.1). For instance, if m ∈ Nd and if we write x = [xi]

mi=1, then x is a vector of size

N(m) whose components xi, i = 1, . . . ,m, are ordered in accordance with (2.1): the first component is x1 = x(1,...,1,1), the secondcomponent is x(1,...,1,2), and so on until the last component, which is xm = x(m1,...,md). Similarly, if

X = [xij ]mi,j=1, (2.2)

then X is a N(m) ×N(m) matrix whose components are indexed by two d-indices i, j, both varying from 1 to m according to thelexicographic ordering (2.1).

• If i, j ∈ Zd are multi-indices, i j means that i precedes (or equals) j in the lexicographic ordering (which is a total ordering on Zd).Moreover, we define

i ∧ j =

i if i j,j if i j. (2.3)

Note that i ∧ j is the minimum among i and j with respect to the lexicographic ordering.

• Given h,k ∈ Zd with h ≤ k, the notation∑kj=h indicates the summation over all j in h, . . . ,k.

• Operations involving d-indices that have no meaning in the vector space Zd must always be interpreted in the componentwise sense. Forinstance, np = (n1p1, . . . , ndpd), αi/j = (αi1/j1, . . . , αid/jd) for all α ∈ C (of course, the division is defined when j1, . . . , jd 6= 0),i2 = (i21, . . . , i

2d), max(i, j) = (max(i1, j1), . . . ,max(id, jd)), imodm = (i1 modm1, . . . , id modmd), and so on.

• When a multi-index appears as subscript or superscript, we often suppress the parentheses to simplify the notation. For instance, thecomponent of the vector x = [xi]

mi=1 corresponding to the multi-index i is denoted by xi or by xi1,...,id , and we preferably avoid the

heavy notation x(i1,...,id).

2.1.2 Matrix-sequencesIn all this work, by a sequence of matrices (or matrix-sequence) we mean a sequence of the form Ann, where:

• n varies in some infinite subset of N;

• n = n(n) is a multi-index in Nd which depends on n, and n→∞ when n→∞;

• An is a square matrix of size N(n).

Recall from the previous section that n→∞ means minj=1,...,d nj →∞. Unless otherwise stated, the multi-index that parameterizes amatrix-sequence is always assumed to be a d-index. We will avoid to repeat this every time, so as to simplify the presentation.

2.2 Multilevel diagonal sampling matricesThree classes of matrix-sequences, which can be regarded as the building blocks of the theory of GLT sequences, will be of particularinterest in the following: zero-distributed sequences, sequences of multilevel diagonal sampling matrices and sequences of multilevelToeplitz matrices. Here, we introduce the multilevel diagonal sampling matrices, while zero-distributed sequences and multilevel Toeplitzmatrices will be considered in more detail in Section 2.7.2 and Chapter 5, respectively.

For n ∈ Nd and a : [0, 1]d → C, we define the d-level diagonal sampling matrix Dn(a) as the following diagonal matrix of sizeN(n):

Dn(a) = diagi=1,...,n

a( in

),

where we recall that i varies from 1 to n according to the lexicographic ordering (2.1). For example, if d = 2 then

Dn(a) = diagi1=1,...,n1

[diag

i2=1,...,n2

a( i1n1,i2n2

)].

Note that Dn(a) can also be defined through a recursive formula: if d = 1, then

Dn(a) = diagi=1,...,n

a( in

);

10

if d > 1, then

Dn(a) = Dn1,...,nd(a) = diagi1=1,...,n1

Dn2,...,nd

(a( i1n1, x2, . . . , xd

)), (2.4)

where a(i1/n1, x2, . . . , xd) : [0, 1]d−1 → C is the (d− 1)-variate function defined by (x2, . . . , xd) 7→ a(i1/n1, x2, . . . , xd).For n ∈ Nd and a : [0, 1]d → C, let Dn(a) be the symmetric matrix defined by

(Dn(a))i,j = (Dn(a))min(i,j),min(i,j) =

(Dn(a))i,i if i ≤ j,(Dn(a))j,j if i > j, i, j = 1, . . . , N(n). (2.5)

In multi-index notation,

(Dn(a))i,j = (Dn(a))i∧j,i∧j =

(Dn(a))i,i if i j,(Dn(a))j,j if i j, , i, j = 1, . . . ,n, (2.6)

which means that Dn(a) can be written in the form (2.2) as follows:

Dn(a) =[a(i ∧ jn

)]ni,j=1

. (2.7)

As we shall see in Chapter 8, the matrix Dn(a) plays an important role in the applications. For d = 1, we have

Dn(a) =

a( 1n ) a( 1

n ) a( 1n ) · · · · · · a( 1

n )

a( 1n ) a( 2

n ) a( 2n ) · · · · · · a( 2

n )

a( 1n ) a( 2

n ) a( 3n ) · · · · · · a( 3

n )

......

...

......

...

......

... a(n−1n ) a(n−1

n )

a( 1n ) a( 2

n ) a( 3n ) a(n−1

n ) a(1)

, (Dn(a))ij = a

(min(i, j)

n

), i, j = 1, . . . , n. (2.8)

2.3 Separable functions and multivariate trigonometric polynomialsLet I1, . . . , Id ⊆ R be measurable sets and let f : I1 × · · · × Id → C. We say that f is separable if there exist measurable functionsfi : Ii → C, i = 1, . . . , d, such that f = f1 ⊗ · · · ⊗ fd. In this case, the functions f1, . . . , fd are called factors of f and f1 ⊗ · · · ⊗ fd issaid to be a factorization of f . Note that the factorization is not unique: it suffices to choose d constants c1, . . . , cd such that c1 · · · cd = 1in order to obtain another factorization f = c1f1 ⊗ · · · ⊗ cdfd. Note also that any separable function is measurable.

Let f : I1 × · · · × Id → C be separable and take a factorization f = f1 ⊗ · · · ⊗ fd. If f ∈ Lp(I1 × · · · × Id) and f is not a.e. equalto 0, then fi ∈ Lp(Ii) for all i = 1, . . . , d. Indeed, for p <∞ we have∫

I1×···×Id|f |p =

d∏i=1

∫Ii

|fi|p.

Since∫Ii|fi|p 6= 0 for all i (otherwise f = 0 a.e., contrary to the assumption), it follows that f ∈ Lp(I1 × · · · × Id) if and only if

fi ∈ Lp(Ii) for all i. For the case p = ∞, we only prove that f1 ∈ L∞(I1), because the proof for the other factors is similar. Since fis not a.e. equal to 0, in particular f2 ⊗ · · · ⊗ fd is not a.e. equal to 0. Hence, µd−1 |f2 ⊗ · · · ⊗ fd| ≥ ε > 0 for some ε > 0. If weassume by contradiction that f1 /∈ L∞(I1), then µ1|f1| ≥ α > 0 for all α > 0. This implies that

µd |f | ≥ α ≥ µd (|f1| ≥ α/ε ∩ |f2 ⊗ · · · ⊗ fd| ≥ ε) = µ1|f1| ≥ α/εµd−1|f2 ⊗ · · · ⊗ fd| ≥ ε > 0,

for all α > 0, which is a contradiction to the assumption that f ∈ L∞(I1 × · · · × Id). In conclusion, we have proved that, for any1 ≤ p ≤ ∞, the factors f1, . . . , fd appearing in any factorization of a separable function f ∈ Lp(I1 × · · · × Id) are themselves in Lp,provided that f is not a.e. equal to 0. In particular, the following lemma holds.

Lemma 2.1. Let f : I1×· · ·×Id → C be a separable function in Lp(I1×· · ·×Id), 1 ≤ p ≤ ∞. Then, there exist functions fi : Ii → C,i = 1, . . . , d, such that f = f1 ⊗ · · · ⊗ fd a.e. and fi ∈ Lp(Ii) for all i = 1, . . . , d.

11

A d-variate trigonometric polynomial is a finite linear combination of the Fourier frequencies eij·θ, j ∈ Zd (here, θ = (θ1, . . . , θd) ∈Rd and j · θ = j1θ1 + . . .+ jdθd). If f(θ) =

∑rj=−r fje

ij·θ is a d-variate trigonometric polynomial, then, as indicated by the notation,the numbers fj , j = −r, . . . , r, are the Fourier coefficients of f .

Let f : Rd → C be a separable d-variate trigonometric polynomial, i.e., a d-variate trigonometric polynomial which is separablein the sense specified above. Let f = f1 ⊗ · · · ⊗ fd be a factorization of f . If f is not identically 0, then f1, . . . , fd are (univariate)trigonometric polynomials. Indeed, since f2, . . . , fd are not identically 0, there exists (ϑ2, . . . , ϑd) such that f2(ϑ2) · · · fd(ϑd) 6= 0. Fromthe definition of d-variate trigonometric polynomials, we see that θ1 7→ f(θ1, ϑ2, . . . , ϑd) = f1(θ1)f2(ϑ2) · · · fd(ϑd) is a (univariate)trigonometric polynomial, and this means that f1 is a trigonometric polynomial. With the same argument, one can show that f2, . . . , fdare trigonometric polynomials as well.

Lemma 2.2. Let f : Rd → C be a separable d-variate trigonometric polynomial. Then, there exist trigonometric polynomials f1, . . . , fd :R→ C such that f = f1 ⊗ · · · ⊗ fd.

We conclude this section by showing that the set of zeros of any d-variate trigonometric polynomial has zero measure. The proof ofthis result can be omitted on first reading.

Lemma 2.3. Let f : Rd → C be a d-variate trigonometrc polynomial with at least one nonzero Fourier coefficient. Thenmdf = 0 = 0.

Proof. The proof proceeds by induction on d. If d = 1, then we can write f(θ) =∑rj=−r fje

ijθ, where the Fourier coefficients fj arenot all equal to 0 by hypothesis. Setting p(z) =

∑rj=−r fjz

j , we have

f(θ) = p(eiθ). (2.9)

The zeros of p(z) over the unit circle S1 = eiθ : θ ∈ [−π, π) coincide with the zeros of the polynomial zrp(z) over S1. It followsfrom the fundamental theorem of Algebra that the number of zeros of p(z) over S1 is finite. Consequently, by (2.9), the number of zerosof f over R is countable, and so m1f = 0 = 0.

Suppose now that d > 1 and assume that the lemma holds for dimensions up to d − 1. Let f(θ) =∑rj=−r fje

ij·θ and setZ = f = 0. By Fubini’s theorem [17, 81],

md(Z) =

∫Z

dθ1 . . . dθd =

∫Rd−1

dθ2 . . . dθd

∫Zθ2,...,θd

dθ1, (2.10)

where, for each fixed (θ2, . . . , θd) ∈ Rd−1, the set Zθ2,...,θd is defined by Zθ2,...,θd = θ1 ∈ R : f(θ1, θ2, . . . , θd) = 0. Write

f(θ) =

r1∑j1=−r1

pj1(θ2, . . . , θd)eij1θ1 , (2.11)

where the pj1 , j1 = −r1, . . . , r1, are (d− 1)-variate trigonometric polynomial,

pj1(θ2, . . . , θd) =

(r2,...,rd)∑(j2,...,jd)=−(r2,...,rd)

fjei(j2,...,jd)·(θ2,...,θd), j1 = −r1, . . . , r1. (2.12)

Let A = (θ2, . . . , θd) ∈ Rd−1 : pj1(θ2, . . . , θd) = 0 for all j1 = −r1, . . . , r1. Since the Fourier coefficients fj are not all equal to 0by assumption, at least one of the pj1 has at least a nonzero Fourier coefficient. Thus, by induction hypothesis, md−1(A) = 0. Moreover,by (2.11),

• if (θ2, . . . , θd) ∈ A, then Zθ2,...,θd = R;

• if (θ2, . . . , θd) ∈ Ac, then there exists a j1 ∈ −r1, . . . , r1 such that pj1(θ2, . . . , θd) 6= 0, and so m1(Zθ2,...,θd) = 0 by inductionhypothesis (or by what we have seen in the first part of the proof).

Going back to (2.10), we see that

md(Z) =

∫A

dθ2 . . . dθd

∫Zθ2,...,θd

dθ1 +

∫Acdθ2 . . . dθd

∫Zθ2,...,θd

dθ1 = 0,

and the proof is complete.

12

2.4 Preliminaries on measure theoryThe convergence in measure is of particular interest in probability theory, and it plays an important role also in the study of GLT sequences.In this section, we recall the definition and provide some basic properties of this convergence.

Definition 2.1 (convergence in measure). Let fm, f : D ⊆ Rk → C be measurable functions. We say that fm → f in measure if, forevery ε > 0,

limm→∞

µk|fm − f | > ε = 0.

We recall that, according to our notation, |fm − f | > ε = x ∈ D : |fm(x)− f(x)| > ε; see Section 2.1.Basic results about the convergence in measure are collected in the next lemmas. Since these results are not so popular, we include

the details of the proofs for the reader’s convenience. However, such proofs are not strictly related to the main subject of interest herein,and so they can be skipped on first reading.

Lemma 2.4. Let fm, gm, f, g : D ⊆ Rk → C be measurable functions.

(i) If fm → f in measure, then |fm| → |f | in measure.

(ii) If fm → f in measure and gm → g in measure, then αfm + βgm → αf + βg in measure for all α, β ∈ C.

(iii) If fm → f in measure, gm → g in measure, and µk(D) <∞, then fmgm → fg in measure.

Proof. (i) We have ||fm| − |f || > ε ⊆ |fm− f | > ε and µk||fm| − |f || > ε ≤ µk|fm− f | > ε. Hence, if fm → f in measure,|fm| → |f | in measure.

(ii) We have

|(αfm + βgm)− (αf + βg)| > ε ⊆ |α||fm − f |+ |β||gm − g| > ε ⊆ |α||fm − f | > ε/2 ∪ |β||gm − g| > ε/2

andµk|(αfm + βgm)− (αf + βg)| > ε ≤ µk|α||fm − f | > ε/2+ µk|β||gm − g| > ε/2.

Hence, if fm → f in measure and gm → g in measure, then αfm + βgm → αf + βg in measure.(iii) For every m and every ε,M > 0, we have

µk|fmgm − fg| > ε ≤ µk|fm − f ||gm|+ |f ||gm − g| > ε ≤ µk|fm − f ||gm| > ε/2+ µk|f ||gm − g| > ε/2= µk|fm − f ||gm| > ε/2, |gm| ≤M+ µk|fm − f ||gm| > ε/2, |gm| > M

+ µk|f ||gm − g| > ε/2, |f | ≤M+ µk|f ||gm − g| > ε/2, |f | > M≤ µk|fm − f |M > ε/2+ µk|gm| > M

+ µkM |gm − g| > ε/2+ µk|f | > M≤ µk|fm − f |M > ε/2+ µk|gm − g| > M/2+ µk|g| > M/2

+ µkM |gm − g| > ε/2+ µk|f | > M. (2.13)

Passing to the limit as m→∞ in (2.13) and using the fact that fm → f in measure and gm → g in measure, we get

lim supm→∞

µk|fmgm − fg| > ε ≤ µk|g| > M/2+ µk|f | > M, (2.14)

for every ε,M > 0. Now we observe that µk|f | > M =∫Dχ|f |>M and χ|f |>M → 0 pointwise when M → ∞. Moreover,

the convergence of χ|f |>M is dominated by the constant 1, which belongs to L1(D) because µk(D) < ∞. Hence, by the dominatedconvergence theorem [17, 81], limM→∞ µk|f | > M = 0. Similarly, limM→∞ µk|g| > M/2 = 0. Passing to the limit as M → ∞in (2.14), we obtain that lim supm→∞ µk|fmgm − fg| > ε = 0, for every ε > 0. Hence, fmgm → fg in measure.

Let K be either R or C and let g : D ⊂ Rk → K be a measurable function defined on a set D with 0 < µk(D) < ∞. Consider thefunctional

φ[g] : Cc(K)→ C, φ[g](F ) =1

µk(D)

∫D

F (g(x))dx. (2.15)

φ[g] is a continuous linear functional on the normed vector space (Cc(K), ‖ · ‖∞), and ‖φ[g]‖ ≤ 1. Indeed, the linearity is obvious andthe continuity, as well as the bound ‖φ[g]‖ ≤ 1, follows from the observation that |φ[g](F )| ≤ ‖F‖∞ for all F ∈ Cc(K). If g is constant,say g = γ a.e., then φ[g] = φ[γ] is the evaluation functional at γ; that is, φ[γ](F ) = F (γ) for every F ∈ Cc(K).

Lemma 2.5. Let K be either R or C, and let gm, g : D ⊂ Rk → K be measurable functions defined on a set D with 0 < µk(D) < ∞.If gm → g in measure, then F gm → F g in L1(D) for all F ∈ Cc(K) and φ[gm] → φ[g] pointwise over Cc(K).

13

Proof. Assume that gm → g in measure. We show that F gm → F g in L1(D) for all F ∈ Cc(K); this immediately implies thatφ[gm] → φ[g] pointwise over Cc(K), because

|φ[gm](F )− φ[g](F )| ≤ 1

µk(D)‖F gm − F g‖L1 .

For every F ∈ Cc(K), every m and every ε > 0,

‖F gm − F g‖L1 =

∫D

∣∣F (gm(x))− F (g(x))∣∣dx

=

∫|gm−g|≥ε

∣∣F (gm(x))− F (g(x))∣∣dx +

∫|gm−g|<ε

∣∣F (gm(x))− F (g(x))∣∣dx

≤ 2‖F‖∞ µk|gm − g| ≥ ε+ ωF (ε), (2.16)

where ωF is the modulus of continuity of F . Since

limm→∞

µk|gm − g| ≥ ε = limε→0

ωF (ε) = 0

(because gm → g in measure and F is uniformly continuous by the Heine-Cantor theorem), passing first to the lim supm→∞ and then tothe limε→0 in (2.16), we conclude that F gm → F g in L1(D).

Lemma 2.5 admits the following converse.

Lemma 2.6. Let K be either R or C, and let gm, g : D ⊂ Rk → K be measurable functions defined on a set D with 0 < µk(D) < ∞.If φ[gm−g] → φ[0] pointwise over Cc(K), then gm → g in measure.

Proof. By hypothesis, for all F ∈ Cc(K) we have

limm→∞

1

µk(D)

∫D

F (gm(x)− g(x))dx = F (0). (2.17)

Suppose by contradiction that gm 6→ g in measure. Then, there exist ε, δ > 0 and a subsequence gmrr such that, for all r,

µk|gmr − g| ≥ ε ≥ δ. (2.18)

Take a real function F ∈ Cc(K) such that F (0) = 1 = maxy∈K F (y) and F (y) = 0 over y ∈ K : |y| ≥ ε. Then, by (2.18), for all rwe have

1

µk(D)

∫D

F (gmr (x)− g(x))dx =1

µk(D)

∫|gmr−g|<ε

F (gmr (x)− g(x))dx ≤ µk|gmr − g| < εµk(D)

≤ µk(D)− δµk(D)

< 1 = F (0),

which is a contradiction to (2.17).

Remark 2.1. Let φ[g] be defined as in (2.15) and assume that φ[g] = φ[0]; then g = 0 a.e. Indeed, if φ[g] = φ[0], then the constantsequence φ[g]m converges pointwise to φ[0] over Cc(K). By Lemma 2.6, this implies that g → 0 in measure, i.e., µk|g| ≥ ε = 0 forevery ε > 0. Hence,

µkg 6= 0 = µk|g| > 0 = µk⋃∞

`=1|g| ≥ 1/`

= 0,

which means that g = 0 a.e.

Lemma 2.7 is the last result we need about measure theory; it will play an important role in the proof of Theorem 7.10. For the proofof Lemma 2.7, we recall that the space generated by the trigonometric monomials

ei ( 2πb1−a1

j1y1 + . . .+ 2πbk−ak

jkyk): j = (j1, . . . , jk) ∈ Zk

is the set of all finite linear combinations of such monomials; we may call it the space of ‘scaled’ k-variate trigonometric polynomials.This space is dense in L1([a1, b1]× · · · × [ak, bk]).

Lemma 2.7. Let κ : [0, 1]d × [−π, π]d → C be a measurable function. Then, there exists a sequence κmm such that κm : [0, 1]d ×[−π, π]d → C is a function of the form

κm(x,θ) =

Nm∑j=−Nm

a(m)j (x)eij·θ, a

(m)j ∈ C∞([0, 1]d), Nm ∈ Nd, (2.19)

and κm → κ a.e.

14

Proof. As observed before the statement of the lemma, the space generated by the trigonometric monomialsei2π`·xeij·θ = ei(2π`1x1+...+2π`dxd+j1θ1+...+jdθd) : `, j ∈ Zd

(2.20)

is dense in L1([0, 1]d × [−π, π]d). The function κm = κχ|κ|≤1/m belongs to L∞([0, 1]d × [−π, π]d) ⊂ L1([0, 1]d × [−π, π]d) andconverges to κ in measure. Indeed, κm → κ pointwise over [0, 1]d × [−π, π]d, and it is known that the convergence a.e. on a set of finitemeasure implies the convergence in measure [17, 81]. Choose a function κm belonging to the space generated by the monomials (2.20),such that ‖κm − κm‖L1 ≤ 1/m. Note that κm is a function of the form (2.19). Then, for all ε > 0,

µ2d|κm − κ| > ε ≤ µ2d|κm − κm| > ε/2+ µ2d|κm − κ| > ε/2 =

∫[0,1]d×[−π,π]d

χ|κm−κm|>ε/2 + µ2d|κm − κ| > ε/2

≤∫

[0,1]d×[−π,π]d

|κm − κm|(ε/2)

+ µ2d|κm − κ| > ε/2 =‖κm − κm‖L1

(ε/2)+ µ2d|κm − κ| > ε/2

which converges to 0 as m → ∞. Hence, κm → κ in measure. Since the convergence in measure on a set of finite measure implies theexistence of a subsequence that converges a.e. [17, 81], passing to a subsequence of κmm (if necessary) we may assume that κm → κa.e.

2.5 Riemann-integrable functionsA function a : [0, 1]d → C is said to be Riemann-integrable if its real and imaginary parts <(a),=(a) : [0, 1]d → R are Riemann-integrable in the classical sense. Recall that any Riemann-integrable function is bounded by definition, so ‖a‖∞ < ∞ for all Riemann-integrable a : [0, 1]d → C. We report below a list of properties possessed by Riemann-integrable functions that will be used in this book,either explicitly or implicitly.

• If α, β ∈ C and a, b : [0, 1]d → C are Riemann-integrable, then αa+ βb is Riemann-integrable.

• If a, b : [0, 1]d → C are Riemann-integrable, then ab is Riemann-integrable.

• If a : [0, 1]d → C is Riemann-integrable and F : C→ C is continuous, then F a : [0, 1]d → C is Riemann-integrable.

• If a : [0, 1]d → C is Riemann-integrable, then a belongs to L∞([0, 1]d) and its Lebesgue and Riemann integrals over [0, 1]d

coincide.

• If a : [0, 1]d → C is bounded, then a is Riemann-integrable if and only if a is continuous a.e., that is, if and only if

µda is discontinuous = µdx ∈ [0, 1]d : a is discontinuous at x = 0.

Note that the last two properties imply the first three ones. A final property of Riemann-integrable functions that will be used throughoutthis book is stated and proved in the next lemma.

Lemma 2.8. Let a : [0, 1]d → R be Riemann-integrable. For each n ∈ Nd, consider the partition of (0, 1]d given by the d-dimensionalrectangles

Ii,n =

(i− 1

n,i

n

]=

(i1 − 1

n1,i1n1

]× · · · ×

(id − 1

nd,idnd

], i = 1, . . . ,n, (2.21)

and let

ai,n ∈

[inf

x∈Ii,na(x), sup

x∈Ii,na(x)

], i = 1, . . . ,n. (2.22)

Thenn∑i=1

ai,nχIi,n → a a.e. in [0, 1]d (2.23)

and

limn→∞

1

N(n)

n∑i=1

ai,n =

∫[0,1]d

a(x)dx. (2.24)

15

Proof. Let x ∈ (0, 1]d be a continuity point of a:

∀ ε > 0 ∃ δ = δ(ε) > 0 : |a(y)− a(x)| ≤ ε ∀y ∈ [0, 1]d such that ‖y − x‖ ≤ δ.

For each ε > 0, let n = n(ε) be such that, for n ≥ n, the sides of every rectangle in (2.21) is smaller than δ/2 (it suffices to choose n sothat min(n) ≥ 2/δ). Then, for every n ≥ n, the unique rectangle of the partition (2.21) that contains x, say Ik,n, is contained in the ballB(x, δ) = y ∈ [0, 1]d : ‖y − x‖ ≤ δ. It follows that |a(y)− a(x)| ≤ ε for all y ∈ Ik,n, and by (2.22) we obtain∣∣∣∣∣

n∑i=1

ai,nχIi,n(x)− a(x)

∣∣∣∣∣ = |ak,n − a(x)| ≤ max(a(x)− inf

y∈Ik,na(y), sup

y∈Ik,na(y)− a(x)

)≤ ε.

Hence,∑ni=1 ai,nχIi,n(x) → a(x) whenever x is a continuity point of a in (0, 1]d, and this implies (2.23) because a is Riemann-

integrable and hence continuous a.e. Eq. (2.24) follows from (2.23) and from the dominated convergence theorem, since∣∣∣∣∣n∑i=1

ai,nχIi,n

∣∣∣∣∣ ≤ ‖a‖∞ <∞, 1

N(n)

n∑i=1

ai,n =

∫[0,1]d

(n∑i=1

ai,nχIi,n

).

2.6 Preliminaries on Linear Algebra and Matrix AnalysisGiven a matrix X ∈ Cm×m, we know from the Singular Value Decomposition (SVD) that

‖X‖ = σmax(X) ≥ |xij |, i, j = 1, . . . ,m. (2.25)

If X is normal, i.e. XX∗ = X∗X , then X is unitarily diagonalizable, meaning that there exist a unitary matrix U and a diagonal matrixD such that X = UDU∗. Using this result and the SVD, it can be shown that the singular values of X are |λj(X)|, j = 1, . . . ,m.

Consequently, ‖X‖ = ρ(X) and ‖X‖p =(∑m

j=1 |λj(X)|p)1/p

for 1 ≤ p < ∞. We observe that, if X is Hermitian (X∗ = X) orskew-Hermitian (X∗ = −X), then X is normal. Since ρ(X) ≤ |X|p for all p ∈ [1,∞] and all matrices X ∈ Cm×m, it is clear that,whenever X is normal, the inequality ‖X‖ ≤ |X|p holds for all p ∈ [1,∞]. If X is not normal, an interesting inequality that replaces theprevious one is the following:

‖X‖ ≤√|X|1|X|∞, X ∈ Cm×m. (2.26)

For the proof, see [14, p. 121] or [58, Corollary 2.3.2]. The inequality (2.26) is particularly useful to estimate the spectral norm of amatrix when we have upper bounds for its components. Indeed, we recall that |X|1 and |X|∞ admit explicit expressions in terms of thecomponents of X . More precisely, |X|1 is the maximum among the 1-norms of the columns of X , and |X|∞ is the maximum among the1-norms of the rows of X .

Given a matrix X ∈ Cm×m, we know from the SVD that rank(X) is the number of nonzero singular values of X . As a consequence,recalling that ‖X‖ = ‖X‖∞ = σmax(X), we obtain the following inequality for the trace-norm of X:

‖X‖1 =

m∑i=1

σi(X) ≤ rank(X)‖X‖ ≤ m‖X‖, X ∈ Cm×m. (2.27)

More generally, let 1 < p, q <∞ be conjugate exponents ( 1p + 1

q = 1); then, using the Hölder inequality, we have

‖X‖1 =

m∑i=1

σi(X) =

rank(X)∑i=1

σi(X) ≤

rank(X)∑i=1

1q

1/qrank(X)∑i=1

σi(X)p

1/p

= rank(X)1/q‖X‖p ≤ m1/q‖X‖p, X ∈ Cm×m.

(2.28)

Other interesting trace-norm inequalities, which provide an upper and lower bound for the trace-norm in terms of the components, are thefollowing:

|trace(X)| ≤ ‖X‖1 ≤m∑

i,j=1

|xij |, X ∈ Cm×m. (2.29)

In particular, the left inequality is Weyl’s majorization theorem for p = 1; see [10, Theorem II.3.6, formula (II.23)]. The proof of (2.29)is simple. Let X = UΣV ∗ be an SVD of X . Then, setting Q = V U∗, the matrix Q is unitary and we have

‖X‖1 = trace(Σ) = trace(U∗XV ) = trace(XQ) ≤m∑i=1

m∑k=1

|xikqki| ≤m∑i=1

maxk=1,...,m

|qki|m∑k=1

|xik| ≤m∑i=1

m∑k=1

|xik|.

16

Moreover, using the Cauchy-Schwarz inequality and the fact that the Euclidean norms of the rows/columns of U and V are 1, we get

|trace(X)| =

∣∣∣∣∣m∑i=1

(UΣV ∗)ii

∣∣∣∣∣ =

∣∣∣∣∣m∑i=1

m∑k=1

σk(X)uikvik

∣∣∣∣∣ =

∣∣∣∣∣m∑k=1

σk(X)

m∑i=1

uikvik

∣∣∣∣∣ ≤m∑k=1

σk(X)

m∑i=1

|uik||vik| ≤m∑k=1

σk(X) = ‖X‖1.

If 1 ≤ p, q ≤ ∞ are conjugate exponents, the following Hölder-type inequality holds for the Schatten norms (see [10, Problem III.6.2and Corollary IV.2.6]):

‖XY ‖1 ≤ ‖X‖p‖Y ‖q, X, Y ∈ Cm×m. (2.30)

An analogous inequality actually holds for all unitarily invariant norms, as shown in [10, Corollary IV.2.6].In the next lemma we provide a variational characterization of Schatten p-norms. A completely analogous characterization actually

holds for all unitarily invariant norms, as proved in [95, Theorem 2.1].

Lemma 2.9. If 1 ≤ p ≤ ∞ and X ∈ Cm×m, then

‖X‖p = sup∣∣(u∗1Xv1, . . . ,u

∗mXvm)

∣∣p, (2.31)

where the supremum is taken over all pairs of orthonormal bases uimi=1, vimi=1 of Cm. If moreover X is HPSD, then

‖X‖p = sup∣∣(u∗1Xu1, . . . ,u

∗mXum)

∣∣p, (2.32)

where the supremum is taken over all orthonormal bases uimi=1 of Cm.

Proof. Let X = UΣV ∗ be an SVD of X , so that U∗XV = diag(σ1(X), . . . , σm(X)). If uimi=1 and vimi=1 are, respectively, thecolumns of U and V , then

‖X‖p =∣∣(σ1(X), . . . , σm(X))

∣∣p

=∣∣(u∗1Xv1, . . . ,u

∗mXvm)

∣∣p.

Hence, ≤ holds in (2.31). On the other hand, suppose that U (with columns uimi=1) and V (with columns vimi=1) are any two unitarymatrices. If Pi = eie

∗i is the orthogonal projection onto the subspace of Cm generated by ei (the i-th vector of the canonical basis),

then the singular values of∑mi=1 PiU

∗XV Pi =∑mi=1(u∗iXvi)Pi are (obviously) |u∗iXvi|, i = 1, . . . ,m. Thus, from the pinching

inequality [10, p. 97, formula (IV.52)], we obtain

∣∣(u∗1Xv1, . . . ,u∗mXv∗m)

∣∣p

=∣∣(|u∗1Xv1|, . . . , |u∗mXvm|)

∣∣p

=

∥∥∥∥∥m∑i=1

PiU∗XV Pi

∥∥∥∥∥p

≤ ‖U∗XV ‖p = ‖X‖p.

Therefore, since uimi=1 and vimi=1 are arbitrary orthogonal bases, we infer that also ≥ holds in (2.31), and this completes the proofof (2.31).

To prove (2.32), we first note that ≥ certainly holds in (2.32) by (2.31). The proof of ≤ is the same as before; it suffices to observethat, since X is HPSD, we have λi(X) = σi(X) for all i = 1, . . . ,m, and, moreover, we can take an SVD of X of the form X = UΣU∗,with Σ = diag(λ1(X), . . . , λm(X)).

In the case p =∞, it is not difficult to see that (2.31)–(2.32) simplify to

σmax(X) = ‖X‖ = ‖X‖∞ = sup‖u‖=‖v‖=1

|u∗Xv|, X ∈ Cm×m, (2.33)

λmax(X) = ‖X‖ = ‖X‖∞ = sup‖u‖=1

u∗Xu, X ∈ Cm×m, X HPSD. (2.34)

In the next theorems, we recall some important interlacing and perturbation theorems for singular values and eigenvalues.

Theorem 2.1 (interlacing theorem for singular values). Let Y = X + E, where X,E ∈ Cm×m and rank(E) ≤ k. Then

σj−k(X) ≥ σj(Y ) ≥ σj+k(X), j = 1, . . . ,m. (2.35)

In (2.35), σ1(X) ≥ . . . ≥ σm(X) and σ1(Y ) ≥ . . . ≥ σm(Y ) are the singular values of X and Y labeled in non-increasing order;moreover, we use the following convention: σj(X) = +∞ if j < 1 and σj(X) = −∞ if j > m.

Theorem 2.2 (interlacing theorem for eigenvalues). Let Y = X + E, where X,E ∈ Cm×m are Hermitian. Let k+, k− ≥ 0 berespectively the number of positive and the number of negative eigenvalues of E:

k+ = #j ∈ 1, . . . ,m : λj(E) > 0, k− = #j ∈ 1, . . . ,m : λj(E) < 0.

17

Thenλj−k+(X) ≥ λj(Y ) ≥ λj+k−(X), j = 1, . . . ,m. (2.36)

In particular, if rank(E) ≤ k thenλj−k(X) ≥ λj(Y ) ≥ λj+k(X), j = 1, . . . ,m. (2.37)

In (2.36)–(2.37), λ1(X) ≥ . . . ≥ λm(X) and λ1(Y ) ≥ . . . ≥ λm(Y ) are the eigenvalues of X and Y labeled in non-increasing order;moreover, we use the following convention: λj(X) = +∞ if j < 1 and λj(X) = −∞ if j > m.

Theorem 2.1 can be seen as a corollary of Theorem 2.2. Indeed, for any A ∈ Cm×m, the eigenvalues of the (2m)× (2m) Hermitianmatrix

A =

[O AA∗ O

]are σj(A), −σj(A), j = 1, . . . ,m; see [10, Exercise II.1.15]. Therefore, applying Theorem 2.2 with Y , X, E in place of Y, X, E, weobtain Theorem 2.1. The proof of Theorem 2.2 can be done by using the result of [10, Exercise III.2.4].

Theorem 2.3 (perturbation theorem for singular values). Let X,Y ∈ Cm×m, then

|σj(X)− σj(Y )| ≤ ‖X − Y ‖, j = 1, . . . ,m.

Here, σ1(X) ≥ . . . ≥ σm(X) and σ1(Y ) ≥ . . . ≥ σm(Y ) are the singular values of X and Y labeled in non-increasing order.

Theorem 2.4 (perturbation theorem for eigenvalues). Let X,Y ∈ Cm×m be Hermitian, then

|λj(X)− λj(Y )| ≤ ‖X − Y ‖, j = 1, . . . ,m.

Here, λ1(X) ≥ . . . ≥ λm(X) and λ1(Y ) ≥ . . . ≥ λm(Y ) are the eigenvalues of X and Y labeled in non-increasing order.

Theorem 2.4 is Weyl’s perturbation theorem [10, Corollary III.2.6]. Theorem 2.3 can be seen as a corollary of Theorem 2.4, byconsidering again the matrices X and Y . Alternatively, Theorem 2.3 (resp. Theorem 2.4) can be proved by using the minimax principlefor singular values (resp. the minimax principle for eigenvalues). We report below these principles, and we also refer the reader to [10,Problem II.6.13] for a general perturbation theorem for singular values, which extends Theorem 2.3.

Theorem 2.5 (minimax principle for singular values). Let X ∈ Cm×m, with singular values σ1(X) ≥ . . . ≥ σm(X); then

σj(X) = maxV subspace of Cm

dimV=j

minx∈V, ‖x‖=1

‖Xx‖ = minV subspace of CmdimV=m−j+1

maxx∈V, ‖x‖=1

‖Xx‖, j = 1, . . . ,m.

In particular,σmax(X) = max

x∈Cm, ‖x‖=1‖Xx‖, σmin(X) = min

x∈Cm, ‖x‖=1‖Xx‖.

Theorem 2.6 (minimax principle for eigenvalues). Let X ∈ Cm×m be Hermitian, with eigenvalues λ1(X) ≥ . . . ≥ λm(X); then

λj(X) = maxV subspace of Cm

dimV=j

minx∈V, ‖x‖=1

x∗Xx = minV subspace of CmdimV=m−j+1

maxx∈V, ‖x‖=1

x∗Xx, j = 1, . . . ,m.

In particular,λmax(X) = max

x∈Cm, ‖x‖=1x∗Xx, λmin(X) = min

x∈Cm, ‖x‖=1x∗Xx.

Theorem 2.5 can be proved by applying Theorem 2.6 to the Hermitian matrixX∗X , whose eigenvalues are the squares of the singularvalues of X . Theorem 2.6 can be found in [10, Corollary III.1.2] or [14, Teorema 6.7]. As a consequence of Theorem 2.6,

Λ(X) ⊆ [λmin(<(X)), λmax(<(X))]× [λmin(=(X)), λmax(=(X))] ⊂ C, X ∈ Cm×m. (2.38)

Indeed, if λ is an eigenvalue of X and x is a corresponding eigenvector with ‖x‖ = 1, then, by Theorem 2.6,

λ = x∗Xx = x∗<(X)x + ix∗=(X)x ∈ [λmin(<(X)), λmax(<(X))]× [λmin(=(X)), λmax(=(X))] .

An important relation between the imaginary parts of the eigenvalues of X and the eigenvalues of =(X) is provided by Theorem 2.7,which is known in the literature as the Ky-Fan theorem.

18

Theorem 2.7. Let X ∈ Cm×m and label the eigenvalues of X and =(X) so that =(λ1(X)) ≥ . . . ≥ =(λm(X)) and λ1(=(X)) ≥. . . ≥ λm(=(X)). Then

k∑j=1

=(λj(X)) ≤k∑j=1

λj(=(X)) (2.39)

for all k = 1, . . . ,m. Moreover, for k = m, the equality holds in (2.39).

The proof of Theorem 2.7 can be found in [10, Proposition III.5.3]. Note that Theorem 2.7 is stated in [10] with ‘< ’ in place of ‘= ’,but this is not an issue because <(X) = =(iX). Using Theorem 2.7, we now prove a result that will be used in Chapter 3.

Lemma 2.10. Let X ∈ Cm×m and ε > 0, then

#j ∈ 1, . . . ,m : |=(λj(X))| > ε ≤ ‖=(X)‖1ε

. (2.40)

In particular, if Λ(<(X)) is contained in the interval I ⊆ R, then

#j ∈ 1, . . . ,m : λj(X) /∈ I × [−ε, ε] ≤ ‖=(X)‖1ε

. (2.41)

Proof. As in Theorem 2.7, label the eigenvalues of X and =(X) so that =(λ1(X)) ≥ . . . ≥ =(λm(X)) and λ1(=(X)) ≥ . . . ≥λm(=(X)). By Theorem 2.7,

∑j:λj(=(X))≥0

λj(=(X)) = maxk=1,...,m

k∑j=1

λj(=(X)) ≥ maxk=1,...,m

k∑j=1

=(λj(X)) =∑

j:=(λj(X))≥0

=(λj(X)). (2.42)

Again by Theorem 2.7, we have∑j:λj(=(X))<0

λj(=(X)) +∑

j:λj(=(X))≥0

λj(=(X)) =∑

j:=(λj(X))<0

=(λj(X)) +∑

j:=(λj(X))≥0

=(λj(X)),

and so (2.42) implies that ∑j:λj(=(X))<0

λj(=(X)) ≤∑

j:=(λj(X))<0

=(λj(X)). (2.43)

Since =(X) is Hermitian, its trace-norm equals the sum of the absolute values of its eigenvalues. Thus, by (2.42)–(2.43),

‖=(X)‖1 =

m∑j=1

|λj(=(X))| =∑

j:λj(=(X))≥0

λj(=(X)) −∑

j:λj(=(X))<0

λj(=(X))

≥∑

j:=(λj(X))≥0

=(λj(X)) −∑

j:=(λj(X))<0

=(λj(X))

=

m∑j=1

|=(λj(X))| ≥∑

j: |=(λj(X))|>ε

|=(λj(X))| ≥ ε ·#j ∈ 1, . . . ,m : |=(λj(X))| > ε.

This proves (2.40). The inequality (2.41) follows from (2.40) and (2.38).

2.6.1 Tensor products and direct sumsIfX,Y are matrices of any dimension, sayX ∈ Cm1×m2 and Y ∈ C`1×`2 , the tensor (Kronecker) product ofX and Y is them1`1×m2`2matrix defined by

X ⊗ Y = [xijY ]i=1,...,m1j=1,...,m2

=

x11Y · · · x1m2Y...

...xm11Y · · · xm1m2

Y

,and the direct sum of X and Y is the (m1 + `1)× (m2 + `2) matrix defined by

X ⊕ Y = diag(X,Y ) =

[X OO Y

].

Tensor products and direct sums possess a lot of nice algebraic properties.

19

(i) Associativity: for all matrices X,Y, Z, (X ⊗ Y ) ⊗ Z = X ⊗ (Y ⊗ Z) and (X ⊕ Y ) ⊕ Z = X ⊕ (Y ⊕ Z). This means that wecan omit parentheses in expressions like X1 ⊗X2 ⊗ · · · ⊗Xd or X1 ⊕X2 ⊕ · · · ⊕Xd.

(ii) The relations (X1 ⊗ Y1)(X2 ⊗ Y2) = (X1X2) ⊗ (Y1Y2) and (X1 ⊕ Y1)(X2 ⊕ Y2) = (X1X2) ⊕ (Y1Y2) hold whenever X1, X2

can be multiplied and Y1, Y2 can be multiplied.

(iii) For all matrices X,Y, (X ⊗ Y )∗ = X∗ ⊗ Y ∗, (X ⊕ Y )∗ = X∗ ⊕ Y ∗ and (X ⊗ Y )T = XT ⊗ Y T , (X ⊕ Y )T = XT ⊕ Y T .

(iv) Bilinearity (of tensor products): (α1X1 + α2X2) ⊗ (β1Y1 + β2Y2) = α1β1(X1 ⊗ Y1) + α1β2(X1 ⊗ Y2) + α2β1(X2 ⊗ Y1) +α2β2(X2 ⊗ Y2) for all α1, α2, β1, β2 ∈ C and for all matrices X1, X2, Y1, Y2 such that X1, X2 are summable and Y1, Y2 aresummable.

(v) Multi-index formula (for tensor products): if we have d matrices Xh ∈ Cmh×mh , h = 1, . . . , d, then

(X1 ⊗X2 ⊗ · · · ⊗Xd)ij = (X1)i1j1(X2)i2j2 · · · (Xd)idjd , i, j = 1, . . . ,m, (2.44)

where m = (m1,m2, . . . ,md). This means that, for all i, j in the multi-index range 1, . . . ,m, the (i, j)-th entry of X1 ⊗X2 ⊗· · · ⊗ Xd is given by (2.44). Note that it makes sense to talk about the (i, j)-th entry of X1 ⊗ X2 ⊗ · · · ⊗ Xd, because we havefixed for the set 1, . . . ,m the lexicographic ordering (2.1). Note also that (2.44) can be rewritten in the form (2.2) as follows:

X1 ⊗ · · · ⊗Xd = [(X1)i1j1(X2)i2j2 · · · (Xd)idjd ]mi,j=1 .

Eq. (2.44) is of fundamental importance and, indeed, it motivates the introduction of multi-indices to index the entries of amatrix formed by a sum of tensor products. To better understand the importance of (2.44), try to write the (i, j)-th entry ofX1 ⊗X2 ⊗ · · · ⊗Xd as a function of two linear indices i, j = 1, . . . , N(m).

From (i)–(v), a lot of other interesting properties follow. For example, if X,Y are invertible, then X ⊗ Y is invertible, with inverseX−1 ⊗ Y −1. If X,Y are normal (resp. Hermitian, symmetric, unitary) then X ⊗ Y is also normal (resp. Hermitian, symmetric,unitary). If X ∈ Cm×m and Y ∈ C`×`, the eigenvalues and singular values of X ⊗ Y (resp. X ⊕ Y ) are λi(X)λj(Y ) : i =1, . . . ,m, j = 1, . . . , ` and σi(X)σj(Y ) : i = 1, . . . ,m, j = 1, . . . , ` (resp. λi(X) : i = 1, . . . ,m ∪ λj(Y ) : j = 1, . . . , `and σi(X) : i = 1, . . . ,m ∪ σj(Y ) : j = 1, . . . , `). In particular, for all X ∈ Cm×m and Y ∈ Cn×n,

‖X ⊕ Y ‖ = max(‖X‖, ‖Y ‖), ‖X ⊗ Y ‖ = ‖X‖ ‖Y ‖, (2.45)

‖X ⊕ Y ‖p =(‖X‖pp + ‖Y ‖pp

)1/p, ‖X ⊗ Y ‖p = ‖X‖p ‖Y ‖p, 1 ≤ p <∞, (2.46)

rank(X ⊕ Y ) = rank(X) + rank(Y ), rank(X ⊗ Y ) = rank(X) rank(Y ). (2.47)

We also highlight the following property: suppose we are given 2d matrices X1, . . . , Xd, Y1, . . . , Yd, with Xi, Yi ∈ Cmi×mi for alli = 1, . . . , d; then,

rank(X1 ⊗ · · · ⊗Xd − Y1 ⊗ · · · ⊗ Yd) ≤d∑i=1

rank(Xi − Yi)m1 · · ·mi−1mi+1 · · ·md = N(m)

d∑i=1

rank(Xi − Yi)mi

, (2.48)

where, of course,m = (m1, . . . ,md). This is true because

rank(X1 ⊗ · · · ⊗Xd − Y1 ⊗ · · · ⊗ Yd) = rank

(d∑i=1

Y1 ⊗ · · · ⊗ Yi−1 ⊗ (Xi − Yi)⊗Xi+1 ⊗ · · · ⊗Xd

)

≤d∑i=1

rank (Y1 ⊗ · · · ⊗ Yi−1 ⊗ (Xi − Yi)⊗Xi+1 ⊗ · · · ⊗Xd)

=

d∑i=1

rank(Y1 ⊗ · · · ⊗ Yi−1) rank(Xi − Yi) rank(Xi+1 ⊗ · · · ⊗Xd)

≤d∑i=1

m1 · · ·mi−1 rank(Xi − Yi)mi+1 · · ·md.

Lemmas 2.11–2.12 show that tensor products and direct sums are ‘almost’ commutative. Their proofs are rather technical and can beomitted on first reading. Take also into account that Lemmas 2.11–2.12 will be used only in the proof of Theorem 6.1, which may beskipped as well on first reading.

20

Lemma 2.11. For all m ∈ Nd and all permutations σ of the set 1, . . . , d, there exists a permutation matrix Πm;σ of size N(m) suchthat

Xσ(1) ⊗Xσ(2) ⊗ · · · ⊗Xσ(d) = Πm;σ(X1 ⊗X2 ⊗ · · · ⊗Xd)ΠTm;σ

for all matrices X1 ∈ Cm1×m1 , X2 ∈ Cm2×m2 , . . . , Xd ∈ Cmd×md .

Proof. The proof proceeds by induction on d. The case d = 1 is trivial. For the case d = 2, the result is clear when σ is the identity [1, 2],so we only have to prove it when σ = [2, 1]. In other words, we have to show that, for every m ∈ N2, there exists a permutation matrixΠm;[2,1] such that

X2 ⊗X1 = Πm;[2,1](X1 ⊗X2)ΠTm;[2,1] (2.49)

for all X1 ∈ Cm1×m1 and X2 ∈ Cm2×m2 . Let Πm;[2,1] be the permutation matrix associated with the permutation ζ of 1, . . . ,m1m2given by

ζ = [1,m2 + 1, 2m2 + 1, . . . , (m1 − 1)m2 + 1, 2,m2 + 2, 2m2 + 2, . . . , (m1 − 1)m2 + 2, . . . . . . ,m2, 2m2, 3m2 . . . ,m1m2],

i.e.,

ζ(i) = ((i− 1) modm1)m2 +

⌊i− 1

m1

⌋+ 1, i = 1, . . . ,m1m2.

In other words, Πm;[2,1] is the matrix whose rows are (in this order) eζ(i), i = 1, . . . ,m1m2, where ei, i = 1, . . . ,m1m2, are the vectorsof the canonical basis of Cm1m2 . It can be verified that the matrix Πm;[2,1] satisfies (2.49) for all X1 ∈ Cm1×m1 and X2 ∈ Cm2×m2 .The verification can be done componentwise, by showing that the (i, j)-th entry of the first matrix in (2.49) is equal to the (i, j)-th entryof the second matrix, for all i, j = 1, . . . ,m1m2. This completes the proof of the lemma in the case d = 2.

For the case d ≥ 3, we assume that the lemma holds for d − 1, and we prove that it also holds for d. Let m ∈ Nd and let σ be apermutation of 1, . . . , d. Let i be the index such that σ(i) = d, and let τ be the permutation of 1, . . . , d− 1 defined by τ(j) = σ(j)for j = 1, . . . , i − 1 and τ(j) = σ(j + 1) for j = i, . . . , d − 1. If i = d, then, by induction hypothesis and the properties of tensorproducts,

Xσ(1) ⊗ · · · ⊗Xσ(d) = Xτ(1) ⊗ · · · ⊗Xτ(d−1) ⊗Xd

=[Π(m1,...,md−1);τ (X1 ⊗ · · · ⊗Xd−1)ΠT

(m1,...,md−1);τ

]⊗Xd

= (Π(m1,...,md−1);τ ⊗ Imd)(X1 ⊗ · · · ⊗Xd−1 ⊗Xd)(ΠT(m1,...,md−1);τ ⊗ Imd),

and the thesis holds with Πm;σ = Π(m1,...,md−1);τ ⊗ Imd . If i < d, then

Xσ(1) ⊗ · · · ⊗Xσ(d) = Xσ(1) ⊗ · · · ⊗Xσ(i−1) ⊗Xd ⊗Xσ(i+1) ⊗ · · · ⊗Xσ(d)

= Xσ(1) ⊗ · · · ⊗Xσ(i−1) ⊗[Π(mσ(i+1)···mσ(d),md);[2,1](Xσ(i+1) ⊗ · · · ⊗Xσ(d) ⊗Xd)Π

T(mσ(i+1)···mσ(d),md);[2,1]

]= (Imσ(1)···mσ(i−1)

⊗Π(mσ(i+1)···mσ(d),md);[2,1])(Xσ(1) ⊗ · · · ⊗Xσ(i−1) ⊗Xσ(i+1) ⊗ · · · ⊗Xσ(d) ⊗Xd)

· (Imσ(1)···mσ(i−1)⊗ΠT

(mσ(i+1)···mσ(d),md);[2,1])

= Pm;σ(Xτ(1) ⊗ · · · ⊗Xτ(d−1) ⊗Xd)PTm;σ, (2.50)

where Pm;σ = Imσ(1)···mσ(i−1)⊗Π(mσ(i+1)···mσ(d),md);[2,1]. By induction hypothesis,

Xτ(1) ⊗ · · · ⊗Xτ(d−1) = Π(m1,...,md−1);τ (X1 ⊗ · · · ⊗Xd−1)ΠT(m1,...,md−1);τ .

Substituting in (2.50) and using the properties of tensor products, we see that the thesis holds with Πm;σ = Pm;σ(Π(m1,...,md−1);τ⊗Imd),which is a permutation matrix, being a product of two permutation matrices.

Lemma 2.11 says that the tensor product operation is ‘almost’ commutative. It is important to notice that the permutation matrixΠm;σ depends only on m and σ, and not on the specific matrices X1, X2, . . . , Xd. Lemma 2.12 is the version of Lemma 2.11 for directsums.

Lemma 2.12. For allm ∈ Nd and all permutations σ of the set 1, . . . , d, there exists a permutation matrix Vm;σ of sizem1 + . . .+md

such thatXσ(1) ⊕Xσ(2) ⊕ · · · ⊕Xσ(d) = Vm;σ(X1 ⊕X2 ⊕ · · · ⊕Xd)V

Tm;σ

for all matrices X1 ∈ Cm1×m1 , X2 ∈ Cm2×m2 , . . . , Xd ∈ Cmd×md .

21

Proof. The proof is done by induction on d. For d = 1, the only possible permutation is σ = [1] and we can take Vm;[1] = Im. For d = 2,the only possible permutations are the identity σ = [1, 2] and the transposition σ = [2, 1], and we can take

Vm;[1,2] = Im1+m2 , Vm;[2,1] =

[O Im2

Im1O

].

For d ≥ 3, let i be the index for which σ(i) = d. Define τ to be the permutation of the set 1, . . . , d − 1 such that τ(j) = σ(j) forj = 1, . . . , i− 1 and τ(j) = σ(j + 1) for j = i, . . . , d− 1. If i = d, then, by induction hypothesis,

Xσ(1) ⊕ · · · ⊕Xσ(d) = Xτ(1) ⊕ · · · ⊕Xτ(d−1) ⊕Xd

= V(m1,...,md−1);τ (X1 ⊕ · · · ⊕Xd−1)V T(m1,...,md−1);τ ⊕Xd

= (V(m1,...,md−1);τ ⊕ Imd)(X1 ⊕ · · · ⊕Xd−1 ⊕Xd)(V(m1,...,md−1);τ ⊕ Imd)T

and the thesis holds with Vm;σ = V(m1,...,md−1);τ ⊕ Imd . If i < d, then

Xσ(1) ⊕ · · · ⊕Xσ(d) = Xσ(1) ⊕ · · ·Xσ(i−1) ⊕Xd ⊕Xσ(i+1) ⊕ · · · ⊕Xσ(d)

= Xσ(1) ⊕ · · ·Xσ(i−1) ⊕[V(mσ(i+1)+...+mσ(d),md);[2,1](Xσ(i+1) ⊕ · · · ⊕Xσ(d) ⊕Xd)V

T(mσ(i+1)+...+mσ(d),md);[2,1]

]= (Imσ(1)+...+mσ(i−1)

⊕ V(mσ(i+1)+...+mσ(d),md);[2,1])(Xσ(1) ⊕ · · · ⊕Xσ(i−1) ⊕Xσ(i+1) ⊕ · · · ⊕Xσ(d) ⊕Xd

)· (Imσ(1)+...+mσ(i−1)

⊕ V(mσ(i+1)+...+mσ(d),md);[2,1])T

= Um;σ

(Xτ(1) ⊕ · · · ⊕Xτ(d−1) ⊕Xd

)UTm;σ, (2.51)

where Um;σ = Imσ(1)+...+mσ(i−1)⊕ V(mσ(i+1)+...+mσ(d),md);[2,1]. Using the induction hypothesis, we obtain

Xτ(1) ⊕ · · · ⊕Xτ(d−1) = V(m1,...,md−1);τ (X1 ⊕ · · · ⊕Xd−1)V T(m1,...,md−1);τ .

Substituting this into (2.51), we see that the thesis holds with Vm;σ = Um;σ(V(m1,...,md−1);τ ⊕ Imd).

Concerning the ‘distributive properties’ of tensor products with respect to direct sums, it follows directly from the definitions that thedistributive law on the right holds without permutation transformations. In other words, for all matrices X1, . . . , Xd, Y we have

(X1 ⊕X2 ⊕ · · · ⊕Xd)⊗ Y = (X1 ⊗ Y )⊕ (X2 ⊗ Y )⊕ · · · ⊕ (Xd ⊗ Y ). (2.52)

As for the distributive law on the left, a result analogous to Lemmas 2.11–2.12 holds, showing that this property holds modulo permutationtransformations which only depend on the dimensions of the involved matrices. More precisely, for all ` ∈ N andm ∈ Nd, there exists apermutation matrix Q`,m of size `(m1 + . . .+md) such that

X ⊗ (Y1 ⊕ Y2 ⊕ · · · ⊕ Yd) = Q`,m[(X ⊗ Y1)⊕ (X ⊗ Y2)⊕ · · · ⊕ (X ⊗ Yd)

]QT`,m

for all matrices X ∈ C`×`, Y1 ∈ Cm1×m1 , Y2 ∈ Cm2×m2 , . . . , Yd ∈ Cmd×md . However, in this book we will only need the distributivelaw on the right displayed in (2.52), and so we do not provide the proof of the distributive law on the left; the interested reader is referredto [43, Lemma 1.4].

2.7 Singular value and eigenvalue distribution of a matrix-sequenceLet K be either R or C. We recall that to any measurable function g : D ⊂ Rk → K we associate the functional φ[g] defined over Cc(K)by eq. (2.15). This functional will play an important role in this section and in the remainder of this book.

Definition 2.2 (singular value and eigenvalue distribution of a matrix-sequence, spectral symbol). Let Ann be a matrix-sequence.

• We say that Ann has an asymptotic singular value distribution described by a functional φ : Cc(R) → C, and we writeAnn ∼σ φ, if, for all F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) = φ(F ). (2.53)

In the case where φ = φ[|f |] for a measurable function f : D ⊂ Rk → C defined on a set D with 0 < µk(D) <∞, the function fis referred to as the singular value symbol of the matrix-sequence Ann, and we write Ann ∼σ f instead of Ann ∼σ φ[|f |].

22

• We say that Ann has an asymptotic eigenvalue (or spectral) distribution described by a functional φ : Cc(C)→ C, and we writeAnn ∼λ φ, if, for all F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑j=1

F (λj(An)) = φ(F ). (2.54)

In the case where φ = φ[f ] for a measurable function f : D ⊂ Rk → C defined on a set D with 0 < µk(D) < ∞, the functionf is referred to as the eigenvalue (or spectral) symbol of the matrix-sequence Ann, and we write Ann ∼λ f instead ofAnn ∼λ φ[f ].

When we write a relation such as Ann ∼σ φ (resp. Ann ∼λ φ), it is understood that φ is a functional onCc(R) (resp. Cc(C)), asin Definition 2.2. Similarly, when we write a relation such as Ann ∼σ f or Ann ∼λ f , it is understood that f is as in Definition 2.2;that is, f is a measurable function defined on a subset D of some Rk with 0 < µk(D) < ∞. Sometimes, for brevity, we will writeAnn ∼σ, λ f to indicate that Ann ∼σ f and Ann ∼λ f .

Remark 2.2. If Ann ∼σ φ, then φ is a continuous linear functional on (Cc(R), ‖ · ‖∞) with ‖φ‖ ≤ 1. Indeed, the linearity ofφ : Cc(R)→ C is a direct consequence of (2.53), while the continuity and the bound ‖φ‖ ≤ 1 follow from the fact that the infinity normof the argument of the limit in (2.53) is bounded by ‖F‖∞ for all F ∈ Cc(R) and all n. A similar argument show that, if Ann ∼λ φ,then φ is a continuous linear functional on (Cc(C), ‖ · ‖∞) with ‖φ‖ ≤ 1.

Remark 2.3. By definition of φ[f ], see eq. (2.15), the spectral distribution relation Ann ∼λ f means that, for all F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑j=1

F (λj(An)) =1

µk(D)

∫D

F (f(x))dx. (2.55)

The informal meaning behind (2.55) is the following. If f is at least Riemann-integrable, n is large enough, andxj,n, j = 1, . . . , N(n)

is an equispaced grid on D, then a suitable ordering λj(An), j = 1, . . . , N(n), of the eigenvalues of An is such that the pairs(

xj,n, λj(An)), j = 1, . . . , N(n)

reconstruct approximately the hypersurface (x, f(x)), x ∈ D. In other words, the spec-

trum of An, except possibly for o(N(n)) outliers, ‘behaves’ (asymptotically) like a uniform sampling of f over D. For instance, ifk = 1, N(n) = n and D = [a, b], then the eigenvalues of An are approximately equal to f(a + i(b − a)/n), i = 1, . . . , n, for nlarge enough. Similarly, if k = 2, N(n) = n2 and D = [a1, b1] × [a2, b2], then the eigenvalues of An are approximately equal tof(a1 + i(b1 − a1)/n, a2 + j(b2 − a2)/n), i, j = 1, . . . , n, for n large enough. A completely analogous meaning can be given also forthe singular value distribution relation Ann ∼σ f , which is equivalent to say that, for all F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) =1

µk(D)

∫D

F (|f(x)|)dx. (2.56)

Remark 2.4. It is clear from Definition 2.2 that Ann ∼σ f is equivalent to Ann ∼σ |f |. Moreover, if every An is normal andAnn ∼λ f , then Ann ∼σ f . Indeed, since An is normal, its singular values coincide with the moduli of the eigenvalues. Therefore,for any fixed F ∈ Cc(R), by applying the eigenvalue distribution relation (2.55) with the test function F (| · |) ∈ Cc(C), we get

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) = limn→∞

1

N(n)

N(n)∑j=1

F (|λj(An)|) =1

µk(D)

∫D

F (|f(x)|)dx.

Hence, Ann ∼σ f .

2.7.1 Clustering and attractionWe introduce in this section the notions of clustering and attraction. Besides being interesting in themselves, these concepts are relatedto the singular value and eigenvalue distribution of a matrix-sequence. Recall that, according to our notation (see Section 2.1), D(S, ε)denotes the ε-expansion of the subset S ⊆ C.

Definition 2.3 (clustering of a matrix-sequence). Let Ann be a matrix-sequence, and let S ⊆ C be a nonempty closed subset of C.We say that Ann is strongly clustered at S in the sense of the eigenvalues (or simply that Ann is strongly clustered at S) if, for everyε > 0, the number of eigenvalues of An outside D(S, ε) is bounded by a constant Cε independent of n. In other words,

#j ∈ 1, . . . , N(n) : λj(An) /∈ D(S, ε) = O(1) as n→∞. (2.57)

We say that An is weakly clustered at S in the sense of the eigenvalues (or simply that An is weakly clustered at S) if, for everyε > 0,

#j ∈ 1, . . . , N(n) : λj(An) /∈ D(S, ε) = o(N(n)) as n→∞. (2.58)

23

If An is strongly or weakly clustered at S and S is not connected, then the connected components of S are called sub-clusters.By replacing ‘eigenvalues’ with ‘singular values’ and λj(An) with σj(An) in (2.57)–(2.58), we obtain the definitions of a matrix-

sequence strongly or weakly clustered at a closed subset of C in the sense of the singular values.

Throughout this book, when we speak of strong/weak cluster, matrix-sequence strongly/weakly clustered, etc., without further spec-ifications, it is understood ‘in the sense of the eigenvalues’; when the clustering is intended in the sense of the singular values, this isspecified every time. It is worth noting that, since the singular values are always nonnegative, any matrix-sequence is strongly clusteredin the sense of the singular values at a certain S ⊆ [0,∞). Similarly, any matrix-sequence formed by matrices with only real eigenvalues(e.g., by Hermitian matrices) is strongly clustered at some S ⊆ R in the sense of the eigenvalues.

Definition 2.4 (spectral attraction). Let Ann be a matrix-sequence, and let z ∈ C. We say that z strongly attracts the spectrumΛ(An) with infinite order if, once we have ordered the eigenvalues of An according to their distance from z, i.e.,

|λ1(An)− z| ≤ |λ2(An)− z| ≤ . . . ≤ |λN(n)(An)− z|,

the following limit relation holds for each fixed j ≥ 1:

limn→∞

|λj(An)− z| = 0.

Given a measurable function f : D ⊂ Rk → C, we recall that the essential range of f , denoted by ER(f), is defined as the set ofpoints z ∈ C such that, for every ε > 0, the measure of f ∈ D(z, ε) is positive. In formulas,

ER(f) = z ∈ C : µkf ∈ D(z, ε) > 0 for all ε > 0.

Note that ER(f) is always closed (the complement is open). Moreover, it can be shown that f(x) ∈ ER(f) for almost every x ∈ D, i.e.,f ∈ ER(f) a.e. In addition, whenever f is continuous and D is sufficiently regular (say, D is contained in the closure of its interior),then ER(f) coincides with the closure of the image of f .

Theorem 2.8. If Ann ∼λ f , then Ann is weakly clustered at the essential range ER(f) and every point of ER(f) strongly attractsthe spectrum Λ(An) with infinite order.

Proof. Denote by D ⊂ Rk the domain of f , set S = ER(f), and fix ε > 0. For any δ > 0, let Fε,δ be a function in Cc(C) such that0 ≤ Fε,δ ≤ 1 over C, Fε,δ = 1 over S ∩D(0, 1/δ) and Fε,δ = 0 outside D(S, ε). Note that such a function exists by Urysohn’s lemma[81], since S ∩D(0, 1/δ) is a compact set contained in the open disk D(S, ε) (recall that S is closed). Clearly, we have

χS∩D(0,1/δ)

≤ Fε,δ ≤ χD(S,ε),

and so

#j ∈ 1, . . . , N(n) : λj(An) /∈ D(S, ε)N(n)

= 1− #j ∈ 1, . . . , N(n) : λj(An) ∈ D(S, ε)N(n)

= 1− 1

N(n)

N(n)∑j=1

χD(S,ε)(λj(An))

≤ 1− 1

N(n)

N(n)∑j=1

Fε,δ(λj(An)).

Passing to the limit as n→∞ and using the assumption Ann ∼λ f , we obtain

lim supn→∞

#j ∈ 1, . . . , N(n) : λj(An) /∈ D(S, ε)N(n)

≤ 1− 1

µk(D)

∫D

Fε,δ(f(x))dx.

To complete the proof that Ann is weakly clustered at S, we show that

limδ→0

∫D

Fε,δ(f(x))dx = µk(D). (2.59)

Since Fε,δ ≤ 1, we have ∫D

Fε,δ(f(x))dx ≤ µk(D).

24

Noting that χS∩D(0,1/δ)

converges pointwise to χS when δ → 0, by the inequality Fε,δ ≥ χS∩D(0,1/δ)

and the dominated convergencetheorem we get ∫

D

Fε,δ(f(x))dx ≥∫D

χS∩D(0,1/δ)

(f(x))dxδ→0−→

∫D

χS(f(x))dx = µk(D),

where the last equality is due to the fact that f ∈ S a.e. (see the discussion before the theorem). This completes the proof of (2.59) andthe proof that Ann is weakly clustered at S.

To show that each point of S strongly attracts Λ(An) with infinite order, fix a point s ∈ S. For any ε > 0, take Fε ∈ Cc(C) such thatFε = 1 over D(s, ε) and Fε = 0 outside D(s, 2ε). Since χD(s,ε) ≤ Fε ≤ χD(s,2ε) and Ann ∼λ f , we see that

#j ∈ 1, . . . , N(n) : λj(An) ∈ D(s, 2ε)N(n)

≥ 1

N(n)

N(n)∑j=1

Fε(λj(An))n→∞−→ 1

µk(D)

∫D

Fε(f(x))dx ≥ µkf ∈ D(s, ε)µk(D)

.

Passing to the limit as n→∞, we obtain

lim infn→∞

#j ∈ 1, . . . , N(n) : λj(An) ∈ D(s, 2ε)N(n)

≥ µkf ∈ D(s, ε)µk(D)

. (2.60)

By definition of essential range, the right-hand side of (2.60) is positive for every ε > 0. This implies that s strongly attracts Λ(An) withinfinite order.

2.7.2 Zero-distributed sequencesA class of matrix-sequences that plays a central role in the framework of the theory of GLT sequences is the class of zero-distributedsequences. A zero-distributed sequence is simply a matrix-sequence Znn such that Znn ∼σ 0. Theorem 2.9 provides a charac-terization of zero-distributed sequences, which will allow us to show in Chapter 6 that any zero-distributed sequence is a sLT sequence.Theorem 2.10 gives a sufficient condition, formulated in terms of Schatten p-norms, that ensures a matrix-sequence Znn to be zero-distributed.

Theorem 2.9. Let Znn be a matrix-sequence. Then, the following conditions are equivalent.

1. Znn ∼σ 0.

2. For every ε > 0, limn→∞

#j ∈ 1, . . . , N(n) : σj(Zn) > εN(n)

= 0.

3. For every n we have Zn = Rn +Nn, where limn→∞

rank(Rn)

N(n)= limn→∞

‖Nn‖ = 0.

Proof. (1⇒ 2) For every ε > 0, take Fε ∈ Cc(R) such that Fε = 1 over [0, ε/2], Fε = 0 over [ε,∞) and 0 ≤ Fε ≤ 1 over [0,∞). Notethat Fε ≤ χ[0,ε] over [0,∞). Since Znn ∼σ 0 by assumption, we have

#j ∈ 1, . . . , N(n) : σj(Zn) > εN(n)

= 1− #j ∈ 1, . . . , N(n) : σj(Zn) ≤ εN(n)

= 1− 1

N(n)

N(n)∑j=1

χ[0,ε](σj(Zn))

≤ 1− 1

N(n)

N(n)∑j=1

Fε(σj(Zn))n→∞−→ 1− Fε(0) = 0.

(2⇒ 3) By assumption, for every ε > 0 the quantity

qn(ε) =#j ∈ 1, . . . , N(n) : σj(Zn) > ε

N(n)

tends to 0 as n→∞. Hence, there exists a sequence εnn of positive numbers such that, when n→∞,

εn → 0, qn(εn) =#j ∈ 1, . . . , N(n) : σj(Zn) > εn

N(n)→ 0.

Let Zn = UnΣnV∗n be an SVD of Zn. Let Σn be the matrix obtained from Σn by setting to 0 all the singular values of Zn that are less

than or equal to εn, and let Σn = Σn − Σn be the matrix obtained from Σn by setting to 0 all the singular values of Zn that exceed εn.Then,

Zn = UnΣnV∗n = UnΣnV

∗n + UnΣnV

∗n = Rn +Nn,

25

where Rn = UnΣnV∗n and Nn = UnΣnV

∗n satisfy

rank(Rn)

N(n)=

#j ∈ 1, . . . , N(n) : σj(Zn) > εnN(n)

= qn(εn)n→∞−→ 0, ‖Nn‖ ≤ εn

n→∞−→ 0.

(3⇒ 1) In order to give an elegant and extremely short proof of this implication, we use some results from Chapter 4. The assumptionin item 3 ensures that ON(n)nm is an approximating class of sequences for Znn according to Definition 4.1. Moreover, it is clearthat ON(n)n ∼σ 0. Hence, Znn ∼σ 0 by Corollary 4.1.

With the terminology of clustering introduced in Section 2.7.1, condition 2 in Theorem 2.9 can be reformulated by saying that Znnis weakly clustered at 0 in the sense of the singular values.

Remark 2.5 (algebra of zero-distributed sequences). From the equivalence 1⇔ 3 in Theorem 2.9, it follows that the set of zero-distributed sequences is a *-algebra over the complex field C. More precisely, fix any sequence of d-indices n = n(n)n ⊆ Nd suchthat n→∞ when n→∞; then,

Z =Znn : Znn ∼σ 0

(2.61)

is a *-algebra over C, with respect to the natural operations of Hermitian transposition, addition, scalar-multiplication and product ofmatrix-sequences:

An∗n = A∗nn, Ann + Bnn = An +Bnn, αAnn = αAnn, AnnBnn = AnBnn. (2.62)

In the statement of the next theorem, we use the natural convention 1/∞ = 0.

Theorem 2.10. Let Znn be a matrix-sequence and suppose that, for some p ∈ [1,∞],

limn→∞

‖Zn‖pN(n)1/p

= 0.

Then Znn ∼σ 0.

Proof. In view of (2.27)–(2.28), for all p ∈ [1,∞] we have

‖Zn‖1N(n)

≤ ‖Zn‖pN(n)1/p

.

Hence, it suffices to prove the theorem under the assumption that

limn→∞

‖Zn‖1N(n)

= 0. (2.63)

What we have to show is that

limn→∞

1

N(n)

N(n)∑j=1

F (σj(Zn)) = F (0) (2.64)

for all F ∈ Cc(R). The proof of (2.64) is easy if F is a real-valued function in C1c (R), because in this case we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Zn))− F (0)

∣∣∣∣∣∣ ≤ 1

N(n)

N(n)∑j=1

|F (σj(Zn))− F (0)| ≤ ‖F′‖∞

N(n)

N(n)∑j=1

σj(Zn) =‖F ′‖∞N(n)

‖Zn‖1,

which tends to 0 by (2.63). In the case of a general F ∈ Cc(R), we approximate (in∞-norm) the real and the imaginary part of F bymeans of real-valued functions in C1

c (R), and we see that (2.64) continues to hold. Let us work out the details. For every ε > 0, let<ε and =ε be real-valued functions in C1

c (R) such that ‖<(F ) − <ε‖∞ ≤ ε and ‖=(F ) − =ε‖∞ ≤ ε. Set Fε = <ε + i=ε. Then,‖F − Fε‖∞ ≤ 2ε and ∣∣∣∣∣∣

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Zn))− F (0)

∣∣∣∣∣∣−∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(σj(Zn))− Fε(0)

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ 4ε.

Since (2.64) holds for <ε, =ε (and hence also for Fε), the previous inequality implies that

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Zn))− F (0)

∣∣∣∣∣∣ ≤ 4ε,

for every ε > 0. Thus, (2.64) holds for F .

26

Exercise 2.1. Let Znn be a matrix-sequence such that limn→∞N(n)−1/p‖Zn‖p = 0 for some p ∈ [1,∞] (as in Theorem 2.10).Show that Znn ∼λ 0.

Exercise 2.2. Provide an example of a sequence Znn such that Znn ∼σ 0 but Znn 6∼λ 0, and an example of a sequence Znnsuch that Znn ∼λ 0 but Znn 6∼σ 0. Conclude that, for every p ∈ [1,∞], the condition ‘limn→∞N(n)−1/p‖Zn‖p = 0 for somep ∈ [1,∞]’ implies but is not equivalent to ‘Znn ∼σ 0’.

27

Chapter 3

Spectral distribution of sequencesof perturbed Hermitian matrices

In this chapter, we present a general result, taken from [54, 57], which is useful for computing the spectral distribution of matrix-sequencesof the form Xn + Ynn, where Xn is Hermitian and Yn is a ‘small’ perturbation of Xn. More precisely, we prove that Xn + Ynnhas an asymptotic spectral distribution if:

1. Xnn is a sequence of Hermitian matrices possessing an asymptotic spectral distribution;

2. the spectral norms ‖Xn‖, ‖Yn‖ are uniformly bounded with respect to n;

3. ‖Yn‖1 = o(N(n)).

Under these assumptions, the functional φ : Cc(C)→ C identifying the spectral distribution in the sense of Definition 2.2 is the same forXnn and Xn + Ynn.

This result should be regarded as a useful addendum to the theory of GLT sequences. In practice, it turns out to be an effective toolfor computing the asymptotic spectral distribution of GLT sequences formed by perturbed Hermitian matrices. In this respect, we pointout that the matrix-sequences coming from a PDE discretization are often GLT sequences formed by symmetric or nearly symmetricmatrices, at least in the case where the higher-order differential operator of the considered PDE is symmetric (self-adjoint).

3.1 Preliminary resultsBefore being able to state and prove the main theorem of this chapter (Theorem 3.3), a lot of work is needed. In this section, we collectall the necessary preliminary results, which are interesting also in themselves.

Lemma 3.1. Let φ : Cc(C)→ C and S ⊆ C. Then, the following conditions are equivalent.

1. ∀F ∈ Cc(C), ∀ ε > 0, ∃ δ = δε,F > 0 : |φ(G)− φ(F )| ≤ ε ∀G ∈ Cc(C) satisfying ‖G− F‖∞,S ≤ δ.

2. φ is continuous on (Cc(C), ‖ · ‖∞) and φ(F ) = φ(G) whenever F = G on S.

Proof. (1⇒ 2) Assume that condition 1 holds. Then, φ is continuous on (Cc(C), ‖·‖∞) because it is continuous at each point F ∈ Cc(C):

∀F ∈ Cc(C), ∀ ε > 0, ∃ δ = δε,F > 0 : |φ(G)− φ(F )| ≤ ε ∀G ∈ Cc(C) satisfying ‖G− F‖∞ ≤ δ. (3.1)

Suppose by contradiction that there exist F,G ∈ Cc(C) such that F = G over S and φ(F ) 6= φ(G). Then, if we choose 0 < ε <|φ(G)− φ(F )|, condition 1 is violated by the pair (F, ε) because ‖G− F‖∞,S = 0 < δ for any δ > 0, and |φ(G)− φ(F )| > ε.

(2⇒ 1) Assume that condition 2 holds. Then φ is continuous on (Cc(C), ‖ · ‖∞), so (3.1) is satisfied. Let us prove that condition 1is satisfied as well. Let F ∈ Cc(C), ε > 0, and take δ = δε,F > 0 as in (3.1). For any G ∈ Cc(C) satisfying ‖G − F‖∞,S ≤ δ, weconstruct a G ∈ Cc(C) such that G = G over S and ‖G − F‖∞ ≤ ‖G − F‖∞,S ≤ δ. Once this is done, by (3.1) and condition 2 weconclude that

|φ(G)− φ(F )| = |φ(G)− φ(F )| ≤ ε ∀G ∈ Cc(C) satisfying ‖G− F‖∞,S ≤ δ,

and condition 1 is proved. Here is the construction of G. Let

ψ(z) =

z if |z| ≤ ‖G− F‖∞,S ,

‖G− F‖∞,Sz

|z|if |z| > ‖G− F‖∞,S .

28

Then ψ is continuous on C, ‖ψ‖∞ ≤ ‖G−F‖∞,S , and ψ(0) = 0. Set G(z) = F (z)+ψ(G(z)−F (z)). Then G ∈ Cc(C), G(z) = G(z)

for each z ∈ S, and ‖G− F‖∞ ≤ ‖ψ‖∞ ≤ ‖G− F‖∞,S ≤ δ.

Definition 3.1. S ⊆ C is said to be a polynomial set if the following condition is met:

∀F ∈ Cc(C) holomorphic in the interior of S, ∀ η > 0, ∀R > 0, ∃ p = pR,η,F ∈ Cc(C) :

p coincides with a polynomial in C[z] over D(0, R), ‖p− F‖∞,S ≤ η.

Lemma 3.2. Let S ⊆ C be closed and assume that there is M0 > 0 such that C\(S ∩D(0,M)) is connected for all M > M0. Then Sis a polynomial set.

Proof. Let F ∈ Cc(C) be holomorphic in the interior of S, let η > 0, and let R > 0. Take M = MR,F > M0 such that D(0,M) ⊇D(0, R) ∪ supp(F ). Since S ∩ D(0,M + 1) is compact, C\(S ∩ D(0,M + 1)) is connected, and F is holomorphic in the interior ofS ∩D(0,M + 1) (which is contained in the interior of S), Mergelyan’s theorem [81] implies the existence of a polynomial q = qR,η,Fin C[z] such that ‖q − F‖∞,S∩D(0,M+1)

≤ η. Let p = pR,η,F be defined as follows: p(z) = q(z)γ(z), where γ is any function in Cc(C)

such that 0 ≤ γ ≤ 1 over C, γ = 1 over D(0,M), and γ = 0 outside D(0,M + 1). Then p ∈ Cc(C), p = q over D(0,M) ⊇ D(0, R),and

‖p− F‖∞,S = ‖p− F‖∞,S∩D(0,M+1)≤ ‖q − F‖∞,S∩D(0,M+1)

≤ η,

where the first equality is justified by the fact that p = F = 0 outside D(0,M + 1), while the first inequality is due to the followingobservation:

|p(z)− F (z)| =|q(z)− F (z)| if z ∈ D(0,M),|p(z)| = |q(z)γ(z)| ≤ |q(z)| = |q(z)− F (z)| if z ∈ D(0,M + 1) \D(0,M).

It follows from Lemma 3.2 that any compact set S ⊆ C whose complement C\S is connected is a polynomial set. Moreover, R is apolynomial set, as well as any straight line in C.

Theorem 3.1. Let Ann be a matrix-sequence and assume that the following conditions are met.

1. Ann is weakly clustered at a polynomial set S ⊆ C.

2. There is a radius R such that limn→∞

1

N(n)

N(n)∑j=1

p(λj(An)) = φ(p) for all p ∈ Cc(C) coinciding on D(0, R) with a polynomial in

C[z], where φ : Cc(C)→ C is a continuous functional on (Cc(C), ‖ · ‖∞) such that φ(F ) = φ(G) whenever F = G on S.

Then, for all F ∈ Cc(C) holomorphic in the interior of S we have

limn→∞

1

N(n)

N(n)∑j=1

F (λj(An)) = φ(F ). (3.2)

In particular, if the interior of S is empty, then Ann ∼λ φ.

Proof. Let F ∈ Cc(C) be holomorphic in the interior of S and let ε, ε > 0. By the hypothesis on φ and Lemma 3.1, there existsδ = δε,F > 0 such that |φ(G)− φ(F )| ≤ ε for all G ∈ Cc(C) with ‖G− F‖∞,S ≤ δ. Without loss of generality, we may assume δ ≤ ε.Since S is polynomial, there exists p = pε,F ∈ Cc(C) coinciding with a polynomial in C[z] over D(0, R) and such that ‖p−F‖∞,S ≤ δ.Then, for all n we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− φ(F )

∣∣∣∣∣∣ ≤∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

p(λj(An))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

p(λj(An))− φ(p)

∣∣∣∣∣∣+ |φ(p)− φ(F )|. (3.3)

29

The second term in the righthand side tends to 0 as n → ∞ (by the second assumption), while the third term is bounded from above byε. For the first term we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

p(λj(An))

∣∣∣∣∣∣ ≤ 1

N(n)

N(n)∑j=1

|F (λj(An))− p(λj(An))|

=1

N(n)

∑j:λj(An)/∈D(S,ε)


+1

N(n)

∑j:λj(An)∈D(S,ε)\S


+1

N(n)

∑j:λj(An)∈S

|F (λj(An))− p(λj(An))|. (3.4)

By definition of D(S, ε), for all n and all j ∈ 1, . . . , N(n) such that λj(An) ∈ D(S, ε) we can find a point µj,n ∈ S such that|λj(An) − µj,n| ≤ ε. Let ωF and ωp denote the moduli of continuity of F and p over C. Then, the three terms in the righthand side of(3.4) can be bounded as follows:

1

N(n)

∑j:λj(An)/∈D(S,ε)

|F (λj(An))− p(λj(An))| ≤ ‖F − p‖∞#j ∈ 1, . . . , N(n) : λj(An) /∈ D(S, ε)

N(n),

1

N(n)


|F (λj(An))− p(λj(An))| ≤ 1

N(n)


(|F (λj(An))− F (µj,n)|+ |F (µj,n)− p(µj,n)|

+ |p(µj,n)− p(λj(An))|)

≤ 1

N(n)


(ωF (ε) + δ + ωp(ε)

)≤ ωF (ε) + ε+ ωp(ε),

1

N(n)

∑j:λj(An)∈S

|F (λj(An))− p(λj(An))| ≤ ‖F − p‖∞,S ≤ δ ≤ ε.

Passing to the limit as n→∞ in (3.3) and recalling that An is weakly clustered at S, we get

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1


∣∣∣∣∣∣ ≤ ωF (ε) + ε+ ωp(ε) + ε+ ε,

and the thesis follows from the fact that the righthand side tends to 0 if we let ε→ 0 and ε→ 0 (we first leave ε tend to 0 because p, andhence ωp, depends on ε).

Lemma 3.3. Let Ann be a matrix-sequence and assume that ‖=(An)‖1 = o(N(n)) for all n. Then Ann is weakly clustered at R.

Proof. By Lemma 2.10, for every ε > 0 we have

#j ∈ 1, . . . , N(n) : λj(An) /∈ D(R, ε) ≤ #j ∈ 1, . . . , N(n) : λj(An) /∈ R× [−ε/2, ε/2] ≤ ‖=(An)‖1ε/2

,

which is o(N(n)) by assumption.

Theorem 3.2. Let Xnn, Ynn be matrix-sequences and set An = Xn + Yn. Assume that the following conditions are met.

1. ‖Xn‖, ‖Yn‖ ≤ C for all n, with C a constant independent of n.

2. Every Xn is Hermitian and there is a radius R such that

limn→∞

1

N(n)

N(n)∑j=1

p(λj(Xn)) = φ(p)

for all p ∈ Cc(C) coinciding on D(0, R) with a polynomial in C[z], where φ : Cc(C) → C is a continuous functional on(Cc(C), ‖ · ‖∞) such that φ(F ) = φ(G) whenever F = G on R.

30

3. ‖Yn‖1 = o(N(n)) as n→∞.

Then Ann ∼λ φ.

Proof. We show that the sequence Ann satisfies the two conditions of Theorem 3.1 with S = R, after which the proof is finished.Actually, we only have to prove that Ann satisfies the second condition, since the first one follows directly from the third assumption,the inequality ‖=(An)‖1 = ‖=(Yn)‖1 ≤ ‖Yn‖1, and Lemma 3.3. We will prove that

limn→∞

1

N(n)

N(n)∑j=1

p(λj(An)) = φ(p)

for all functions p ∈ Cc(C) coinciding with a polynomial over the disk D(0, 2C +R). For a monomial zk, k ≥ 0, we have

Akn = (Xn + Yn)k = Xkn +Rn,k,

where Rn,k = (Xn+Yn)k −Xkn satisfies ‖Rn,k‖1 = o(N(n)). This follows from the third assumption, from the fact that ‖Xn‖, ‖Yn‖

are bounded by a constant C independent of n, and from the Hölder-type inequality ‖XY ‖1 ≤ ‖X‖ ‖Y ‖1 satisfied by the trace-norm(see (2.30)). Therefore, for every polynomial q(z) = q0 + q1z + . . . + qmz

m ∈ C[z] we have q(An) = q(Xn) + Rn,q(z), whereRn,q(z) =

∑mk=0 qkRn,k satisfies ‖Rn,q(z)‖1 = o(N(n)). By (2.29) we obtain

|trace(q(An))− trace(q(Xn))| = |trace(q(An)− q(Xn))| = |trace(Rn,q(z))| ≤ ‖Rn,q(z)‖1 = o(N(n)),

implying that the sequence

1

N(n)trace(q(An)) =

1

N(n)

N(n)∑j=1

q(λj(An))

converges to the limit of the sequence

1

N(n)trace(q(Xn)) =

1

N(n)

N(n)∑j=1

q(λj(Xn))

(provided the latter exists). To conclude, we note that Λ(An), Λ(Xn) ⊆ D(0, 2C +R) for all n. Hence, for all p ∈ Cc(C) coincidingover D(0, 2C +R) with a polynomial qp(z) ∈ C[z], we have

limn→∞

1

N(n)

N(n)∑j=1

p(λj(An)) = limn→∞

1

N(n)

N(n)∑j=1

qp(λj(An)) = limn→∞

1

N(n)

N(n)∑j=1

qp(λj(Xn)) = limn→∞

1

N(n)

N(n)∑j=1

p(λj(Xn)) = φ(p),

where the last equality follows from the second assumption.

3.2 Main resultsWe are now in the position of stating and proving the main result of this chapter, already summarized at the beginning. Most of the workhas been done in Section 3.1, and Theorem 3.3 is now obtained as a corollary of Theorem 3.2.

Theorem 3.3. Let Xnn, Ynn be matrix-sequences and set An = Xn + Yn. Assume that the following conditions are met.


2. Every Xn is Hermitian and Xnn ∼λ φ.

3. ‖Yn‖1 = o(N(n)) as n→∞.

Then Ann ∼λ φ.

Proof. The proof reduces to show that the functional φ : Cc(C)→ C appearing in the statement of the theorem is such that φ(F ) = φ(G)whenever F = G on R. Once this is proved, all the hypotheses of Theorem 3.2 are satisfied and the thesis follows. Let F,G ∈ Cc(C)such that F = G on R. By the second assumption, the eigenvalues of Xn are real and

φ(F ) = limn→∞

1

N(n)

N(n)∑j=1

F (λj(Xn)) = limn→∞

1

N(n)

N(n)∑j=1

G(λj(Xn)) = φ(G),

which yields φ(F ) = φ(G).

31

Corollary 3.1. Let Xnn, Ynn be matrix-sequences and set An = Xn + Yn. Assume that the following conditions are met.


2. Every Xn is Hermitian and Xnn ∼λ f .

3. ‖Yn‖1 = o(N(n)) as n→∞.

Then Ann ∼λ f .

32

Chapter 4

Approximating classes of sequences (a.c.s.)

In this section, we introduce the fundamental definition on which the theory of GLT sequences is based: the notion of approximatingclasses of sequences, first introduced in [87]. This notion lays the foundations for a (spectral) approximation theory for matrix-sequencesand provides general tools (Theorems 4.3 and 4.5) for computing the asymptotic spectral or singular value distribution of a ‘difficult’matrix-sequence Ann from the one of ‘simpler’ matrix-sequences Bn,mnm that approximate Ann in a suitable sense whenm→∞. We refer the reader to the introduction of [56] for a more detailed discussion on this subject.

Definition 4.1 (approximating class of sequences). Let Ann be a matrix-sequence. An approximating class of sequences (a.c.s.) forAnn is a sequence of matrix-sequences Bn,mnm with the following property: for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),(4.1)

where the quantities nm, c(m), ω(m) depend only on m, and

limm→∞

c(m) = limm→∞

ω(m) = 0.

Roughly speaking, Bn,mnm is an a.c.s. for Ann if, for large m, the sequence Bn,mn approximates Ann in the sense thatAn is eventually equal to Bn,m plus a small-rank matrix (with respect to the matrix size N(n)) plus a small-norm matrix.

Remark 4.1. An equivalent definition of a.c.s. is obtained by replacing, in Definition 4.1, ‘for every m’ with ‘for every sufficiently largem’ (i.e., ‘for every m greater than or equal to some number M ’). Indeed, suppose that the splitting (4.1) holds for m ≥M . For m < M ,define nm = 1, c(m) = 1, ω(m) = 0 and Rn,m = An,m −Bn,m, Nn,m = ON(n). Then, we see that (4.1) actually holds for every m.

4.1 The a.c.s. topologyFix a sequence of d-indices n = n(n)n ⊆ Nd such that n→∞ as n→∞, and consider the set of all matrix-sequences

E =Ann : Ann is a matrix-sequence

. (4.2)

Note that E is a *-algebra with respect to the natural operations (2.62). In this section, we will see that the a.c.s. notion is relatedto a topology defined on E . In other words, there exists a topology τa.c.s. on E such that the notion of convergence in the topologicalspace (E , τa.c.s.) coincides precisely with the a.c.s. notion; that is, a sequence of matrix-sequences Bn,mnm ⊂ E converges toanother matrix-sequence Ann ∈ E in the topology τa.c.s. if and only if Bn,mnm is an a.c.s. for Ann. Moreover, there existsa (pseudo)metric da.c.s. on E which induces the topology τa.c.s., i.e., τa.c.s. is metrizable. In order to fully understand the content of thissection, it is necessary that the reader possesses some background on general topology. However, since this section is not essential to theremainder of the book, the reader may also decide to skip everything and go to Section 4.2; we only recommend that the reader keepsin mind the convergence notation (4.7) which will be sometimes used throughout this book to indicate that Bn,mnm is an a.c.s. forAnn.

For every matrix-sequence Ann ∈ E we define

p(Ann) = inf

lim supn→∞

rank(Rn)

N(n)+ lim sup

n→∞‖Nn‖ : Rnn, Nnn ∈ E , Rnn + Nnn = Ann

. (4.3)

Theorem 4.1. Let Ann be a matrix-sequence and let Bn,mnm be a sequence of matrix-sequences. Then, the following conditionsare equivalent.

33

1. Bn,mnm is an a.c.s. for Ann.

2. For every ε > 0 there exists M(ε) with the following property: for m ≥M(ε) there exists n(ε)m such that, for n ≥ n(ε)

m ,

An = Bn,m +R(ε)n,m +N (ε)

n,m

rank(R(ε)n,m) ≤ εN(n), ‖N (ε)

n,m‖ ≤ ε.

3. For every ε > 0 there exists Mε such that p(An −Bn,mn) ≤ ε for m ≥Mε.

4. p(An −Bn,mn)→ 0 as m→∞.

Proof. (1⇒ 2) Suppose that Bn,mnm is an a.c.s. for Ann: for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where limm→∞ c(m) = limm→∞ ω(m) = 0. To see that condition 2 is met, for every ε > 0 it suffices to take M(ε) so thatmax(c(m), ω(m)) ≤ ε for m ≥M(ε), and to set n(ε)

m = nm, R(ε)n,m = Rn,m, N (ε)

n,m = Nn,m.(2⇒ 1) Suppose that condition 2 is met. Then, for ε = 1 there exists M(1) such that the property expressed in 2 holds; without loss

of generality, we may assume that M(1) = 0 because it is clear that the property expressed in 2 holds for all m when ε = 1 (it suffices totake R(1)

n,m = An −Bn,m and N (1)n,m = O). For ε = 1/2 there exists M(1/2) such that the property expressed in 2 holds; without loss of

generality, we may assume that M(1/2) > M(1). For ε = 1/3 there exists M(1/3) such that the property expressed in 2 holds; withoutloss of generality, we may assume that M(1/3) > M(1/2). Continuing in this way, we construct an increasing sequence of integers0 = M(1) < M(1/2) < M(1/3) < . . . with the following property: for every k ≥ 1 and every M(1/k) ≤ m < M(1/(k + 1)), thereexists n(1/k)

m such that, for n ≥ n(1/k)m ,

An = Bn,m +R(1/k)n,m +N (1/k)

n,m

rank(R(1/k)n,m ) ≤ N(n)/k, ‖N (1/k)

n,m ‖ ≤ 1/k.

For every m we define the number

k(m) =

1 if 0 = M(1) ≤ m < M(1/2),2 if M(1/2) ≤ m < M(1/3),3 if M(1/3) ≤ m < M(1/4),. . . . . . . . .

i.e., k(m) = k whenever M(1/k) ≤ m < M(1/(k + 1)), k = 1, 2, . . . Then, k(m) is a sequence of integers such that k(m) → ∞as m → ∞, and moreover M(1/k(m)) ≤ m < M(1/(k(m) + 1)) for all m. Hence, for every m there exists n(1/k(m))

m such that, forn ≥ n(1/k(m))

m ,

An = Bn,m +R(1/k(m))n,m +N (1/k(m))

n,m ,

rank(R(1/k(m))n,m ) ≤ N(n)/k(m), ‖N (1/k(m))

n,m ‖ ≤ 1/k(m).

This proves that Bn,mnm is an a.c.s. for Ann.(2⇒ 3) Suppose that condition 2 is met and fix ε > 0. By condition 2, for m ≥M(ε/2) and n ≥ n(ε/2)

m we have

An −Bn,m = R(ε/2)n,m +N (ε/2)

n,m

rank(R(ε/2)n,m )

N(n)<ε

2, ‖N (ε/2)

n,m ‖ <ε

2,

which implies that, for m ≥M(ε/2),

lim supn→∞

rank(R(ε/2)n,m )

N(n)≤ ε

2, lim sup

n→∞‖N (ε/2)

n,m ‖ ≤ε

2.

Thus, for m ≥M(ε/2), by the definition of p(·) in (4.3) we have

p(An −Bn,mn) ≤ lim supn→∞

rank(R(ε/2)n,m )

N(n)+ lim sup

n→∞‖N (ε/2)

n,m ‖ ≤ ε,

34

and condition 3 is proved with Mε = M(ε/2).(3⇒ 2) By the definition of p(·) in (4.3), for any fixed m and ε, the inequality p(An − Bn,mn) ≤ ε implies the existence of two

matrix-sequences Rn,m,εn, Nn,m,εn and of a number nm,ε such that Rn,m,εn+Nn,m,εn = An−Bn,mn and, for n ≥ nm,ε,

rank(Rn,m,ε) ≤ 2εN(n), ‖Nn,m,ε‖ ≤ 2ε.

Therefore, if condition 3 is met, then also condition 2 is met with M(ε) = Mε/2, n(ε)m = nm,ε/2, R(ε)

n,m = Rn,m,ε/2 and N (ε)n,m =

Nn,m,ε/2.(3⇔ 4) Obvious.

We now study some properties of the function p(·) defined in (4.3). We shall see that p(·) allows us to define a distance da.c.s. on Ewhich induces the topology associated with the a.c.s. notion.

Theorem 4.2. The function p(·) in (4.3) satisfies the following properties.

1. 0 ≤ p(Ann) ≤ 1 for all matrix-sequences Ann ∈ E .

2. p(Ann) = 0 if and only if Ann is a zero-distributed sequence.

3. p(Ann + Bnn) ≤ p(Ann) + p(Bnn) for all matrix-sequences Ann, Bnn ∈ E .

4. p(Ann) can be defined in terms of the singular values of Ann, as follows:

p(Ann) = inf

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) ≥ αnN(n)

+ lim supn→∞

maxi=1,...,N(n)σi(An)<αn

σi(An) : αnn ⊂ (0,∞)

, (4.4)

for every Ann ∈ E , where we assume that the maximum of the empty set is 0.

Proof. 1. It is clear that p(Ann) ≥ 0. Moreover, by taking Rnn = Ann and Nnn = ON(n)n in (4.3), we see thatp(Ann) ≤ 1.

2. If Ann is a zero-distributed sequence, then p(Ann) = 0 by Theorem 2.9. Conversely, suppose that p(Ann) = 0. Then, by(4.3), for everym ∈ N there exist two matrix-sequences Rn,mn, Nn,mn and a number nm such that Rn,mn+Nn,mn = Annand, for n ≥ nm,

rank(Rn,m)

N(n)≤ 1

m, ‖Nn,m‖ ≤

1

m.

This implies that ON(n)nm is an a.c.s. of Ann, and moreover it is clear that ON(n)n ∼σ 0. Hence, by Corollary 4.1, Ann iszero-distributed.

3. Let pA = p(Ann), pB = p(Bnn), and pA+B = p(Ann + Bnn). Based on the definition of pA and pB , for every m ∈ Nthere exist four matrix-sequences RAn,mn, NA

n,mn, RBn,mn, NBn,mn such that RAn,mn + NA

n,mn = Ann, RBn,mn +

NBn,mn = Bnn, and

lim supn→∞

rank(RAn,m)

N(n)+ lim sup

n→∞‖NA

n,m‖ ≤ pA +1

m, lim sup

n→∞

rank(RBn,m)

N(n)+ lim sup

n→∞‖NB

n,m‖ ≤ pB +1

m.

It follows that for every m ∈ N there exist two matrix-sequences RAn,m+RBn,mn and NAn,m+NB

n,mn such that RAn,m+RBn,mn+

NAn,m +NB

n,mn = An +Bnn and

pA+B ≤ lim supn→∞

rank(RAn,m +RBn,m)

N(n)+ lim sup

n→∞‖NA

n,m +NBn,m‖

≤ lim supn→∞

rank(RAn,m) + rank(RBn,m)

N(n)+ lim sup

n→∞

(‖NA

n,m‖+ ‖NBn,m‖

)≤ lim sup

n→∞

rank(RAn,m)

N(n)+ lim sup

n→∞

rank(RBn,m)

N(n)+ lim sup

n→∞‖NA

n,m‖+ lim supn→∞

‖NBn,m‖

≤ pA + pB +2

m.

Thus, pA+B ≤ pA + pB .

35

4. Let αnn ⊂ (0,∞) be any sequence. Let An = UnΣnV∗n be a SVD of An and set

Rn = UnΣ≥αnn V ∗n , Nn = UnΣ<αn

n V ∗n ,

where Σ≥αnn is the matrix obtained from Σn by setting to 0 all the singular values ofAn that are less than αn, and Σ<αn

n = Σn−Σ≥αnn is

the matrix obtained from Σn by setting to 0 all the singular values ofAn that are greater or equal to αn. Then, Ann = Rnn+Nnnand

rank(Rn) = #i ∈ 1, . . . , N(n) : σi(An) ≥ αn, ‖Nn‖ = maxi=1,...,N(n)σi(An)<αn

σi(An).

By choosing the pair of matrix-sequences Rnn and Nnn in (4.3), we obtain

p(Ann) ≤ lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) ≥ αnN(n)

+ lim supn→∞


σi(An),

and this is true for all sequences αnn ⊂ (0,∞). Passing to the infimum over all sequences αnn ⊂ (0,∞), we see that p(Ann) isless than or equal to the right-hand side of (4.4).

To prove the other inequality, let Rnn and Nnn be any two matrix-sequences such that Ann = Rnn + Nnn. By theminimax principle for singular values (Theorem 2.5), for all i = 1, . . . , N(n) we have

σi(An) = minV subspace of CN(n)

dimV=N(n)−i+1

maxx∈V, ‖x‖=1

‖Rnx +Nnx‖ ≤ minV subspace of CN(n)

dimV=N(n)−i+1

maxx∈V, ‖x‖=1

‖Rnx‖+ ‖Nn‖ = σi(Rn) + ‖Nn‖.

In particular, setting ρn = rank(Rn), all the singular values of An with index i > ρn are bounded by ‖Nn‖. Let ρn be the first indexin 0, . . . , N(n) such that all the singular values of An with index i > ρn are bounded by ‖Nn‖. By the previous observation we haveρn ≤ ρn, and by definition of ρn we have σρn(An) > ‖Nn‖ ≥ σρn+1(An).1 Let An = Un diag(σ1(An), . . . , σN(n)(An))V ∗n be aSVD of An, and let

Rn = Un diag(σ1(An), . . . , σρn(An), 0, . . . , 0)V ∗n , Nn = Un diag(0, . . . , 0, σρn+1(An), . . . , σN(n)(An))V ∗n .

Define

αn =

σρn(An) if ρn ∈ 1, . . . , N(n),‖Nn‖+ 1 if ρn = 0.

Then, Ann = Rnn + Nnn,

rank(Rn) = ρn ≥ ρn = rank(Rn) = #i ∈ 1, . . . , N(n) : σi(An) ≥ αn,‖Nn‖ ≥ σρn+1(An) = ‖Nn‖ = max

i=1,...,N(n)σi(An)<αn

σi(An),

and

lim supn→∞

rank(Rn)

N(n)+ lim sup

n→∞‖Nn‖ ≥ lim sup

n→∞

#i ∈ 1, . . . , N(n) : σi(An) ≥ αnN(n)

+ lim supn→∞


σi(An). (4.5)

We have then proved that, for every pair of matrix-sequences Rnn and Nnn such that Rnn + Nnn = Ann we can finda sequence αnn ⊂ (0,∞) such that (4.5) is satisfied. This shows that p(Ann) is greater than or equal to the right-hand side of(4.4).

As a consequence of Theorem 4.2, the function da.c.s. : E × E → [0, 1],

da.c.s.(Ann, Bnn) = p(Ann − Bnn) = p(An −Bnn), (4.6)

is a pseudometric on E , which turns E into a pseudometric space. Recall that the difference between pseudometric and metric spaces issimply that the distance between two points in a pseudometric space can be zero even if the two points do not coincide. However, thisis not so disturbing as it is not so hard to accept that the distance between different points can be zero. For example, if the purpose of adistance is to quantify the money that are necessary to go from a place to another, it may certainly happen that the distance between twodistinct places is zero (e.g., because the two places are so close to each other that one can cover the spatial distance between them by feet,

1We are tacitly assuming that σ0(An) =∞ and σN(n)+1 = 0.

36

without spending money). In our specific case, the distance between two points Ann, Bnn ∈ E is zero if (and only if) An−Bnnis a zero-distributed sequence:

da.c.s.(Ann, Bnn) = 0 ⇐⇒ p(An −Bnn) = 0 ⇐⇒ An −Bnn ∼σ 0.

In the pseudometric space (E , da.c.s.), a sequence Bn,mnm converges to a point Ann if and only if da.c.s.(Bn,mn, Ann)→ 0as m → ∞, i.e., by Theorem 4.1, if and only if Bn,mnm is an a.c.s. for Ann. We have then proved all the statements reportedat the beginning of this section. In particular, the ‘a.c.s. topology’ τa.c.s. mentioned at the beginning is the topology induced on E by thedistance da.c.s.. Recall from general topology that a basis for τa.c.s. is given by the open balls

B(An, δ) =Bnn : da.c.s.(Bnn, Ann) < δ

, Ann ∈ E , δ > 0,

and a local basis at Ann is given by

B(An, δ) =Bnn : da.c.s.(Bnn, Ann) < δ

, δ > 0.

We remark that the topology τa.c.s. is not Housdorff, since any sequence Bn,mnm converging to Ann actually converges to everypoint of the form An + Znn, Znn ∼σ 0. In particular, the fixed point Ann converges to An + Znn for every zero-distributedsequence Znn. In view of the results of this section, the convergence notation

Bn,mna.c.s.−→ Ann (4.7)

is the most expressive to indicate that Bn,mnm is an a.c.s. for Ann. We will often use this notation in the remainder of the book.

4.2 The a.c.s. machinery as a tool for computing singular value and eigenvalue distribu-tions

The importance of the a.c.s. notion resides in Theorems 4.3 and 4.5. Theorem 4.3 provides a general tool for determining the singularvalue distribution of a ‘difficult’ matrix-sequence Ann, starting from the knowledge of the singular value distribution of ‘simpler’matrix-sequences Bn,mn. For the proof of this theorem, we need the following fundamental result about a.c.s.

Lemma 4.1. Let Ann be a matrix-sequence and let Bn,mnm be an a.c.s. for Ann. Then, for every F ∈ Cc(R),

limm→∞

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣ = 0. (4.8)

Proof. We first observe that it suffices to prove (4.8) for all real-valued functionsF ∈ C1c (R). Indeed, suppose that (4.8) holds for this kind

of functions and fix any F ∈ Cc(R). For every ε > 0, choose two real-valued functions <ε,=ε ∈ C1c (R) such that ‖<(F ) − <ε‖∞ ≤ ε

and ‖=(F )−=ε‖∞ ≤ ε, and set Fε = <ε + i=ε. Then, we have ‖F − Fε‖∞ ≤ 2ε and, for all m,n,∣∣∣∣∣∣∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣−∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(σj(An))− 1

N(n)

N(n)∑j=1

Fε(σj(Bn,m))

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ 4ε.

It follows that, for every m,

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣ ≤ lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(σj(An))− 1

N(n)

N(n)∑j=1

Fε(σj(Bn,m))

∣∣∣∣∣∣+ 4ε.

Passing to the limit as m→∞, and taking into account that (4.8) holds for <ε, =ε (and hence also for Fε), we obtain

limm→∞

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣ ≤ 4ε,

which is true for every ε > 0. Thus, (4.8) holds for F .Now, fix a real-valued function F ∈ C1

c (R). By hypothesis, Bn,mnm is an a.c.s. for Ann. Hence, for every m there existsnm such that, for n ≥ nm,



37

where limm→∞ c(m) = limm→∞ ω(m) = 0. For every m and every n ≥ nm,∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣ . (4.10)

We will consider separately the two terms in the right-hand side of (4.10), and we will show that each of them is bounded by a quantitydepending only on m and tending to 0 as m→∞. After this, the thesis is proved.

In order to estimate the first term in the right-hand side of (4.10), we use the perturbation theorem for singular values (Theorem 2.3).Assuming that the singular values are labeled in non-increasing order as in Theorem 2.3, we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))

∣∣∣∣∣∣ ≤ 1

N(n)

N(n)∑j=1

|F (σj(An))− F (σj(Bn,m +Rn,m))|

≤ 1

N(n)

N(n)∑j=1

‖F ′‖∞ |σj(An)− σj(Bn,m +Rn,m)| ≤ ‖F ′‖∞‖An −Bn,m −Rn,m‖ = ‖F ′‖∞‖Nn,m‖ ≤ ‖F ′‖∞ω(m),

which tends to 0 as m→∞.In order to estimate the second term in the right-hand side of (4.10), we use the interlacing theorem for singular values (Theorem 2.1);

for simplifity, we also adopt the convention of Theorem 2.1 about the meaning of σj(X) when j < 1 or j exceeds the size of X . We firstobserve that F can be expressed as the difference between two nonnegative, non-decreasing, bounded functions:

F = H −K, H(x) =

∫ x

−∞(F ′)+(t)dt, K(x) =

∫ x

−∞(F ′)−(t)dt,

where (F ′)+ = max(F ′, 0) and (F ′)− = max(−F ′, 0). Hence, for the second term in the right-hand side of (4.10) we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

H(σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

H(σj(Bn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

K(σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

K(σj(Bn,m))

∣∣∣∣∣∣ . (4.11)

Let rn,m = rank(Rn,m) ≤ c(m)N(n). By Theorem 2.1,

σj−rn,m(Bn,m) ≥ σj(Bn,m +Rn,m) ≥ σj+rn,m(Bn,m), j = 1, . . . , N(n).

Moreover, it is clear from our notation that

σj−rn,m(Bn,m) ≥ σj(Bn,m) ≥ σj+rn,m(Bn,m), j = 1, . . . , N(n).

Let H(∞) = limx→∞H(x). Recalling the monotonicity and nonnegativity of H , we get∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

H(σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

H(σj(Bn,m))

∣∣∣∣∣∣ ≤ 1

N(n)

N(n)∑j=1

|H(σj(Bn,m +Rn,m))−H(σj(Bn,m))|

≤ 1

N(n)

N(n)∑j=1

∣∣H(σj−rn,m(Bn,m))−H(σj+rn,m(Bn,m))∣∣ =

1

N(n)

N(n)∑j=1

H(σj−rn,m(Bn,m))− 1

N(n)

N(n)∑j=1

H(σj+rn,m(Bn,m))

38

=1

N(n)

N(n)−rn,m∑j=1−rn,m

H(σj(Bn,m))− 1

N(n)

N(n)+rn,m∑j=1+rn,m

H(σj(Bn,m))

=1

N(n)

min(rn,m, N(n)−rn,m)∑j=1−rn,m

H(σj(Bn,m))− 1

N(n)

N(n)+rn,m∑j=1+max(rn,m, N(n)−rn,m)

H(σj(Bn,m))

≤ 1

N(n)

rn,m∑j=1−rn,m

H(σj(Bn,m)) ≤ 2rn,mH(∞)

N(n)≤ 2c(m)‖H‖∞.

Similarly, one can show that the second term in the right-hand side of (4.11) is bounded by 2c(m)‖K‖∞, implying that the quantity inthe left-hand side of (4.11), namely the second term in the right-hand side of (4.10), is less than or equal to 2(‖H‖∞ + ‖K‖∞)c(m).Since the latter tends to 0 as m→∞, the thesis is proved.

Theorem 4.3. Let Ann be a matrix-sequence and let φ be a functional on Cc(R). Assume that:

1. Bn,mna.c.s.−→ Ann;

2. Bn,mn ∼σ φm for every m;

3. φm → φ pointwise over Cc(R).

Then Ann ∼σ φ.

Proof. Let F ∈ Cc(R). For all n,m we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− φ(F )

∣∣∣∣∣∣ ≤∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m))− φm(F )

∣∣∣∣∣∣+ |φm(F )− φ(F )|. (4.12)

By hypothesis, the second term in the right-hand side tends to 0 for n → ∞, while the third one tends to 0 for m → ∞. Therefore,passing first to the lim supn→∞ and then to the limm→∞ in (4.12), and using Lemma 4.1, we get the thesis.

Theorem 4.3 admits the following interesting converse, which we report for future use.

Theorem 4.4. Let Ann be a matrix-sequence. Assume that:

1. Ann ∼σ φ;

2. Bn,mna.c.s.−→ Ann;

3. Bn,mn ∼σ φm for every m;

Then φm → φ pointwise over Cc(R).

Proof. Let F ∈ Cc(R). For all n,m we have

|φm(F )− φ(F )| ≤

∣∣∣∣∣∣φm(F )− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m))− 1

N(n)

N(n)∑j=1

F (σj(An))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− φ(F )

∣∣∣∣∣∣ . (4.13)

By hypothesis, the first term in the right-hand side tends to 0 for m→∞, while the third one tends to 0 for n→∞. Therefore, passingfirst to the lim supn→∞ and then to the limm→∞ in (4.13), and using Lemma 4.1, we get the thesis.

Remark 4.2. Let Ann, Bn,mn be matrix-sequences and let φ, φm : Cc(R) → C be functionals. Consider the following fourconditions:

(1) Ann ∼σ φ;

39

(2) Bn,mn ∼σ φm for every m;

(3) Bn,mna.c.s.−→ Ann;

(4) φm → φ pointwise over Cc(R).

Theorems 4.3–4.4 show that ‘(1)∧ (2)∧ (3)⇒ (4)’ and ‘(2)∧ (3)∧ (4)⇒ (1)’.The implication ‘(1)∧ (2)∧ (4)⇒ (3)’ is false in general. As a counterexample, take An = In and Bn,m = diagi=1,...,n(−1)i. Then,

Ann ∼σ 1 and Bn,mn ∼σ 1, so that φ = φm = φ[1] is the evaluation functional at 1, φ[1](F ) = F (1). Moreover, it can bedirectly verified that An − Bn,mn ∼σ ϕ(F ) = 1

2

[F (0) + F (2)

]. Therefore, Bn,mnm cannot be an a.c.s. for Ann, because

otherwise An − Bn,mnm would be an a.c.s. of Onn and so, considering that Onn ∼σ φ[0], by Theorem 4.4 we would haveAn −Bn,mn ∼σ φ[0].

The implication ‘(1)∧ (3)∧ (4)⇒ (2)’, written in this way, is meaningless. However, a natural modification reads as follows:‘(1)∧ (3)⇒ there exists a functional φm such that φm → φ pointwise over Cc(R) and Bn,mn ∼σ φm for all sufficiently largem’. This statement is false in general. As a counterexample, take An = On and Bn,m = (1 + (−1)n) 1

mIn. Then, ‖Bn,m‖ ≤ 2m , so

Bn,mnm is an a.c.s. of Onn. Nevertheless, the limit

limn→∞

1

n

n∑j=1

F (σj(Bn,m)) = limn→∞

F ((1 + (−1)n) 1m )

does not exist for any function F ∈ Cc(R) such that F (0) 6= F ( 2m ). Therefore, the relation Bn,mn ∼σ φm cannot hold for any

functional φm.

In Theorem 4.5 we prove the analogue of Theorem 4.3 for the case of the eigenvalues, but we need to add the assumption that An andBn,m are Hermitian. Theorem 4.5 is then a general tool for determining the spectral distribution of a ‘difficult’ matrix-sequence Annformed by Hermitian matrices, starting from the spectral distribution of simpler matrix-sequences Bn,mn, again formed by Hermitianmatrices.

The next lemma shows that, whenever An and Bn,m are Hermitian, the small-rank matrix Rn,m and the small-norm matrix Nn,min the splitting (4.1) may be supposed to be Hermitian. In the following, we say that Bn,mnm is an a.c.s. for Ann formed byHermitian matrices if Bn,mnm is an a.c.s. for Ann and every Bn,m is Hermitian.

Lemma 4.2. Let Ann be a sequence of Hermitian matrices, and let Bn,mnm be an a.c.s. for Ann formed by Hermitianmatrices. Then, for every m there exists nm such that, for n ≥ nm,



where Rn,m, Nn,m are Hermitian, the quantities nm, c(m), ω(m) depend only on m, and

limm→∞

c(m) = limm→∞

ω(m) = 0.

Proof. Take the real part in (4.1) and use the inequalities rank(<(X)) ≤ 2 rank(X) and ‖<(X)‖ ≤ ‖X‖ to conclude that, by replacingRn,m, Nn,m with <(Rn,m), <(Nn,m) (if necessary), we can assume Rn,m, Nn,m to be Hermitian.

Lemma 4.3 is the ‘eigenvalue version’ of Lemma 4.1.

Lemma 4.3. Let Ann be a sequence of Hermitian matrices and let Bn,mnm be an a.c.s. for Ann formed by Hermitianmatrices. Then, for every F ∈ Cc(C),

limm→∞

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

F (λj(Bn,m))

∣∣∣∣∣∣ = 0. (4.14)

Proof. The proof is essentially the same as the proof of Lemma 4.1; we just outline the main steps, emphasizing the analogies withLemma 4.1 and pointing out where we need the assumption that An and Bn,m are Hermitian.

Noting that all the eigenvalues λj(An), λj(Bn,m), j = 1, . . . , N(n), are real, it suffices to prove (4.14) for all real-valued functionsF ∈ C1

c (R). Indeed, suppose that (4.14) holds for this kind of functions and fix any F ∈ Cc(C). For every ε > 0, choose two real-valuedfunctions <ε, =ε ∈ C1

c (R) such that ‖<(F ) − <ε‖∞,R ≤ ε and ‖=(F ) − =ε‖∞,R ≤ ε, and set Fε = <ε + i=ε. Then, we have‖F − Fε‖∞,R ≤ 2ε and, for all m,n,∣∣∣∣∣∣

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

F (λj(Bn,m))

∣∣∣∣∣∣−∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(λj(An))− 1

N(n)

N(n)∑j=1

Fε(λj(Bn,m))

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ 4ε.

40

It follows that, for every m,

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

F (λj(Bn,m))

∣∣∣∣∣∣ ≤ lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(λj(An))− 1

N(n)

N(n)∑j=1

Fε(λj(Bn,m))

∣∣∣∣∣∣+ 4ε.

Passing to the limit as m→∞, and taking into account that (4.14) holds for <ε, =ε (and hence also for Fε), we obtain

limm→∞

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

F (λj(Bn,m))

∣∣∣∣∣∣ ≤ 4ε,

which is true for every ε > 0. Thus, (4.14) holds for F .Now we fix a real-valued function F ∈ C1

c (R). Since Bn,mnm is a.c.s. for Ann and An, Bn,m are Hermitian, we canuse Lemma 4.2 to see that for every m there exists nm such that, for n ≥ nm, the splitting (4.9) holds with Hermitian Rn,m, Nn,m.Following the proof of Lemma 4.1, we arrive at the inequality (4.10), with ‘σj’ replaced by ‘λj’, and the thesis is proved if we are able tobound the two terms in the right-hand side by a quantity depending only on m and tending to 0 as m→∞.

The first term is bounded exactly as in Lemma 4.1, using the perturbation theorem for eigenvalues (Theorem 2.4) instead of theperturbation theorem for singular values (Theorem 2.3). Note that the perturbation theorem for eigenvalues, contrary to the perturbationtheorem for singular values, applies only to Hermitian matrices.

Also the second term is bounded exactly as in Lemma 4.1, using the interlacing theorem for eigenvalues (Theorem 2.2) instead ofthe interlacing theorem for singular values (Theorem 2.1). Even in this case, the interlacing theorem for eigenvalues, contrary to theinterlacing theorem for singular values, applies only to Hermitian matrices.

Theorem 4.5. Let Ann be a sequence of Hermitian matrices and let φ be a functional on Cc(C). Assume that:

1. Bn,mna.c.s.−→ Ann with every Bn,m Hermitian;

2. Bn,mn ∼λ φm for every m;

3. φm → φ pointwise over Cc(C).

Then Ann ∼λ φ.

Proof. Let F ∈ Cc(C). For all n,m we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1


∣∣∣∣∣∣ ≤∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

F (λj(Bn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(Bn,m))− φm(F )

∣∣∣∣∣∣+ |φm(F )− φ(F )|. (4.15)

By hypothesis, the second term in the right-hand side tends to 0 for n → ∞, while the third one tends to 0 for m → ∞. Therefore,passing first to the lim supn→∞ and then to the limm→∞ in (4.15), and using Lemma 4.3, we get the thesis.

Two important corollaries of Theorems 4.3 and 4.5 are given in the following. They will be used in Section 7.2 to prove the asymptoticspectral and singular value distribution results for GLT sequences.

Corollary 4.1. Let Ann be a matrix-sequence. Assume that:

1. Bn,mnm is an a.c.s. for Ann;

2. for every m, Bn,mn ∼σ fm for some measurable function fm : D ⊂ Rk → C;

3. |fm| → |f | in measure over D, being f : D → C another measurable function.

Then Ann ∼σ f .

Proof. Apply Theorem 4.3 with φm = φ[|fm|] and φ = φ[|f |]. Note that φm → φ pointwise over Cc(R) by Lemma 2.5.

Corollary 4.2. Let Ann be a sequence of Hermitian matrices. Assume that:

1. Bn,mna.c.s.−→ Ann with every Bn,m Hermitian;

2. for every m, Bn,mn ∼λ fm for some measurable function fm : D ⊂ Rk → C;

3. fm → f in measure over D, being f : D → C another measurable function.

Then Ann ∼λ f .

Proof. Apply Theorem 4.5 with φm = φ[fm] and φ = φ[f ]. Note that φm → φ pointwise over Cc(C) by Lemma 2.5.

41

4.3 The a.c.s. algebraIn this section, we investigate the algebraic properties possessed by the approximating classes of sequences. These properties form thebasis of the so-called GLT algebra, which will be studied later on, in Section 7.5. We begin with the following observation, whose proofis very simple and is left to the reader.

Remark 4.3. Let Bn,mnm be an a.c.s. for Ann. Then B∗n,mnm is an a.c.s. for A∗nn.

Proposition 4.1. Let Ann, A′nn be matrix-sequences, and let

• Bn,mnm an a.c.s. for Ann,

• B′n,mnm an a.c.s. for A′nn.

Then αBn,m + βB′n,mnm is an a.c.s. for αAn + βA′nn, for all α, β ∈ C.

Proof. By hypothesis,

• for every m there exists nm such that, for n ≥ nm,



where limm→∞ c(m) = limm→∞ ω(m) = 0;

• for every m there exists n′m such that, for n ≥ n′m,

A′n = B′n,m +R′n,m +N ′n,m,

rank(R′n,m) ≤ c′(m)N(n), ‖N ′n,m‖ ≤ ω′(m),

where limm→∞ c′(m) = limm→∞ ω′(m) = 0.

Hence, for every m and every n ≥ max(nm, n′m),

αAn + βA′n = (αBn,m + βB′n,m) + (αRn,m + βR′n,m) + (αNn,m + βN ′n,m),

rank(αRn,m + βR′n,m) ≤(c(m) + c′(m)

)N(n), ‖αNn,m + βN ′n,m‖ ≤ |α|ω(m) + |β|ω′(m),

wherelimm→∞

(c(m) + c′(m)

)= limm→∞

(|α|ω(m) + |β|ω′(m)

)= 0.

Hence, by Definition 4.1, αBn,m + βB′n,mnm is an a.c.s. for αAn + βA′nn.

Proposition 4.1 addresses the case of a linear combination αAn + βA′nn of two matrix-sequences Ann and A′nn. We wouldlike to prove an analogous result for the product AnA′nn. To this end, an additional (mild) assumption on Ann and A′nn isrequired, namely that Ann and A′nn are sparsely unbounded.

Definition 4.2 (sparsely unbounded matrix-sequence). Let Ann be a matrix-sequence. We say that Ann is sparsely unbounded(s.u.) if for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

where limM→∞ r(M) = 0.

The following proposition provides equivalent characterizations of sparsely unbounded matrix-sequences.

Proposition 4.2. Let Ann be a matrix-sequence. The following conditions are equivalent.

1. Ann is s.u.

2. limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 0.

42

3. For every M > 0 there exists nM such that, for n ≥ nM ,

An = An,M + An,M ,

rank(An,M ) ≤ r(M)N(n), ‖An,M‖ ≤M,


Note that condition 2 can be rewritten as

limM→∞

lim supn→∞

1

N(n)

N(n)∑i=1

χ(M,∞)(σi(An)) = 0.

Proof. (1⇒ 2) Suppose that Ann is s.u. Then, for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

where limM→∞ r(M) = 0. Therefore,

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M)

and, consequently,

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 0.

(2⇒ 1) Suppose that condition 2 is met. For every M > 0, define

δ(M) = lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

∈ [0, 1]

and note that (obviously)

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

< δ(M) +1

M.

Hence, by definition of lim sup, for every M > 0 the sequence #i∈1,...,N(n):σi(An)>MN(n) is eventually less than r(M) = δ(M) + 1

M ,i.e., there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M).

Since r(M)→ 0 as M →∞, item 1 is proved.(1⇒ 3) Suppose that Ann is s.u.: for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

where limM→∞ r(M) = 0. Let An = UnΣnV∗n be an SVD of An. Let Σn,M be the matrix obtained from Σn by setting to 0 all the

singular values of An that are less than or equal to M , and let Σn,M = Σn − Σn,M be the matrix obtained from Σn by setting to 0 allthe singular values of An that exceed M . Then,

An = UnΣnV∗n = UnΣn,MV

∗n + UnΣn,MV

∗n = An,M + An,M ,

where An,M = UnΣn,MV∗n and An,M = UnΣn,MV

∗n satisfy, for n ≥ nM ,

rank(An,M ) = #i ∈ 1, . . . , N(n) : σi(An) > M ≤ r(M)N(n), ‖An,M‖ = σmax(An,M ) ≤M.

(3⇒ 1) Suppose that condition 3 holds. Then, for every M > 0 there exists nM such that, for n ≥ nM ,

An = An,M + An,M ,

rank(An,M ) ≤ r(M)N(n), ‖An,M‖ ≤M,

43

where limM→∞ r(M) = 0. By the minimax principle for singular values (Theorem 2.5), for all i = 1, . . . , N(n) we have

σi(An) = maxdimV=i

minx∈V, ‖x‖=1

‖Anx‖ ≤ maxdimV=i

minx∈V, ‖x‖=1

(‖An,Mx‖+ ‖An,Mx‖

)≤ max

dimV=imin

x∈V, ‖x‖=1

(‖An,Mx‖+ ‖An,M‖

)= σi(An,M ) + ‖An,M‖ ≤ σi(An,M ) +M. (4.16)

Since rank(An,M ) ≤ r(M)N(n), An,M has at most r(M)N(n) nonzero singular values. Therefore, by (4.16), An has at mostr(M)N(n) singular values greater than M , i.e., #i ∈ 1, . . . , N(n) : σi(An) > M ≤ r(M)N(n), or, equivalently,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M).

This implies that Ann is s.u.

We now show that any matrix-sequence enjoying an asymptotic singular value distribution is s.u. A proof of this useful result,analogous to the one that we are going to see, appeared in [96, Proposition 2.7].

Proposition 4.3. If Ann ∼σ f then Ann is s.u.

Proof. Let D ⊂ Rk be the domain of the function f . If we could choose F = χ(M,∞) as a test function in (2.56), then we would obtain

limn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= limn→∞

1

N(n)

N(n)∑i=1

χ(M,∞)(σi(An)) =1

µk(D)

∫D

χ(M,∞)(|f(x)|)dx

=µk|f | > M

µk(D). (4.17)

Since µk|f | > M → 0 asM →∞ (by the dominated convergence theorem), eq. (4.17) would imply that condition 2 in Proposition 4.2is met and the proof would be over. However, χ(M,∞) cannot be chosen as a test function in (2.56), because it does not belong to Cc(R).Hence, to obtain the thesis we need some work.

Fix M > 0 and take FM ∈ Cc(R) such that FM = 1 over [0,M/2], FM = 0 over [M,∞) and 0 ≤ FM ≤ 1 over R. Note thatFM ≤ χ[0,M ] over [0,∞). Then,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 1− #i ∈ 1, . . . , N(n) : σi(An) ≤MN(n)

= 1− 1

N(n)

N(n)∑i=1

χ[0,M ](σi(An))

≤ 1− 1

N(n)

N(n)∑i=1

FM (σi(An))n→∞−→ 1− 1

µk(D)

∫D

FM (|f(x)|)dx

and

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ 1− 1

µk(D)

∫D

FM (|f(x)|)dx.

Since FM (|f(x)|)→ 1 a.e. and |FM (|f(x)|)| ≤ 1, by the dominated convergence theorem we get

limM→∞

∫D

FM (|f(x)|)dx = µk(D),

and so

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 0.

This means that condition 2 in Proposition 4.2 is satisfied, i.e., Ann is s.u.

We now prove the analogue of Proposition 4.1 for the case of the product of two a.c.s. This important result appeared for the first timein [87].

Proposition 4.4. Let Ann, A′nn be s.u. matrix-sequences, and let

• Bn,mnm an a.c.s. for Ann,

• B′n,mnm an a.c.s. for A′nn.

44

Then, Bn,mB′n,mnm is an a.c.s. for AnA′nn.

Proof. By hypothesis, for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m, A′n = B′n,m +R′n,m +N ′n,m,

rank(Rn,m), rank(R′n,m) ≤ c(m)N(n), ‖Nn,m‖, ‖N ′n,m‖ ≤ ω(m),

where limm→∞ c(m) = limm→∞ ω(m) = 0. Hence,

AnA′n = Bn,mB

′n,m +Bn,mR

′n,m + Bn,mN

′n,m +Rn,mA

′n + Nn,mA

′n .

Since Ann and A′nn are s.u., for every M > 0 there exists n(M) such that, for n ≥ n(M),

An = An,M + An,M , A′n = A′n,M + A′n,M ,

rank(An,M ), rank(A′n,M ) ≤ r(M)N(n), ‖An,M‖, ‖A′n,M‖ ≤M,

where limM→∞ r(M) = 0. Setting Mm = [ω(m)]−1/2, for every m and every n ≥ max(nm, n(Mm)) we have

Bn,mN′n,m +Nn,mA

′n = (An −Rn,m −Nn,m)N ′n,m +Nn,m(A′n,Mm

+ A′n,Mm)

= (An,Mm+ An,Mm

−Rn,m −Nn,m)N ′n,m +Nn,mA′n,Mm

+Nn,mA′n,Mm

= An,MmN ′n,m + An,Mm

N ′n,m −Rn,mN ′n,m −Nn,mN ′n,m +Nn,mA′n,Mm

+Nn,mA′n,Mm

,

and so

AnA′n = Bn,mB

′n,m +Bn,mR

′n,m + Bn,mN

′n,m +Rn,mA

′n + Nn,mA

′n

= Bn,mB′n,m +Bn,mR

′n,m +Rn,mA

′n + An,Mm

N ′n,m + An,MmN ′n,m −Rn,mN ′n,m −Nn,mN ′n,m

+Nn,mA′n,Mm

+Nn,mA′n,Mm

,

whererank(Bn,mR

′n,m +Rn,mA

′n + An,Mm

N ′n,m −Rn,mN ′n,m +Nn,mA′n,Mm

) ≤ [3c(m) + 2r(Mm)]N(n),

‖An,MmN ′n,m −Nn,mN ′n,m +Nn,mA

′n,Mm

‖ ≤ 2[ω(m)]1/2 + [ω(m)]2.

Thus, Bn,mB′n,mnm is an a.c.s. for AnA′nn.

Before concluding this section, we would like to mention two further properties of a.c.s., which will not be used in this book, but areanyway interesting to know.

• Let Ann be a s.u. matrix-sequence formed by Hermitian matrices and let Bn,mnm be an a.c.s. of Ann formed byHermitian matrices. Then, f(Bn,mnm is an a.c.s. of f(An)n for all continuous functions f : R → R. This result can befound in [91, Theorem 2.2].

• Let Ann be a sparsely vanishing matrix-sequence (see Definition 7.2) and let Bn,mnm be an a.c.s. of Ann. ThenB†n,mnm is an a.c.s. of A†nn. For the proof, see [93, Proposition 2.3].

4.4 Some criterions to identify a.c.s.In practical applications, it often happens that a sequence of matrix-sequences Bn,mnm is given together with a matrix-sequenceAnn, and one would like to show that Bn,mnm is an a.c.s. for Ann, without constructing the splitting (4.1). In this section, weprovide two useful criterions to solve this problem.

The first criterion, expressed in Theorem 4.6 and Corollary 4.3, is formulated in terms of Schatten p-norms. The second criterion,expressed in Theorem 4.7 and Corollary 4.4, is formulated in terms of the singular value distribution.

Lemma 4.4. Let C be a square matrix of size s. Suppose that

‖C‖pp ≤ εs

for some p ∈ [1,∞). ThenC = R+N,

withrank(R) ≤ ε

1p+1 s, ‖N‖ ≤ ε

1p+1 .

45

Proof. Since ‖C‖pp =∑si=1[σi(C)]p ≤ εs, the number of singular values of C that exceed ε

1p+1 cannot be larger than ε

1p+1 s. Let

C = UΣV ∗ be an SVD of C and writeC = UΣV ∗ = U ΣV ∗ + U ΣV ∗,

where Σ is obtained from Σ by setting to 0 all the singular values that are less than or equal to ε1p+1 , while Σ = Σ− Σ is obtained from

Σ by setting to 0 all the singular values that exceed ε1p+1 . Then

C = R+N,

where R = U ΣV ∗ and N = U ΣV ∗ satisfy rank(R) ≤ ε1p+1 s and ‖N‖ ≤ ε

1p+1 .

By Definition 4.1, a sequence of matrix-sequences Cn,mnm is an a.c.s. of ON(n)n if and only if the following condition ismet: for every m there exists nm such that, for n ≥ nm,

Cn,m = Rn,m +Nn,m,


where limm→∞ c(m) = limm→∞ ω(m) = 0. Moreover, it is clear that Bn,mnm is an a.c.s. for Ann if and only if An −Bn,mnm is an a.c.s. of ON(n)n.

Theorem 4.6. Let Cn,mnm be a sequence of matrix-sequences and let 1 ≤ p <∞. Suppose that for every m there exists nm suchthat, for n ≥ nm,

‖Cn,m‖pp ≤ ε(m,n)N(n),

where limm→∞

lim supn→∞

ε(m,n) = 0. Then Cn,mnm is an a.c.s. of ON(n)n.

Proof. By Lemma 4.4, for every m and every n ≥ nm we have

Cn,m = Rn,m +Nn,m,

rank(Rn,m) ≤ [ε(m,n)]1p+1N(n), ‖Nn,m‖ ≤ [ε(m,n)]

1p+1 .

Letε(m) = lim sup

n→∞ε(m,n).

By definition of lim sup, for every m there exists n′m such that, for n ≥ n′m,

ε(m,n) ≤ ε(m) +1

m.

Setting nm = max(nm, n′m), for every m and every n ≥ nm we have

Cn,m = Rn,m +Nn,m,

rank(Rn,m) ≤(ε(m) +

1

m

) 1p+1

N(n), ‖Nn,m‖ ≤(ε(m) +

1

m

) 1p+1

.

Since ε(m)→ 0 by assumption, Cn,mnm is an a.c.s. of ON(n)n.

Corollary 4.3. Let Ann be a matrix-sequence, let Bn,mnm be a sequence of matrix-sequences, and let 1 ≤ p < ∞. Supposethat for every m there exists nm such that, for n ≥ nm,

‖An −Bn,m‖pp ≤ ε(m,n)N(n),

where limm→∞

lim supn→∞

ε(m,n) = 0. Then Bn,mnm is an a.c.s. for Ann.

Exercise 4.1. Fix 1 ≤ p < ∞. By Corollary 4.3, a sufficient condition to ensure that Bn,mnm is an a.c.s. for Ann is thefollowing:

‖An −Bn,m‖p ≤ ε(m,n)N(n)1/p for every m and n, with limm→∞

lim supn→∞

ε(m,n) = 0. (4.19)

Is there any α > 1/p such that the following weaker condition,

‖An −Bn,m‖p ≤ ε(m,n)N(n)α for every m and n, with limm→∞

lim supn→∞

ε(m,n) = 0, (4.20)

still ensures that Bn,mnm is an a.c.s. for Ann?

46

Theorem 4.7. Let Cn,mnm be a sequence of matrix-sequences. Suppose that Cn,mn ∼σ gm for some measurable functiongm : D ⊂ Rk → C such that gm → 0 in measure as m→∞. Then Cn,mnm is an a.c.s. of ON(n)n.

Proof. For any ` ∈ N, let F` ∈ Cc(R) such that F` = 1 over [0, 1/2`], F` = 0 outside [−1/`, 1/`], and 0 ≤ F` ≤ 1 over R. Note thatF` ≤ χ[0,1/`] over [0,∞). For every m, `, we have

#i ∈ 1, . . . , N(n) : σi(Cn,m) > 1/`N(n)

= 1− #i ∈ 1, . . . , N(n) : σi(Cn,m) ≤ 1/`N(n)

= 1− 1

N(n)

N(n)∑i=1

χ[0,1/`](σi(Cn,m))

≤ 1− 1

N(n)

N(n)∑i=1

F`(σi(Cn,m))n→∞−→ c(m, `), (4.21)

wherec(m, `) = 1− 1

µk(D)

∫D

F`(|gm(x)|)dx.

By Lemma 2.5, c(m, `)→ 0 as m→∞, for every fixed `. Hence, there exists a sequence `mm of natural numbers such that

limm→∞

`m =∞, limm→∞

c(m, `m) = 0.

By (4.21), for each m we have

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(Cn,m) > 1/`mN(n)

≤ c(m, `m). (4.22)

Let Cn,m = Un,mΣn,mV∗n,m be an SVD of Cn,m. Let Σn,m be the matrix obtained from Σn,m by setting to 0 all the singular vaules

that are less than or equal to 1/`m, and let Σn,m = Σn,m− Σn,m be the matrix obtained from Σn,m by setting to 0 all the singular valuesthat exceed 1/`m. Then we can write

Cn,m = Rn,m +Nn,m,

where Rn,m = Un,mΣn,mV∗n,m and Nn,m = Un,mΣn,mV

∗n,m. By definition, ‖Nn,m‖ ≤ 1/`m. Moreover, (4.22) says that

lim supn→∞

rank(Rn,m)

N(n)≤ c(m, `m),

implying the existence of a nm such that, for n ≥ nm,

rank(Rn,m) ≤(c(m, `m) + 1/m

)N(n).

This shows that Cn,mnm is an a.c.s. of ON(n)n.

Corollary 4.4. Let Ann be a matrix-sequence and let Bn,mnm be a sequence of matrix-sequences. Suppose that An −Bn,mn ∼σ gm for some measurable function gm : D ⊂ Rk → C such that gm → 0 in measure as m →∞. Then Bn,mnm is ana.c.s. for Ann.

We conclude this section with the following converse of Theorem 4.7.

Proposition 4.5. Let Cn,mnm be an a.c.s. of ON(n)n and suppose that Cn,mn ∼σ gm for every m. Then gm → 0 in measure.

Proof. Since ON(n)n ∼σ 0, by Theorem 4.4 we have φ[gm] → φ[0] pointwise over Cc(R). By Lemma 2.6, this implies that gm → 0in measure.

4.5 An extension of the concept of a.c.s.We provide in this section an extension of the definition of a.c.s. that will be used to define LT and GLT sequences in Chapters 6–7.The extension is plain. The underlying idea is that, in Definition 4.1, one could choose to approximate Ann by a class of sequencesBn,αnα∈A parameterized by a not necessarily integer parameter α. For example, one may want to use a parameter ε > 0 and toclaim that a given class of sequences Bn,εnε>0 is an a.c.s. for Ann when ε → 0. Intuitively, this assertion should have thefollowing meaning: for every ε > 0 there exists nε such that, for n ≥ nε,

An = Bn,ε +Rn,ε +Nn,ε,

rank(Rn,ε) ≤ c(ε)N(n), ‖Nn,ε‖ ≤ ω(ε),

where the quantities nε, c(ε), ω(ε) depend only on ε and limε→0 c(ε) = limε→0 ω(ε) = 0. Actually, this is the correct meaning.

47

Definition 4.3 (approximating class of sequences for ε→ 0). Let Ann be a matrix-sequence. We say that Bn,εnε>0 is an a.c.s.of Ann for ε→ 0 if the following property holds: for every ε > 0 there exists nε such that, for n ≥ nε,

An = Bn,ε +Rn,ε +Nn,ε,

rank(Rn,ε) ≤ c(ε)N(n), ‖Nn,ε‖ ≤ ω(ε),

where the quantities nε, c(ε), ω(ε) depend only on ε and

limε→0

c(ε) = limε→0

ω(ε) = 0.

Definition 4.3 will be used to define GLT sequences in Chapter 7. Note that, if Bn,εnε>0 is an a.c.s. of Ann for ε → 0, thenBn,ε(m)nm is an a.c.s. for Ann (in the sense of the classical Definition 4.1) for all sequences of positive numbers ε(m)m suchthat ε(m)→ 0 as m→∞.

Exercise 4.2. Show that Bn,εnε>0 is an a.c.s. of Ann for ε → 0 if and only if da.c.s.(Bn,εn, Ann) → 0 as ε → 0, whereda.c.s. is the ‘a.c.s. distance’ defined in (4.6). This means that we could reformulate Definition 4.3 as follows: Bn,εnε>0 is an a.c.s.of Ann for ε→ 0 if Bn,εn converges to Ann as ε→ 0 in the pseudometric space (E , da.c.s.) considered in Section 4.1.

For the definition of LT sequences, we do not need the concept of a.c.s. parameterized by a positive ε → 0. On the contrary, weneed the concept of a.c.s. parameterized by a multi-index m → ∞. Since any m ∈ N is a special multi-index, the definition of a.c.s.parameterized by a multi-index m → ∞ (Definition 4.4) is a true extension of Definition 4.1. In the following, a multi-index sequenceof matrix-sequences is any class of sequences of the form Bn,mnm∈M, where:

1. M⊆ Nq for some q ≥ 1, andM∩ i ∈ Nq : i ≥ h 6= ∅ for every h ∈ Nq . We express the latter condition by saying that∞ isan accumulation point forM. This is required to ensure thatm can tend to∞ insideM;

2. for everym ∈M, Bn,mn is a matrix-sequence.

Definition 4.4 (approximating class of sequences). Let Ann be a matrix-sequence. An approximating class of sequences (a.c.s.) forAnn is a multi-index sequence of matrix-sequences Bn,mnm∈M with the following property: for everym ∈M there exists nmsuch that, for n ≥ nm,



where the quantities nm, c(m), ω(m) depend only onm, and

limm→∞

c(m) = limm→∞

ω(m) = 0.

Definition 4.4 extends the classical definition of a.c.s. (Definition 4.1). Indeed, a classical a.c.s. Bn,mnm for Ann is an a.c.s.also in the sense of Definition 4.4 (takeM as the infinite subset of N where m varies).

Remark 4.4. If Bn,mnm∈M is an a.c.s. for Ann in the sense of Definition 4.4, then Bn,mnm is an a.c.s. for Ann (in thesense of the classical Definition 4.1) for all sequences m = m(m)m ⊆M such thatm→∞ when m→∞.

Remark 4.5. An equivalent definition of a.c.s. is obtained by replacing, in Definition 4.4, ‘for all m ∈ M’ with ‘for all sufficientlylarge m ∈ M’ (i.e., ‘for every m ∈ M that is greater than or equal to some m ∈ Nq , being M ⊆ Nq’). Indeed, suppose that thesplitting (4.23) and the related conditions on Rn,m and Nn,m hold for m ≥ m; then, defining nm = 1, c(m) = 1, ω(m) = 0 andRn,m = An −Bn,m, Nn,m = O for all the other values ofm, we see that they actually hold for everym ∈M.

Remark 4.6. Suppose that Bn,mnm∈M is an a.c.s. for Ann and B′n,mnm∈M is an a.c.s. for A′nn. Then:

1. B∗n,mnm∈M is an a.c.s. for A∗nn;

2. αBn,m + βB′n,mnm∈M is an a.c.s. for αAn + βA′nn, for all α, β ∈ C;

3. if Ann and A′nn are s.u., then Bn,mB′n,mnm∈M is an a.c.s. for AnA′nn.

The proof of these results is omitted, because it is essentially the same as the proof of the analogous results for standard a.c.s.; seeRemark 4.3 and Propositions 4.1, 4.4.

48

Chapter 5

Multilevel Toeplitz matrices

In this chapter, we provide the definition and some key properties of multilevel Toeplitz matrices. Of course, we do not pretend to coverhere, in a single chapter, all the details of this extensive topic, which was the subject of several books [19, 20, 24, 25, 59]. Nevertheless,we will present all the results that are needed herein, so as to keep the present book as much self-contained as possible. In particular, wewill give an a.c.s.-based proof of the multilevel version of Szego’s first limit theorem and of the Avram–Parter theorem. These theoremsconcern the spectral and singular value distribution of multilevel Toeplitz matrices. Needless to say, they play a fundamental role in thetheory of GLT sequences.

5.1 DefinitionsGiven a d-index n = (n1, . . . , nd) ∈ Nd, a matrix of the form

[ai−j ]ni,j=1 ∈ CN(n)×N(n), (5.1)

whose (i, j)-th entry depends only on the difference between the d-indices i and j, is called a multilevel Toeplitz matrix (or, moreprecisely, a d-level Toeplitz matrix). In the case d = 1, the matrix (5.1) becomes

[ai−j ]ni,j=1 =

a0 a−1 a−2 · · · · · · a−(n−1)

a1. . . . . . . . .

...

a2. . . . . . . . . . . .

......

. . . . . . . . . . . . a−2

.... . . . . . . . . a−1

an−1 · · · · · · a2 a1 a0

,

which means that a 1-level Toeplitz matrix is just a matrix whose entries are constant along each diagonal. 1-level Toeplitz matrices areoften referred to as Toeplitz matrices, without further specifications. In the case d = 2, the matrix (5.1) can be written as

[ai−j ]ni,j=1 =

[[ai1−j1,i2−j2 ]

n2

i2,j2=1

]n1

i1,j1=1= [ai1−j1 ]

n1

i1,j1=1 =

a0 a−1 a−2 · · · · · · a−(n1−1)

a1. . . . . . . . .

...

a2. . . . . . . . . . . .

......

. . . . . . . . . . . . a−2

.... . . . . . . . . a−1

an1−1 · · · · · · a2 a1 a0

, (5.2)

49

where

ak = [ak,i2−j2 ]n2

i2,j2=1 =

ak,0 ak,−1 ak,−2 · · · · · · ak,−(n2−1)

ak,1. . . . . . . . .

...

ak,2. . . . . . . . . . . .

......

. . . . . . . . . . . . ak,−2

.... . . . . . . . . ak,−1

ak,n2−1 · · · · · · ak,2 ak,1 ak,0

, k = −(n1 − 1), . . . , n1 − 1. (5.3)

A matrix of this form, which we call a 2-level Toeplitz matrix, is also known as a block Toeplitz matrix with Toeplitz blocks, or BTTBmatrix. More generally, we could say that a d-level Toeplitz matrix is a block Toeplitz matrix with (d− 1)-level Toeplitz blocks.

The next lemma provides a useful expression for the d-level Toeplitz matrix (5.1). For n ∈ N and k ∈ Z, we denote by J (k)n the n×n

matrix whose (i, j)-th entry equals 1 if i− j = k and 0 otherwise; that is,

(J (k)n )ij = δi−j−k, i, j = 1, . . . , n, n ∈ N, k ∈ Z, (5.4)

where δr = 1 if r = 0 and δr = 0 otherwise. Observe that J (k)n =

(J

(1)n

)kfor k ≥ 0 and J (k)

n =(J

(−1)n

)−kfor k ≤ 0. For n ∈ Nd and

k ∈ Zd, we setJ (k)n = J (k1)

n1⊗ J (k2)

n2⊗ · · · ⊗ J (kd)

nd. (5.5)

Lemma 5.1. The d-level Toeplitz matrix (5.1) admits the following expression:

[ai−j ]ni,j=1 =

n−1∑k=−(n−1)

akJ(k)n , (5.6)

where J (k)n is defined in (5.5).

Proof. Eq. (5.6) is proved componentwise, by showing that the (i, j)-th entry of the matrix in the right-hand side is equal to ai−j . By(5.4) and by the fundamental property (2.44), for all i, j = 1, . . . ,n we have

(J (k)n )ij = (J (k1)

n1)i1j1(J (k2)

n2)i2j2 · · · (J (kd)

nd)idjd = δi1−j1−k1δi2−j2−k2 · · · δid−jd−kd = δi−j−k,

where δr = 1 if r = 0 and δr = 0 otherwise. Therefore, n−1∑k=−(n−1)

akJ(k)n

ij

=

n−1∑k=−(n−1)

ak(J (k)n )ij =

n−1∑k=−(n−1)

akδi−j−k = ai−j ,

and (5.6) is proved.

Given a function f : [−π, π]d → C belonging to L1([−π, π]d), its Fourier coefficients are denoted by

fk =1

(2π)d

∫[−π,π]d

f(θ)e−ik·θdθ, k ∈ Zd. (5.7)

The n-th multilevel Toeplitz matrix associated with f is defined as

Tn(f) = [fi−j ]ni,j=1 =

n−1∑j=−(n−1)

fjJ(k)n , (5.8)

where in the last equality we used Lemma 5.1. We call Tn(f)n∈Nd the family of multilevel Toeplitz matrices associated with f , which,in turn, is called the generating function of Tn(f)n∈Nd .

50

5.2 Properties of multilevel Toeplitz matricesIn this section, we fix a d-index n ∈ Nd and we study the properties of the Toeplitz map Tn(·) : L1([−π, π]d) → CN(n)×N(n). Thismap is linear, i.e.,

Tn(αf + βg) = αTn(f) + βTn(g), α, β ∈ C, f, g ∈ L1([−π, π]d), n ∈ Nd. (5.9)

Eq. (5.9) follows from the relation (αf + βg)k = αfk + βgk, k ∈ Zd, which is a consequence of the linearity of the integral in (5.7).Another nice property of the Toeplitz map Tn(·), which is a direct consequence of its definition, is that

Tn(1) = IN(n), n ∈ Nd. (5.10)

For every f ∈ L1([−π, π]d), the Fourier coefficients of f are related to the Fourier coefficients of its conjugate f by

fj =1

(2π)d

∫[−π,π]d

f(θ)e−ij·θdθ =1

(2π)d

∫[−π,π]d

f(θ)eij·θdθ = (f)−j , j ∈ Zd.

Therefore, for all i, j = 1, . . . ,n, [Tn(f)

]ij

= (f)i−j = fj−i =[Tn(f)∗

]ij,

i.e.,Tn(f)∗ = Tn(f), f ∈ L1([−π, π]d), n ∈ Nd. (5.11)

From this identity, we infer that, if f is real or a.e. real,1 then all the multilevel Toeplitz matrices Tn(f) are Hermitian.The next lemma provides an integral expression for the sesquilinear form u∗Tn(f)v, where u,v ∈ CN(n). This expression, in

combination with the minimax principle for eigenvalues (Theorem 2.6), will be used in Theorem 5.1 to localize the spectrum of Tn(f) inthe case where f is real a.e. (so that Tn(f) is Hermitian).

Lemma 5.2. For every f ∈ L1([−π, π]d) and every n ∈ Nd,

u∗Tn(f)v =1

(2π)d

∫[−π,π]d

f(θ)u∗U(θ)v dθ, u,v ∈ CN(n), (5.12)

where U(θ) =[e−i(i−j)·θ]n

i,j=1. The matrix U(θ) satisfies

u∗U(θ)u =

∣∣∣∣∣∣n∑j=1

ujeij·θ

∣∣∣∣∣∣ , θ ∈ [−π, π]d, u ∈ CN(n), (5.13)

and so U(θ) is HPSD for every θ ∈ [−π, π]d; moreover,

1

(2π)d

∫[−π,π]d

u∗U(θ)u dθ = ‖u‖2, u ∈ CN(n). (5.14)

Proof. Instead of using the traditional linear indexing, in this proof it is convenient to index the components of any vector u ∈ CN(n)

by means of a d-index i = 1, . . . ,n, so that u = [ui]ni=1. Since also the components of Tn(f) are naturally indexed by two d-indices

i, j = 1, . . . ,n (see (5.8)), for any u,v ∈ CN(n) we obtain

u∗Tn(f)v =

n∑i,j=1

fi−juivj =

n∑i,j=1

(1

(2π)d

∫[−π,π]d

f(θ)e−i(i−j)·θdθ

)uivj =

1

(2π)d

∫[−π,π]d

f(θ)

n∑i,j=1

e−i(i−j)·θuivj dθ

=1

(2π)d

∫[−π,π]d

f(θ)u∗U(θ)v dθ,

where U(θ) is defined in the statement of the lemma. This proves (5.12). Eq. (5.14) follows from (5.12) by taking v = u and f = 1(recalling that Tn(1) = IN(n)). Finally,

u∗U(θ)u =

n∑i,j=1

e−i(i−j)·θuiuj =

n∑i=1

uie−ii·θn∑j=1

ujeij·θ =

∣∣∣∣∣∣n∑j=1

ujeij·θ

∣∣∣∣∣∣2

,

which proves (5.13).1Note that two functions f, g ∈ L1([−π, π]d) which coincide a.e. give rise to the same multilevel Toeplitz matrices Tn(f) = Tn(g), n ∈ Nd, because the Fourier

coefficients of f and g coincide.

51

If f : D ⊆ Rk → C is any measurable function which is real a.e., the essential infimum/supremum of f are defined as theinfimum/supremum of the essential range of f . Note that this makes sense, because f(x) ∈ R a.e. and, consequently, ER(f) ⊆ R. Tosimplify the notation, we will use mf and Mf to denote, respectively, the essential infimum and supremum of f . In formulas,

mf = ess infθ∈[−π,π]d

f(θ) = inf(ER(f)

),

Mf = ess supθ∈[−π,π]d

f(θ) = sup(ER(f)

).

An equivalent definition of mf and Mf is the following:

mf = infα ∈ R : mdf > α > 0

,

Mf = supβ ∈ R : mdf < β > 0

.

Theorem 5.1. Assume that f ∈ L1([−π, π]d) is real a.e. Then Λ(Tn(f)) ⊆ [mf ,Mf ] for all n ∈ Nd. If moreover mf < Mf , thenΛ(Tn(f)) ⊂ (mf ,Mf ) for all n ∈ Nd.

We note that the case mf = Mf is trivial, because we have f = mf a.e. and, consequently, Tn(f) = mf IN(n). Whenever f is notconstant a.e., mf < Mf and the spectrum of the multilevel Toeplitz matrices Tn(f) is contained in the open interval (mf ,Mf ).

Proof. By Lemma 5.2, for every u ∈ CN(n) such that ‖u‖ = 1 we have

u∗Tn(f)u =1

(2π)d

∫[−π,π]d

f(θ)u∗U(θ)u dθ,1

(2π)d

∫[−π,π]d

u∗U(θ)u dθ = 1.

Since mf ≤ f(θ) ≤Mf for a.e. θ ∈ [−π, π]d, we obtain

mf ≤ u∗Tn(f)u ≤Mf ,

and the inclusion Λ(Tn(f)) ⊆ [mf ,Mf ] follows from the minimax principle for eigenvalues (Theorem 2.6).Suppose now that mf < Mf . In this case, we show that, for all u ∈ CN(n) satisfying ‖u‖ = 1,

mf < u∗Tn(f)u < Mf . (5.15)

Once this is proved, the inclusion Λ(Tn(f)) ⊂ (mf ,Mf ) follows again from the minimax principle for eigenvalues. We only prove theleft inequality in (5.15), because the proof of the right inequality is completely analogous. By contradiction, suppose that there existsu ∈ CN(n) such that ‖u‖ = 1 and u∗Tn(f)u = mf . By Lemma 5.2,

0 = u∗Tn(f)u−mf =1

(2π)d

∫[−π,π]d

(f(θ)−mf ) u∗U(θ)u dθ =1

(2π)d

∫[−π,π]d

(f(θ)−mf )

∣∣∣∣∣∣n∑j=1

ujeij·θ

∣∣∣∣∣∣ dθ. (5.16)

Considering that the integrand in (5.16) is nonnegative a.e. (by definition of mf ) and that∣∣∑n

j=1 ujeij·θ∣∣ > 0 a.e. (by Lemma 2.3),

it follows from (5.16) and from [81, Theorem 1.39, p. 30] that f(θ) − mf = 0 a.e. This is a contradiction to the assumption thatmf < Mf .

Two important corollaries of Theorem 5.1 are given below. The first follows from Theorem 5.1 and from the observation that everynonnegative function f which does not vanish a.e. satisfies mf < Mf . The second follows from Theorem 5.1 and the linearity of theToeplitz map; see (5.9).

Corollary 5.1. Assume that f ∈ L1([−π, π]d) is nonnegative and not a.e. equal to 0. Then Tn(f) > O for all n ∈ Nd.

Corollary 5.2. For any fixed n ∈ Nd, Tn(·) : L1([−π, π]d) → CN(n)×N(n) is a linear positive operator (LPO), i.e., it is linear andsatisfies Tn(f) ≥ O whenever f ≥ 0 a.e. In particular, Tn(·) is monotone, i.e.,

f ≥ g a.e. ⇒ Tn(f) ≥ Tn(g). (5.17)

The last result of this section provides an important relation between tensor products and multilevel Toeplitz matrices.

Lemma 5.3. Let f1, . . . , fd ∈ L1([−π, π]) and n ∈ Nd. Then,

Tn1(f1)⊗ · · · ⊗ Tnd(fd) = Tn(f1 ⊗ · · · ⊗ fd). (5.18)

Note that the tensor-product function f1 ⊗ · · · ⊗ fd : [−π, π]d → C belongs to L1([−π, π]d) by Fubini’s theorem.

52

Proof. The proof is simple if we use the fundamental property (2.44). The Fourier coefficients of f1 ⊗ · · · ⊗ fd are given by

(f1 ⊗ · · · ⊗ fd)k = (f1)k1 · · · (fd)kd , k ∈ Zd.

Hence, for all i, j = 1, . . . ,n,

[Tn1(f1)⊗ · · · ⊗ Tnd(fd)]ij = [Tn1

(f1)]i1j1 · · · [Tnd(fd)]idjd = (f1)i1−j1 · · · (fd)id−jd = (f1 ⊗ · · · ⊗ fd)i−j= [Tn(f1 ⊗ · · · ⊗ fd)]ij ,

and (5.18) follows.

5.3 Schatten p-norms of multilevel Toeplitz matricesImportant inequalities involving multilevel Toeplitz matrices and Schatten p-norms are provided in Theorem 5.2. They originally appearedin [95, Corollary 4.2] and were generalized in [88, Corollary 3.5]. To prove Theorem 5.2, we need a couple of intermediate lemmas,which are interesting also in themselves. They combine results from [95, Theorems 3.2 and 3.3], which hold in the more general contextof ‘LPOs and unitarily invariant norms’. We recall that the Toeplitz map is an LPO (Corollary 5.2) and that the Schatten p-norms areunitarily invariant.

Lemma 5.4. For every f ∈ L1([−π, π]d) and every n ∈ Nd,

|u∗Tn(f)v| ≤√u∗Tn(|f |)u · v∗Tn(|f |)v ≤ 1

2u∗Tn(|f |)u +

1

2v∗Tn(|f |)v, u,v ∈ CN(n). (5.19)

Proof. The proof is based on Lemma 5.2 and on the Cauchy-Schwarz inequality applied first in CN(n) and then in L2([−π, π]d). Usingthese ingredients, we obtain

|u∗Tn(f)v|2 =

∣∣∣∣∣ 1

(2π)d

∫[−π,π]d

f(θ)u∗U(θ)v dθ

∣∣∣∣∣2

≤

(1

(2π)d

∫[−π,π]d

|f(θ)| |u∗U(θ)v| dθ

)2

=

(1

(2π)d

∫[−π,π]d

|f(θ)|∣∣∣(U(θ)

1/2u)∗(

U(θ)1/2

v)∣∣∣ dθ)2

≤

(1

(2π)d

∫[−π,π]d

|f(θ)|√u∗U(θ)u

√v∗U(θ)v dθ

)2

=

(1

(2π)d

∫[−π,π]d

√|f(θ)|u∗U(θ)u

√|f(θ)|v∗U(θ)v dθ

)2

≤ 1

(2π)d

∫[−π,π]d

|f(θ)|u∗U(θ)u dθ1

(2π)d

∫[−π,π]d

|f(θ)|v∗U(θ)v dθ

= u∗Tn(|f |)u · v∗Tn(|f |)v.

This proves the first inequality in (5.19). The second one is just the geometric-mean – arithmetic-mean inequality.

Lemma 5.5. Let f ∈ L1([−π, π]d), n ∈ Nd and 1 ≤ p ≤ ∞. Then

‖Tn(f)‖p ≤ ‖Tn(|f |)‖p. (5.20)

Proof. For any pair of orthonormal bases uiN(n)i=1 , viN(n)

i=1 of CN(n), by Lemma 5.4 we have

∣∣∣[u∗i Tn(f)vi]N(n)i=1

∣∣∣p≤

∣∣∣∣∣[

1

2u∗i Tn(|f |)ui +

1

2v∗i Tn(|f |)vi

]N(n)

i=1

∣∣∣∣∣p

≤ 1

2

∣∣∣[u∗i Tn(|f |)ui]N(n)i=1

∣∣∣p

+1

2

∣∣∣[v∗i Tn(|f |)vi]N(n)i=1

∣∣∣p.

Passing to the supremum over all pairs of orthonormal bases uiN(n)i=1 , viN(n)

i=1 , and taking into account that Tn(|f |) is HPSD (Corol-lary 5.2), by Lemma 2.9 we get (5.20).

53

Theorem 5.2. Let f ∈ Lp([−π, π]d) and n ∈ Nd. Then,

‖Tn(f)‖ = ‖Tn(f)‖∞ ≤ ‖f‖L∞ , (5.21)

‖Tn(f)‖p ≤1

(2π)d/p‖f‖Lp N(n)1/p, 1 ≤ p <∞. (5.22)

In particular, using the natural convention 1/∞ = d/∞ = 0, the inequalities

‖Tn(f)‖p ≤N(n)1/p

(2π)d/p‖f‖Lp ≤ N(n)

1/p‖f‖Lp (5.23)

hold for all p ∈ [1,∞].

Proof. In view of Lemma 5.5, it suffices to prove (5.21)–(5.22) in the case where f ≥ 0. In this case, we know from Corollary 5.2 thatTn(f) is HPSD. In particular, ‖Tn(f)‖ = λmax(Tn(f)) and (5.21) follows directly from Theorem 5.1.

Suppose now that 1 ≤ p <∞. By Lemma 5.2 and Jensen’s inequality [81], for every u ∈ CN(n) such that ‖u‖ = 1 we have

(u∗Tn(f)u)p

=

(1

(2π)d

∫[−π,π]d

f(θ)u∗U(θ)u dθ

)p≤ 1

(2π)d

∫[−π,π]d

f(θ)pu∗U(θ)u dθ = u∗Tn(fp)u.

Hence, by Lemma 2.9,

‖Tn(f)‖pp = supuiN(n)

i=1 orthonormalbasis of CN(n)

∣∣∣[u∗i Tn(f)ui]N(n)i=1

∣∣∣pp≤ supuiN(n)

i=1 orthonormalbasis of CN(n)

∣∣∣[u∗i Tn(fp)ui]N(n)i=1

∣∣∣1

= ‖Tn(fp)‖1.

To conclude, we observe that, since Tn(fp) is HPSD, its singular values coincide with its eigenvalues. Thus,

‖Tn(fp)‖1 = trace(Tn(fp)) =

n∑i=1

(fp)0 = N(n)1

(2π)d

∫[−π,π]d

f(θ)pdθ = N(n)

1

(2π)d‖f‖pLp .

Lemma 5.6 will be used in Section 6.1.1 to study the Locally Toeplitz operator. The result of this lemma is already in [24, Lemma 5.16]for d = 1 and f, g ∈ L∞([−π, π]), and can be found in [36, Proposition 2] for an arbitrary d ≥ 1 and f, g ∈ L∞([−π, π]d). By Hölder’sinequality [81], if f ∈ Lp([−π, π]d), g ∈ Lq([−π, π]d), and p, q ∈ [1,∞] are conjugate exponents, then fg ∈ L1([−π, π]d). In this case,we can consider the three matrices Tn(f), Tn(g), Tn(fg).

Lemma 5.6. Let f ∈ Lp([−π, π]d) and g ∈ Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugate exponents. Then,

limn→∞

‖Tn(f)Tn(g)− Tn(fg)‖1N(n)

= 0. (5.24)

Proof. If f, g are in L∞([−π, π]d), eq. (5.24) holds by [36, Proposition 2]. In the general case where f ∈ Lp([−π, π]d) and g ∈Lq([−π, π]d), the proof requires some work.

Take two sequences fmm and gmm such that fm, gm ∈ L∞([−π, π]d) for all m, fm → f in Lp([−π, π]d) and gm → g inLq([−π, π]d); for example, one can choose fm = f χ|f |≤m and gm = g χ|g|≤m. By the linearity of Tn(·) and the inequalities(2.30), (5.23), for every m and every n ∈ Nd we have

‖Tn(f)Tn(g)− Tn(fg)‖1≤ ‖Tn(f − fm)Tn(g)‖1 + ‖Tn(fm)Tn(g − gm)‖1 + ‖Tn(fm)Tn(gm)− Tn(fmgm)‖1 + ‖Tn(fmgm − fg)‖1≤ N(n)

1/p‖f − fm‖LpN(n)1/q‖g‖Lq +N(n)

1/p‖fm‖LpN(n)1/q‖g − gm‖Lq

+ ‖Tn(fm)Tn(gm)− Tn(fmgm)‖1 +N(n)‖fmgm − fg‖L1

≤ N(n)

[‖f − fm‖Lp‖g‖Lq + sup

i‖fi‖Lp‖g − gm‖Lq +

‖Tn(fm)Tn(gm)− Tn(fmgm)‖1N(n)

+ ‖fmgm − fg‖L1

]. (5.25)

Note that supi ‖fi‖Lp < ∞, because fi → f in Lp([−π, π]d) and ‖fi‖Lp → ‖f‖Lp . Since fm, gm ∈ L∞([−π, π]d), by [36, Proposi-tion 2] we have

limn→∞

‖Tn(fm)Tn(gm)− Tn(fmgm)‖1N(n)

= 0.

54

Dividing (5.25) by N(n) and passing to the limit as n→∞, we obtain

lim supn→∞

‖Tn(f)Tn(g)− Tn(fg)‖1N(n)

≤ ‖f − fm‖Lp‖g‖Lq + supi‖fi‖Lp‖g − gm‖Lq + ‖fmgm − fg‖L1 . (5.26)

This relation holds for every m. When m → ∞, fm → f in Lp and gm → g in Lq by construction. Moreover, fmgm → fg inL1([−π, π]d) by Hölder’s inequality; indeed,

‖fg − fmgm‖L1 ≤ ‖(f − fm)g‖L1 + ‖fm(g − gm)‖L1 ≤ ‖f − fm‖Lp‖g‖Lq + ‖fm‖Lp‖g − gm‖Lq≤ ‖f − fm‖Lp‖g‖Lq + sup

i‖fi‖Lp‖g − gm‖Lq .

Passing to the limit as m→∞ in (5.26), we get the thesis.

5.4 Multilevel circulant matricesGiven n ∈ Nd, a matrix of the form [

a(i−j) modn]ni,j=1

∈ CN(n)×N(n), (5.27)

is called a multilevel circulant matrix, (or, more precisely, a d-level circulant matrix). Since the (i, j)-th entry a(i−j) modn depends onlyon the difference i− j, it is clear that any multilevel circulant matrix is in particular a multilevel Toeplitz matrix. In the case d = 1, thematrix (5.27) becomes

[a(i−j) modn

]ni,j=1

=

a0 an−1 an−2 · · · · · · a1

a1. . . . . . . . .

...

a2. . . . . . . . . . . .

......

. . . . . . . . . . . . an−2

.... . . . . . . . . an−1

an−1 · · · · · · a2 a1 a0

, (5.28)

which is the expression of the generic 1-level circulant matrix. 1-level circulant matrices are often referred to as circulant matrices,without further specifications. In the case d = 2, the matrix (5.27) can be written as[

a(i−j) modn]ni,j=1

=[[a(i1−j1) modn1,(i2−j2) modn2

]n2

i2,j2=1

]n1

i1,j1=1=[a(i1−j1) modn1

]n1

i1,j1=1

=

a0 an1−1 an1−2 · · · · · · a1

a1. . . . . . . . .

...

a2. . . . . . . . . . . .

......

. . . . . . . . . . . . an1−2

.... . . . . . . . . an1−1

an1−1 · · · · · · a2 a1 a0

, (5.29)

where

ak =[ak,(i2−j2) modn2

]n2

i2,j2=1=

ak,0 ak,n2−1 ak,n2−2 · · · · · · ak,1

ak,1. . . . . . . . .

...

ak,2. . . . . . . . . . . .

......

. . . . . . . . . . . . ak,n2−2

.... . . . . . . . . ak,n2−1

ak,n2−1 · · · · · · ak,2 ak,1 ak,0

, k = 0, . . . , n1 − 1. (5.30)

A matrix of this form, which we call a 2-level circulant matrix, is also known as a block circulant matrix with circulant blocks, or BCCBmatrix. Similarly, we could say that a d-level circulant matrix is a block circulant matrix with (d− 1)-level circulant blocks.

55

The main theorem of this section is Theorem 5.3, which provides the complete spectral decomposition of multilevel circulant matrices.For the proof of this theorem, we need Lemma 5.7, which is the version for multilevel circulant matrices of Lemma 5.1. If n ∈ N, wedenote by Cn the n× n matrix whose (i, j)-th entry equals 1 if (i− j) modn = 1 and 0 otherwise:

Cn =

0 1

1. . .. . . . . .

. . . . . .. . . . . .

1 0

. (5.31)

The matrix Cn is called the generator of circulant matrices of order n. This name is due to the fact that the powers of Cn are

C2n =

0 1 0

0. . . 1

1. . . . . .. . . . . . . . .

. . . . . . . . .1 0 0

, C3

n =

0 1 0 0

0. . . 1 0

0. . . . . . 1

1. . . . . . . . .. . . . . . . . . . . .

1 0 0 0

, . . . , Cnn = In, (5.32)

and so the generic circulant matrix (5.28) can be written as a linear combination of nonnegative powers of Cn:

[a(i−j) modn]ni,j=1 =

n−1∑k=0

akCkn. (5.33)

In particular, any finite linear combination of powers of Cn (i.e., any matrix of the form∑rk=−r ckC

kn) is a circulant matrix, because it

can be written in the form (5.33) by using the identity C−1n = Cn−1

n . Lemma 5.7 generalizes (5.33) to multilevel circulant matrices. Forn ∈ Nd and k ∈ Zd, we set

Ckn = Ck1n1⊗ Ck2n2

⊗ · · · ⊗ Ckdnd . (5.34)

Lemma 5.7. The d-level circulant matrix (5.27) admits the following expression:

[a(i−j) modn

]ni,j=1

=

n−1∑k=0

akCkn, (5.35)

where Ckn is defined in (5.34).

Proof. The proof follows the same pattern as the proof of Lemma 5.1. Eq. (5.35) is proved componentwise, by showing that the (i, j)-thentry of the matrix in the right-hand side is equal to a(i−j) modn. Let δr = 1 if r = 0 and δr = 0 otherwise. By (5.31)–(5.32), we canexpress the (i, j)-th entry of Ckn as follows:

(Ckn)ij = δ(i−j−k) modn, i, j = 1, . . . , n, n ∈ N, k ∈ Z. (5.36)

By the fundamental property (2.44), for all i, j = 1, . . . ,n we have

(Ckn)ij = (Ck1n1)i1j1(Ck2n2

)i2j2 · · · (Ckdnd)idjd = δ(i1−j1−k1) modn1δ(i2−j2−k2) modn2

· · · δ(id−jd−kd) modnd = δ(i−j−k) modn,

where δr = 1 if r = 0 and δr = 0 otherwise. Therefore,(n−1∑k=0

akCkn

)ij

=n−1∑k=0

ak(Ckn)ij =

n−1∑k=0

akδ(i−j−k) modn = a(i−j) modn,


56

As a consequence of Lemma 5.7, any finite linear combination of the form∑rk=−r ckC

kn is a multilevel circulant matrix, because it

can be written in the form (5.35) by using the identity C−1n = Cn−1

n (see (5.32)). Theorem 5.3 gives the spectral decomposition of anylinear combination

∑rk=−r ckC

kn and hence, by Lemma 5.7, of any multilevel circulant matrix. For n ∈ Nd, we denote by

Fn = Fn1 ⊗ · · · ⊗ Fnd (5.37)

the unitary d-level Fourier transform, where

Fn =1√n

(e−2πijk/n)n−1

j,k=0 =1√n

(e−2πi(j−1)(k−1)/n)n

j,k=1 (5.38)

is the classical unitary (1-level) Fourier transform of order n (F ∗nFn = In).

Theorem 5.3. Let r ∈ Nd and ck ∈ C for k = −r, . . . , r; then,r∑

k=−r

ckCkn = Fn diag

j=0,...,n−1

[c(2πj

n

)]F ∗n, (5.39)

where c(θ) =∑rk=−r ckeik·θ and Fn is defined in (5.37). In particular,

∑rk=−r ckC

kn is a normal matrix whose spectrum is given by

Λ

(r∑

k=−r

ckCkn

)=

c(2πj

n

): j = 0, . . . ,n− 1

.

Proof. The spectral decomposition of Cn is known and is given by

Cn = FnDnF∗n , Dn = diag

j=0,...,n−1(e2πij/n) = diag

j=1,...,n(e2πi(j−1)/n).

This can be verified by direct computation: for all i, j = 1, . . . , n, from (5.36) and (5.38) we have

(F ∗nCn)ij =1√n

n∑`=1

e2πi(i−1)(`−1)/nδ(`−j−1) modn =1√n

e2πi(i−1)j/n = (DnF∗n)ij .

Therefore, also the spectral decomposition of the matrix Ckn in (5.34) is known. Indeed, using the properties of tensor products inSection 2.6.1, we obtain

Ckn = Ck1n1⊗ Ck2n2

⊗ · · · ⊗ Ckdnd = (Fn1Dk1n1F ∗n1

)⊗ (Fn2Dk2n2F ∗n2

)⊗ · · · ⊗ (FndDkdndF ∗nd)

= (Fn1⊗ Fn2

⊗ · · · ⊗ Fnd)(Dk1n1⊗Dk2

n2⊗ · · · ⊗Dkd

nd)(Fn1

⊗ Fn2⊗ · · · ⊗ Fnd)∗ = FnD

knF∗n, (5.40)

whereDkn = Dk1

n1⊗Dk2

n2⊗ · · · ⊗Dkd

nd= diagj=0,...,n−1

(e2πi∑dr=1 jrkr/nr ) = diag

j=0,...,n−1(e2πi(j/n)·k). (5.41)

By (5.40)–(5.41), we getr∑

k=−r

ckCkn =

r∑k=−r

ckFnDknF∗n = Fn

(r∑

k=−r

ckDkn

)F ∗n = Fn

(r∑

k=−r

ck diagj=0,...,n−1

(e2πi(j/n)·k)

)F ∗n

= Fn diagj=0,...,n−1

(r∑

k=−r

cke2πi(j/n)·k

)F ∗n,

and the thesis follows from the observation that∑rk=−r cke2πi(j/n)·k = c(2πj/n).

5.5 Spectral and singular value distribution of multilevel Toeplitz matrices: an a.c.s.-based proof

Theorem 5.4 is the multilevel version of Szego’s first limit theorem and of the Avram–Parter theorem in the form proved by Tilli [100].This theorem is a fundamental result on multilevel Toeplitz matrices. For the eigenvalues it goes back to Szego [59], and for the singularvalues it was established by Avram [4] and Parter [76]. They assumed that d = 1 and f ∈ L∞([−π, π]); see [24, Sections 5 and6] and also [25, Section 10.14] for more on the subject in the case of L∞ generating functions. The extension to any d ≥ 1 andf ∈ L1([−π, π]d) was performed by Tyrtyshnikov and Zamarashkin [103, 104, 108] and Tilli [100]. For the eigenvalues in the case ofd ≥ 1 and f ∈ L∞([−π, π]d), Theorem 5.4 can also be derived from [23, Corollary 22]. In this section, we are going to see a proofof Theorem 5.4 based on the notion of a.c.s. Our proof is essentially the same as the one appeared in [56], with the only difference thatpaper [56] focused on the 1-level case for simplicity, while here we will address the general d-level setting.

57

Theorem 5.4. If f ∈ L1([−π, π]d) and Tn(f)n is any matrix-sequence extracted from Tn(f)n∈Nd , then Tn(f)n ∼σ f . Ifmoreover f is real a.e., then Tn(f)n ∼λ f .

In Theorem 5.4, it is understood that the multi-index n parameterizing the sequence Tn(f)n depends on n and, moreover, n→∞when n→∞. We omitted this specification only because it is satisfied, by definition, for every matrix-sequence Ann; see Section 2.1.2.

Proof. The proof consists of two steps. In the first step, we show that the theorem holds if f is a d-variate trigonometric polynomial. Inthe second step, using an approximation argument, we show that the theorem holds for every f ∈ L1([−π, π]d).

1. Suppose that f is a d-variate trigonometric polynomial, so that f(θ) =∑rk=−r fkeik·θ for some r ∈ Nd. Consider the multilevel

circulant matrix

Cn(f) =

r∑k=−r

fkCkn, (5.42)

where Ckn = Ck1n1⊗ Ck2n2

⊗ · · · ⊗ Ckdnd (as in (5.34)) and Cn is defined in (5.31). Note that Cn(f) is Hermitian whenever f is real,by Theorem 5.3. We are going to show that Cn(f)nm is an a.c.s. of Tn(f)n and Cn(f)n ∼σ, λ f . Once this is proved, byCorollary 4.1 we immediately get Tn(f)n ∼σ f ; moreover, if f is real, by Corollary 4.2 and the Hermitianess of Tn(f) and Cn(f),we obtain Tn(f)n ∼λ f .

To prove that Cn(f)nm is an a.c.s. of Tn(f)n, it is enough to show that, for n ≥ r + 1,

rank(Tn(f)− Cn(f)) ≤ N(2r + 1)N(n)

d∑i=1

rini. (5.43)

By definition of Tn(f) (see (5.8)) and by Lemma 5.1, for n ≥ r + 1 we can write

Tn(f) =

n−1∑k=−(n−1)

fkJ(k)n =

r∑k=−r

fkJ(k)n , (5.44)

where J (k)n = J

(k1)n1 ⊗ J (k2)

n2 ⊗ · · · ⊗ J (kd)nd (as in (5.5)) and J (k)

n is the n × n matrix whose (i, j)-th entry equals 1 if i − j = k and 0otherwise. Since the nonzero rows of Ckn − J

(k)n are at most |k|, we have

rank(Ckn − J (k)n ) ≤ |k|, k ∈ Z, n ∈ N.

Hence, by (2.48),

rank(Ckn − J (k)n ) ≤ N(n)

d∑i=1

|ki|ni, k ∈ Zd, n ∈ Nd.

Comparing (5.42) and (5.44), we see that, for n ≥ r + 1,

rank(Cn(f)− Tn(f)) ≤r∑

k=−r

rank(Ckn − J (k)n ) ≤

r∑k=−r

N(n)

d∑i=1

|ki|ni≤ N(2r + 1)N(n)

d∑i=1

rini,

and (5.43) is proved. It follows that Cn(f)nm is an a.c.s. of Tn(f)n.To prove that Cn(f)n ∼σ, λ f , we use Theorem 5.3. Since Cn(f) is normal, it is enough to show that Cn(f)n ∼λ f (see

Remark 2.4). By Theorem 5.3, the eigenvalues of Cn(f) are given by f(2πj/n), j = 0, . . . ,n− 1. Hence, for every F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑j=1

F (λj(Cn(f))) = limn→∞

1

N(n)

n−1∑j=0

F(f(2πj

n

))=

1

(2π)d

∫[0,2π]d

F (f(θ))dθ =1

(2π)d

∫[−π,π]d

F (f(θ))dθ.

(5.45)In (5.45), the last equality holds because f is a d-variate trigonometric polynomial (so, it is periodic in each direction with period 2π); andthe second equality is due to the fact that (2π)d

N(n)

∑n−1j=0 F (f( 2πj

n )) is a Riemann sum for∫

[0,2π]dF (f(θ))dθ and converges to this integral

when n→∞, because the function F (f(θ)) is continuous (and hence Riemann-integrable) over [0, 2π]d. Thus, Cn(f)n ∼λ f .

2. Let f be a function in L1([−π, π]d). Since the set of d-variate trigonometric polynomials is dense in L1([−π, π]d), there exists asequence of d-variate trigonometric polynomials fm such that fm → f in L1([−π, π]d). By replacing fm with <(fm) (if necessary),we may assume that, if f is real a.e., then each fm is real. In this way, when f is real a.e., all the matrices Tn(f) and Tn(fm) areHermitian. By the first part of the proof, Tn(fm)n ∼σ fm and Tn(fm)n ∼λ fm whenever fm is real. Moreover, Tn(fm)nm isan a.c.s. of Tn(f)n by Corollary 4.3, because, by Theorem 5.2,

‖Tn(f)− Tn(fm)‖1 = ‖Tn(f − fm)‖1 ≤ N(n)‖f − fm‖L1 .

Thus, the distribution relations Tn(f)n ∼σ, λ f follow from Corollaries 4.1–4.2.

58

5.6 Extreme eigenvalues of Hermitian multilevel Toeplitz matricesSuppose that f ∈ L1([−π, π]d) is real a.e. In this case, the matrix-sequence Tn(f)n is formed by Hermitian matrices whose spectrumis contained in [mf ,Mf ] by Theorem 5.1. We label the eigenvalues of Tn(f) in non-increasing order:

λ1(Tn(f)) ≥ . . . ≥ λN(n)(Tn(f)).

By Theorems 2.8 and 5.4, each point of the essential range ER(f) ⊆ [mf ,Mf ] strongly attracts the spectrum Λ(Tn(f)) with infiniteorder. If mf ,Mf are finite (this happens when f ∈ L∞([−π, π]d)), then, by definition, they belong to the essential range ER(f). As aconsequence, for each fixed j ≥ 1 we have

limn→∞

λj(Tn(f)) = Mf , limn→∞

λN(n)−j+1(Tn(f)) = mf . (5.46)

The relations (5.46) are actually satisfied even if mf and/or Mf are infinite. Indeed, suppose by contradiction that there exists a fixedj ≥ 1 such that

lim infn→∞

λj(Tn(f)) = ` < Mf .

Passing, if necessary, to a subsequence of Tn(f)n, we can assume that

limn→∞

λj(Tn(f)) = ` < Mf .

This means that all the eigenvalues of Tn(f) (except at most j − 1) are eventually smaller than ` + ε for some positive ε such that`+ ε < Mf . By definition of Mf , we can find a point x ∈ ER(f) which is closer to Mf than `+ ε. Clearly, the point x cannot stronglyattract Λ(Tn(f)) with infinite order. This contradiction proves the left limit relation in (5.46); the right limit relation is proved in thesame way. We collect in the next proposition the result we have just proved.

Proposition 5.1. Let f ∈ L1([−π, π]d) be real a.e. Then, for each fixed j ≥ 1,

limn→∞

λj(Tn(f)) = Mf , limn→∞

λN(n)−j+1(Tn(f)) = mf .

For classical (1-level) Toeplitz matrices Tn(f), the convergence rate of the extreme eigenvalues λmax(Tn(f)) and λmin(Tn(f)) totheir respective limits Mf and mf was analyzed in [18, 42, 77, 85, 86, 105]. Actually, the analysis reduces to studying the convergenceof λmin(Tn(f)) to 0 under the assumption that mf = 0. Indeed, the linearity of the Toeplitz map and the identity Tn(1) = In implythat λmax(Tn(f)) = −λmin(Tn(−f)) and λmin(Tn(f)) − mf = λmin(Tn(f − mf )). Assuming mf = 0, if f ∈ L∞([−π, π]) thenλmax(Tn(f)) is bounded by Mf for all n (Theorem 5.1) and converges to Mf as n→∞ (Proposition 5.1); consequently, the asymptoticgrowth of the spectral condition number

κ(Tn(f)) = ‖Tn(f)‖ ‖Tn(f)−1‖ =λmax(Tn(f))

λmin(Tn(f))

is dictated by the asymptotics of λmin(Tn(f)), and vice versa. In other words, the asymptotic analysis of λmin(Tn(f)) is equivalent to theasymptotic analysis of κ(Tn(f)) whenever f ∈ L∞([−π, π]) and mf = 0.

The theoretical result obtained in the aforementioned papers concerning the asymptotics of λmin(Tn(f)) can be roughly summarizedas follows:2

the convergence rate of λmin(Tn(f)) to mf is determined by the maximum order of the essential zeros of f −mf .

Of course, this assertion is rather vague and should be made more precise; in particular, one should define the ‘order’ for the essentialzeros of f −mf . An attempt to achieve such a definition in the case of a generic function f ∈ L∞([−π, π]) was made in [18, 42], whereestimates of λmin(Tn(f)) and κ(Tn(f)) were provided, showing the dependence on the orders of the essential zeros of f . However, themathematical carification of the above statement goes beyond the scope of this book. In the following discussion, we give neverthelessan idea of its meaning by considering specific examples and by providing precise pointers to the literature. Of course, this part is notnecessary to understand the remainder of this book and can be omitted on first reading.

Consider the function |θ|α, α > 0. This is the prototype of a function with a (unique) zero of order α. In 1961, on the basis of theresults obtained by Widom in [105], Parter proved a theorem [77, p. 264] that implies as a particular case the existence of a constantK(α) > 0 such that

λmin(Tn(|θ|α))n→∞∼ K(α)

1

nα, (5.47)

2If g : D ⊆ Rk → C is a measurable function, we say that x is an essential zero of g if ess inf(g|B(x,ε)) = 0 for every ε > 0, where g|B(x,ε) is the restriction of g tothe set B(x, ε) = y ∈ D : ‖y − x‖ < ε.

59

i.e., limn→∞ nαTn(|θ|α) = K(α). From (5.47), it is clear how the order of the zero of |θ|α determines the asymptotics of λmin(Tn(|θ|α)).We refer the reader to [42, Theorem 2.1] for two concrete positive constants A(α) and B(α) such that

A(α)1

nα≤ λmin(Tn(|θ|α)) ≤ B(α)

1

nα

for all n.3 If f(θ) is another (continuous) nonnegative function with a unique zero of order α at θ = 0, then we can find constants c, C > 0such that c|θ|α ≤ f(θ) ≤ C|θ|α over [−π, π]. By the monotonicity of the Toeplitz map (Corollary 5.2) we obtain

c Tn(|θ|α) ≤ Tn(f) ≤ C Tn(|θ|α),

and by the minimax principle for eigenvalues (Theorem 2.6) we get

cλmin(Tn(|θ|α)) ≤ λmin(Tn(f)) ≤ Cλmin(Tn(|θ|α)).

This means that also the minimum eigenvalue of Tn(f) converges to 0 like 1/nα. We note that the location of the zero is not an issue.Indeed, if we set fα(θ) = |θ|α : [−π, π]→ R and we denote by f#

α the 2π-periodic extension of fα to R, then it is not difficult to showthat

Tn(f#α (θ − ζ)) = DnTn(fα(θ))D−1

n = DnTn(|θ|α)D∗n = DnTn(|θ|α)D−1n , Dn = diag

i=1,...,ne−iiζ , ζ ∈ R. (5.48)

It follows that λmin(Tn(f#α (θ − ζ))) = λmin(Tn(|θ|α)) for any ζ ∈ R and any α > 0. If f(θ) is a (continuous) nonnegative function

with a zero of exponential order, like, e.g., the function e−1/|θ|, it is not difficult to guess from the above results that λmin(Tn(f)) goesto 0 exponentially. This is in fact the case [18, 42, 86]. However, a surprising result obtained in [86, Section 3.1] shows that, for everynon-pathological nonnegative function f ∈ L1([−π, π]) (i.e., for every nonnegative f ∈ L1([−π, π]) such that, for some positive δ, theset f ≥ δ contains a non-trivial interval), the minimum eigenvalue λmin(Tn(f)) cannot go to 0 faster than sn(n−1)/2, where s ∈ (0, 1)

is related to the measure of the set f = 0. For example, λmin(Tn(e−1/|θ|3)) cannot go to 0 like e−n3

. We conclude this discussionby noting that the case where the function f (is even and) possesses more than one zero (with even orders) was addressed in [85]; seealso [20, Theorem 10.1] for an alternative approach. We refer the reader to [20, pp. 256–259] for the history and further references aboutextreme eigenvalues of Hermitian Toeplitz matrices.

3The exact asymptotics of K(2η) in the case where η ∈ N can be found in [26, eq. (9)]. Note that, in view of the equations

1 = limθ→0

(2− 2 cos θ)η

θ2η= limθ→0

d2η

dθ2η(2− 2 cos θ)η

(2η)!=

1

(2η)!

d2η

dθ2η(2− 2 cos θ)η

∣∣∣∣θ=0

,

which are consequences of De l’Hôpital’s theorem, Parter’s theorem [77, p. 264] implies that limn→∞ n2ηλmin(Tn((2−2 cos θ)η))) = K(2η); hence,K(2η) coincideswith the constant cη of [26].

60

Chapter 6

LT and sLT sequences

In this section, we develop the theory of Locally Toeplitz sequences. We first introduce and analyze the Locally Toeplitz operator inSection 6.1. Then, in Section 6.2, we define the Locally Toeplitz and separable Locally Toeplitz sequences, and we study their properties.The results contained in this section are of fundamental importance for the theory of Generalized Locally Toeplitz sequences, which willbe the subject of Chapter 7.

6.1 The Locally Toeplitz operatorDefinition 6.1.

• Let m,n ∈ N, let a : [0, 1]→ C, and let f : [−π, π]→ C in L1([−π, π]). Then, we define the n× n matrix

LTmn (a, f) = Dm(a)⊗ Tbn/mc(f) ⊕ Onmodm = diagj=1,...,m

[a( jm

)Tbn/mc(f)

]⊕ Onmodm

= diagj=1,...,m

a( jm

)Tbn/mc(f) ⊕ Onmodm.

It is understood that LTmn (a, f) = On when n < m and that the term Onmodm is not present when n is a multiple of m. Moreover,here and in the following, the tensor product operation ⊗ is always applied before the direct sum ⊕, exactly as in the case ofnumbers, where multiplication is always applied before addition. Note also that in the last equality we deliberately removed thesquare brackets, so as to illustrate a notation that will be used hereinafter to simplify the presentation (roughly speaking, we areassuming that the ‘diag operator’ is applied before the direct sum ⊕).

• Letm,n ∈ Nd, let a : [0, 1]d → C, and let f1, . . . , fd : [−π, π]→ C in L1([−π, π]). Then, we define the N(n)×N(n) matrix

LTmn (a, f1 ⊗ · · · ⊗ fd) = LTm1,...,mdn1,...,nd

(a(x1, . . . , xd), f1 ⊗ · · · ⊗ fd)

= diagj1=1,...,m1

Tbn1/m1c(f1)⊗ LTm2,...,mdn2,...,nd

(a( j1m1

, x2, . . . , xd

), f2 ⊗ · · · ⊗ fd

)⊕ O(n1 modm1)n2···nd .

This is a recursive definition, whose base case has been considered in the previous item. For example, in the case d = 2 we have

LTm1,m2n1,n2

(a, f1 ⊗ f2) = diagj1=1,...,m1

Tbn1/m1c(f1)⊗

[diag

j2=1,...,m2

a( j1m1

,j2m2

)Tbn2/m2c(f2) ⊕ On2 modm2

]⊕ O(n1 modm1)n2

.

In this section, especially in Subsection 6.1.1, we investigate the properties of the Locally Toeplitz operator LTmn (a, f) that will beof interest later on. We write LTmn (a, f) instead of LTmn (a, f1 ⊗ · · · ⊗ fd) because we are going to see that LTmn (a, f) is well-defined(in a unique way) for any function f ∈ L1([−π, π]d); see Definition 6.2. In particular, if f is separable, the definition is independent ofthe factorization of f as a tensor product of the form f1 ⊗ · · · ⊗ fd, f1, . . . , fd ∈ L1([−π, π]).

The main result about the Locally Toeplitz operator is stated in Theorem 6.1. It shows that LTmn (a, f1 ⊗ · · · ⊗ fd) coincides withDm(a) ⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ O up to a permutation transformation Πmn which only depends on m,n and not on the specificfunctions a, f1, . . . , fd. This result allows us to extend the definition of the Locally Toeplitz operator as in Definition 6.2. With thisextension, we will be able to define in Section 6.2 the notion of Locally Toeplitz sequences in the multilevel setting. The proof ofTheorem 6.1 is rather technical and can be omitted on first reading.

61

Theorem 6.1. For anym,n ∈ Nd there exists a permutation matrix Πmn of size N(n) such that

LTmn (a, f1 ⊗ · · · ⊗ fd) = Πmn

[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n)−N(m)N(bn/mc)

](Πmn )T

for every a : [0, 1]d → C and every f1, . . . , fd ∈ L1([−π, π]).

Proof. The proof is done by induction on d. For d = 1 the result holds with Πmn = In. For d ≥ 2, set ν = (n2, . . . , nd) and

µ = (m2, . . . ,md). By definition,

LTmn (a, f1 ⊗ · · · ⊗ fd) = diagj1=1,...,m1

Tbn1/m1c(f1)⊗ LTµν(a( j1m1

, ·), f2 ⊗ · · · ⊗ fd

)⊕ O(n1 modm1)n2···nd , (6.1)

where a(j1/m1, ·) : [0, 1]d−1 → C is the function (x2, . . . , xd) 7→ a(j1/m1, x2, . . . , xd). By induction hypothesis, setting N(ν,µ) =N(ν)−N(µ)N(bν/µc), we have

LTµν

(a( j1m1

, ·), f2 ⊗ · · · ⊗ fd

)= Πµν

[Dµ

(a( j1m1

, ·))⊗ Tbν/µc(f2 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)

](Πµν )T . (6.2)

Let us work on the argument of the ‘diag operator’ in (6.1). From Lemma 2.11, eq. (6.2) and the properties of tensor products (seeSection 2.6.1), we get


, ·), f2 ⊗ · · · ⊗ fd

)= Π(bn1/m1c,N(ν));[2,1]

LTµν

(a( j1m1

, ·), f2 ⊗ · · · ⊗ fd

)⊗ Tbn1/m1c(f1)

(Π(bn1/m1c,N(ν));[2,1])

T

= Π(bn1/m1c,N(ν));[2,1]

Πµν

[Dµ

(a( j1m1

, ·))⊗ Tbν/µc(f2 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)

](Πµν )T

⊗ Tbn1/m1c(f1)

(Π(bn1/m1c,N(ν));[2,1])

T

= Π(bn1/m1c,N(ν));[2,1](Πµν ⊗ Ibn1/m1c)

[Dµ

(a( j1m1

, ·))⊗ Tbν/µc(f2 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)

]⊗ Tbn1/m1c(f1)

(Πµν ⊗ Ibn1/m1c)

T (Π(bn1/m1c,N(ν));[2,1])T . (6.3)

Using eq. (2.52), Lemma 2.11, Lemma 5.3 and the properties of tensor products and direct sums (see Section 2.6.1), we obtain[Dµ

(a( j1m1

, ·))⊗ Tbν/µc(f2 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)

]⊗ Tbn1/m1c(f1)

= Dµ

(a( j1m1

, ·))⊗ Tbν/µc(f2 ⊗ · · · ⊗ fd)⊗ Tbn1/m1c(f1) ⊕ ON(ν,µ)bn1/m1c

= Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2]

[Dµ

(a( j1m1

, ·))⊗ Tbn1/m1c(f1)⊗ Tbν/µc(f2 ⊗ · · · ⊗ fd)

]· (Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2])

T ⊕ ON(ν,µ)bn1/m1c

= Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2]

[Dµ

(a( j1m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd)

]· (Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2])

T ⊕ ON(ν,µ)bn1/m1c

= (Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2] ⊕ IN(ν,µ)bn1/m1c)

[Dµ

(a( j1m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)bn1/m1c

]· (Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2] ⊕ IN(ν,µ)bn1/m1c)

T . (6.4)

Substituting (6.4) into (6.3), we arrive at


, ·), f2 ⊗ · · · ⊗ fd

)= Pmn

[Dµ

(a( j1m1


](Pmn )T , (6.5)

62

where Pmn = Π(bn1/m1c,N(ν));[2,1](Πµν ⊗ Ibn1/m1c)(Π(N(µ),bn1/m1c,N(bν/µc));[1,3,2]⊕ IN(ν,µ)bn1/m1c). Combining (6.5) and (6.1), we

obtain

LTmn (a, f1 ⊗ · · · ⊗ fd)

=( m1⊕j1=1

Pmn

)diag

j1=1,...,m1

[Dµ

(a( j1m1


]( m1⊕j1=1

Pmn

)T⊕ O(n1 modm1)n2···nd .

From Lemma 2.12,

diagj1=1,...,m1

[Dµ

(a( j1m1


]=

m1⊕j1=1

[Dµ

(a( j1m1


]

= Vmn

m1⊕j1=1

[Dµ

(a( j1m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd)

]⊕ ON(ν,µ)bn1/m1cm1

(Vmn )T

= Vmn[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)bn1/m1cm1

](Vmn )T ,

where

Vmn = Vh(m,n);σ,

σ = [1,m1 + 1, 2,m1 + 2, . . . ,m1, 2m1],

h(m,n) =(N(µ)N(bn/mc), N(ν,µ)bn1/m1c︸︷︷︸

1

, . . . , N(µ)N(bn/mc), N(ν,µ)bn1/m1c︸︷︷︸m1

).

Thus,

LTmn (a, f1 ⊗ · · · ⊗ fd)

=( m1⊕j1=1

Pmn

)Vmn

[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)bn1/m1cm1

](Vmn )T

( m1⊕j1=1

Pmn

)T⊕ O(n1 modm1)n2···nd

=

( m1⊕j1=1

Pmn

)Vmn ⊕ I(n1 modm1)n2···nd

[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(ν,µ)bn1/m1cm1+(n1 modm1)n2···nd]

·

(Vmn )T( m1⊕j1=1

Pmn

)T⊕ I(n1 modm1)n2···nd

.This concludes the proof; note that the permutation matrix Πmn is given by

Πmn =( m1⊕j1=1

Pmn

)Vmn ⊕ I(n1 modm1)n2···nd

and, moreover, N(ν,µ)bn1/m1cm1 + (n1 modm1)n2 · · ·nd = N(n)−N(m)N(bn/mc).

As a consequence of Theorem 6.1, we can extend Definition 6.1 in the following way.

Definition 6.2. Letm,n ∈ Nd, let a : [0, 1]d → C and let f ∈ L1([−π, π]d). Then, we define the Locally Toeplitz operator

LTmn (a, f) = Πmn[Dm(a)⊗ Tbn/mc(f) ⊕ ON(n)−N(m)N(bn/mc)

](Πmn )T ,

where Πmn is the permutation matrix appearing in Theorem 6.1.

Remark 6.1. Note that LTmn (a, f) = LTmn (a, g) whenever f = g a.e. Moreover, suppose that f = f1⊗· · ·⊗fd a.e., with f1, . . . , fd ∈L1([−π, π]); then LTmn (a, f) is equal to LTmn (a, f1 ⊗ · · · ⊗ fd), as defined by Definition 6.1.

63

6.1.1 Properties of the Locally Toeplitz operatorWe now derive a lot of interesting properties of LTmn (a, f) that we shall use in the study of LT, sLT and GLT sequences. We first notethat, for any n,m ∈ Nd and any pair of functions a : [0, 1]d → C and f ∈ L1([−π, π]d),

[LTmn (a, f)]∗ = LTmn (a, f). (6.6)

This follows from Definition 6.2, from the relations (X⊗Y )∗ = X∗⊗Y ∗, (X⊕Y )∗ = X∗⊕Y ∗, and from the equality [Tk(f)]∗ = Tk(f)(see Chapter 5).

Proposition 6.1. Letm,n ∈ Nd, let a : [0, 1]d → C and let f ∈ L1([−π, π]d). Then,

‖LTmn (a, f)‖ = ‖Dm(a)‖ ‖Tbn/mc(f)‖ = maxj=1,...,m

∣∣∣a( jm

)∣∣∣‖Tbn/mc(f)‖, (6.7)

‖LTmn (a, f)‖p = ‖Dm(a)‖p ‖Tbn/mc(f)‖p =

m∑j=1

∣∣∣∣a( jm)∣∣∣∣p1/p

‖Tbn/mc(f)‖p, 1 ≤ p <∞. (6.8)

Proof. Use Definition 6.2, the invariance of ‖ · ‖ and ‖ · ‖p by unitary transformations (such as permutations), and eqs. (2.45)–(2.46).

We denote by C[0,1]d the vector space of all functions a : [0, 1]d → C.

Proposition 6.2. Letm,n ∈ Nd. Then, the map

LTmn (a, ·) : L1([−π, π]d)→ CN(n)×N(n)

is linear for any a : [0, 1]d → C, and the map

LTmn (·, f) : C[0,1]d → CN(n)×N(n)

is linear for any f ∈ L1([−π, π]d).

Proof. Use Definition 6.2, the linearity of the maps Dm(·) and Tbn/mc(·), and the bilinearity of tensor products.

By Hölder’s inequality [81], if f ∈ Lp([−π, π]d) and f ∈ Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugate exponents, then ff ∈L1([−π, π]d). In this case, for any a, a : [0, 1]d → C, we can consider the three matrices LTmn (a, f), LTmn (a, f) and LTmn (aa, f f). InProposition 6.3 we show that LTmn (a, f)LTmn (a, f) is ‘close’ to LTmn (aa, f f).

Proposition 6.3. Let a, a : [0, 1]d → C be bounded, and let f ∈ Lp([−π, π]d) and f ∈ Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugateexponents. Then, for every n,m ∈ Nd,

‖LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)‖1 ≤ ε(bn/mc)N(n), (6.9)

where

ε(k) = ‖aa‖∞‖Tk(f)Tk(f)− Tk(ff)‖1

N(k)

and limk→∞

ε(k) = 0 by Lemma 5.6. In particular, for everym ∈ Nd there exists nm ∈ Nd such that, for n ≥ nm,

‖LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)‖1 ≤N(n)

N(m), (6.10)

LTmn (a, f)LTmn (a, f) = LTmn (aa, f f) +Rn,m +Nn,m, rank(Rn,m) ≤ N(n)√N(m)

, ‖Nn,m‖ ≤1√N(m)

. (6.11)

Proof. By Definition 6.2 and the properties of tensor products and direct sums,

LTmn (a, f)LTmn (a, f)− LTmn (aa, f f) = Πmn

[Dm(aa)⊗

(Tbn/mc(f)Tbn/mc(f)− Tbn/mc(ff)

)⊕ O

](Πmn )T .

Hence,

‖LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)‖1 = ‖Dm(aa)‖1 ‖Tbn/mc(f)Tbn/mc(f)− Tbn/mc(ff)‖1

≤ N(n)‖aa‖∞‖Tbn/mc(f)Tbn/mc(f)− Tbn/mc(ff)‖1

N(bn/mc),

and (6.9) is proved. Since ε(k)→ 0 when k→∞, for everym ∈ Nd there exists nm ∈ Nd such that, for n ≥ nm, (6.10) holds. (6.11)follows directly from (6.10) and Lemma 4.4.

64

Theorems 6.2–6.3 provide information about the asymptotic singular value and eigenvalue distribution of a finite sum of the form∑pi=1 LT

mn (ai, fi). Together with Theorems 4.3, 4.5 and Corollaries 4.1–4.2, they play a central role in the computation of the singular

value and eigenvalue distribution of GLT sequences.

Theorem 6.2. Let a1, . . . , ap : [0, 1]d → C and let f1, . . . , fp ∈ L1([−π, π]d). Then, for everym ∈ Nd and every F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑r=1

F(σr

( p∑i=1

LTmn (ai, fi)))

= φm(F ) =1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(∣∣∣ p∑i=1

ai

( jm

)fi(θ)

∣∣∣)dθ. (6.12)

Moreover, if a1, . . . , ap are Riemann-integrable, then, for every F ∈ Cc(R),

limm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F(∣∣∣ p∑i=1

ai(x)fi(θ)∣∣∣)dxdθ. (6.13)

Proof. By Definition 6.2,

(Πmn )T

(p∑i=1

LTmn (ai, fi)

)Πmn =

(p∑i=1

Dm(ai)⊗ Tbn/mc(fi)

)⊕ ON(n)−N(m)N(bn/mc). (6.14)

Recalling that Dm(ai) = diagj=1,...,m ai(j/m), for every j = 1, . . . ,m the j-th diagonal block of size N(bn/mc) of the matrix(6.14) is given by

p∑i=1

ai

( jm

)Tbn/mc(fi) = Tbn/mc

( p∑i=1

ai

( jm

)fi

).

It follows that the singular values of∑pi=1 LT

mn (ai, fi) are

σk

(Tbn/mc

( p∑i=1

ai

( jm

)fi

)), k = 1, . . . , N(bn/mc), j = 1, . . . ,m,

plus further N(n)−N(m)N(bn/mc) singular values equal to 0. Therefore, by Theorem 5.4, for any F ∈ Cc(R) we have

limn→∞

1

N(n)

N(n)∑r=1

F(σr

( p∑i=1

LTmn (ai, fi)))

= limn→∞

N(m)N(bn/mc)N(n)

1

N(m)

m∑j=1

1

N(bn/mc)

N(bn/mc)∑k=1

F(σk

(Tbn/mc

( p∑i=1

ai

( jm

)fi

)))=

1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(∣∣∣ p∑i=1

ai

( jm

)fi(θ)

∣∣∣)dθ. (6.15)

This proves (6.12).If a1, . . . , ap are Riemann-integrable, then the function x 7→ F (|

∑pi=1 ai (x) fi(θ)|) is Riemann-integrable for each fixed θ ∈

[−π, π]d, being the composition of a continuous function with a Riemann-integrable function. Hence,

limm→∞

1

N(m)

m∑j=1

F(∣∣∣ p∑i=1

ai

( jm

)fi(θ)

∣∣∣) =

∫[0,1]d

F(∣∣∣ p∑i=1

ai(x)fi(θ)∣∣∣)dx.

Passing to the limit asm→∞ in (6.15), and using the dominated convergence theorem, we get (6.13).

Theorem 6.3. Let a1, . . . , ap : [0, 1]d → C and let f1, . . . , fp ∈ L1([−π, π]d). Then, for everym ∈ Nd and every F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑r=1

F(λr

(<( p∑i=1

LTmn (ai, fi))))

= φm(F ) =1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(<( p∑i=1

ai

( jm

)fi(θ)

))dθ. (6.16)

Moreover, if a1, . . . , ap are Riemann-integrable, then, for every F ∈ Cc(C),

limm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F(<( p∑i=1

ai(x)fi(θ)))dxdθ. (6.17)

65

Proof. The proof follows the same pattern as the proof of Theorem 6.2. By (6.6) and Definition 6.2,

(Πmn )T

(<( p∑i=1

LTmn (ai, fi)))

Πmn = (Πmn )T

(1

2

(p∑i=1

LTmn (ai, fi) +

p∑i=1

LTmn (ai, fi)

))Πmn

=1

2

(p∑i=1

Dm(ai)⊗ Tbn/mc(fi) +

p∑i=1

Dm(ai)⊗ Tbn/mc(fi)

)⊕ ON(n)−N(m)N(bn/mc).

The j-th block of this matrix, 1 ≤ j ≤m, is given by

1

2

(p∑i=1

ai

( jm

)Tbn/mc(fi) +

p∑i=1

ai

( jm

)Tbn/mc(fi)

)= Tbn/mc

(<( p∑i=1

ai

( jm

)fi

)).

It follows that the eigenvalues of < (∑pi=1 LT

mn (ai, fi)) are

λk

(Tbn/mc

(<( p∑i=1

ai

( jm

)fi

))), k = 1, . . . , N(bn/mc), j = 1, . . . ,m,

plus further N(n)−N(m)N(bn/mc) eigenvalues equal to 0. Therefore, by Theorem 5.4, for any F ∈ Cc(C) we have

limn→∞

1

N(n)

N(n)∑r=1

F(λr

(<( p∑i=1

LTmn (ai, fi))))

= limn→∞

N(m)N(bn/mc)N(n)

1

N(m)

m∑j=1

1

N(bn/mc)

N(bn/mc)∑k=1

F(λk

(Tbn/mc

(<( p∑i=1

ai

( jm

)fi

))))=

1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(<( p∑i=1

ai

( jm

)fi(θ)

))dθ. (6.18)

This proves (6.16).If a1, . . . , ap are Riemann-integrable, then the function x 7→ F (< (

∑pi=1 ai (x) fi(θ))) is Riemann-integrable for each fixed θ ∈

[−π, π]d, and so

limm→∞

1

N(m)

m∑j=1

F(<( p∑i=1

ai

( jm

)fi(θ)

))=

∫[0,1]d

F(<( p∑i=1

ai(x)fi(θ)))dx.

Passing to the limit asm→∞ in (6.18), and using the dominated convergence theorem, we get (6.17).

6.2 Definition of LT and sLT sequencesWe recall that, unless otherwise specified, the multi-index that parameterizes a matrix-sequence Ann is always assumed to be a d-index, n = (n1, . . . , nd). This convention, which was adopted from the beginning of this work (see Section 2.1.2), must be kept in mindespecially in the present section and in Chapter 7. We will not repeat this anymore (although, sometimes, we will add the specificationthat n ∈ Nd, as a reminder for the reader).

Definition 6.3 (LT sequence). Let Ann be a matrix-sequence, n ∈ Nd, and let a : [0, 1]d → C be a Riemann-integrable functionand f ∈ L1([−π, π]d). We say that Ann is a Locally Toeplitz (LT) sequence with symbol a ⊗ f , and we write Ann ∼LT a ⊗ f , ifLTmn (a, f)nm∈Nd is an a.c.s. for Ann. This means that, for allm ∈ Nd there is nm such that, for n ≥ nm,

An = LTmn (a, f) +Rn,m +Nn,m,


where the quantities nm, c(m), ω(m) are independent of n, and limm→∞ c(m) = limm→∞ ω(m) = 0. The functions a and f are,respectively, the weight function and the generating function of Ann.1

Definition 6.4 (sLT sequence). Let Ann be a matrix-sequence, n ∈ Nd. We say that Ann is a separable Locally Toeplitz (sLT)sequence if Ann ∼LT a ⊗ f for some Riemann-integrable function a : [0, 1]d → C and some separable function f ∈ L1([−π, π]d).In this case, we write Ann ∼sLT a⊗ f .

1We refer the reader to the introduction of Tilli’s paper [101] for the origin and the meaning of this terminology.

66

It is clear from the definition that a sLT sequence is just a LT sequence with separable generating function. From now on, if we writeAnn ∼LT a⊗ f (resp. Ann ∼sLT a⊗ f ), it is understood that a : [0, 1]d → C is Riemann-integrable and f ∈ L1([−π, π]d) (resp.f ∈ L1([−π, π]d) is separable).

Exercise 6.1. Show that, if A(i)n n ∼LT a⊗fi for i = 1, . . . , r, then

∑ri=1A

(i)n n ∼LT a⊗(

∑ri=1 fi). Show also that, if A(i)

n n ∼LT

ai ⊗ f for i = 1, . . . , r, then ∑ri=1A

(i)n n ∼LT (

∑ri=1 ai)⊗ f .

6.3 Zero-distributed sequences, sequences of multilevel diagonal sampling matrices andsequences of multilevel Toeplitz matrices

Let us now provide basic examples of LT sequences: zero-distributed sequences, sequences of multilevel diagonal sampling matrices, andsequences of multilevel Toeplitz matrices. These may be regarded as the building blocks of the theory of GLT sequences, because, startingfrom them, we can construct through algebraic operations a lot of other matrix-sequences which will be seen in Chapter 7 to be GLTsequences. We recall that, according to our terminology,∞ is an accumulation point for a subsetM⊆ Nq ifM∩i ∈ Nq : i ≥ h 6= ∅for all h ∈ Nq .

Theorem 6.4. Let Znn be a matrix-sequence. LetM be a subset of some Nq such that∞ is an accumulation point forM. Then, thefollowing conditions are equivalent.

1. Znn ∼σ 0.

2. ON(n)nm∈M is an a.c.s. for Znn.

3. Znnm∈M is an a.c.s. for ON(n)n.

In particular, a matrix-sequence Znn is zero-distributed if and only if Znn ∼sLT 0.

Proof. The equivalence 2⇔ 3 is obvious from Definition 4.4.Let us prove that 1⇒ 2. By Theorem 2.9, we can write Zn = Rn +Nn for all n, where

limn→∞

rank(Rn)

N(n)= limn→∞

‖Nn‖ = 0.

It follows that ON(n)nm∈M is an a.c.s. for Znn. To see this, take, in (4.23), Rn,m = Rn, Nn,m = Nn, c(m) and ω(m) anytwo positive functions of m that converge to 0 as m → ∞ (for instance, c(m) = ω(m) = 1/min(m)), and nm any integer such thatrank(Rn)/N(n) ≤ c(m) and ‖Nn‖ ≤ ω(m) for n ≥ nm.

Let us now prove that 2⇒ 1. By assumption, ON(n)nm∈M is an a.c.s. for Znn. Hence, if we take any sequence m =m(m)m ⊆ M such that m → ∞ as m → ∞, ON(n)nm is a (classical) a.c.s. for Znn. Moreover, it is clear thatON(n)n ∼σ 0. Hence, Znn ∼σ 0 by Corollary 4.1.

The fact that Znn is zero-distributed if and only if Znn ∼sLT 0 follows from the equivalence 1⇔ 2 (applied withM = Nd) andfrom the observation that LTmn (0, 0)nm∈Nd = ON(n)nm∈Nd and 0⊗ 0 = 0.

For the proof of Theorems 6.5–6.6, we need the following technical lemma, whose proof can be skipped on first reading.

Lemma 6.1. Let N be an infinite subset of N. Let x(·, ·) : N× Nd → R be any function satisfying

limm→∞

limh→∞

x(m,h) = ξ ∈ R.

Then, there exists a function m(·) : Nd → N such that, when h→∞, m(h)→∞ and x(m(h),h)→ ξ.

Proof. Let N = m1,m2,m3, . . .. We denote by m+ (m−) the number that follows (preceeds) m in N. Set

x(m) = limh→∞

x(m,h), m ∈ N.

Since x(m) → ξ by assumption, x(m) is eventually a real number (different from −∞ or +∞). Suppose, for instance, that x(m) ∈ Rfor all m ≥ mr. We construct an injective function h(·) : mr,mr+1,mr+2, . . . → Nd as follows: we set h(mr) = 1 and, for everym ∈ N\mr, we choose h(m) > h(m−) such that, for h ≥ h(m),

|x(m,h)− x(m)| ≤ 1

m⇒ |x(m,h)− ξ| ≤ |x(m)− ξ|+ 1

m.

67

Hence, we have constructed a sequence

1 = h(mr) < h(mr+1) < h(mr+2) < h(mr+3) < . . .

and this sequence has the following property: if m ∈ mr+1,mr+2, . . . and h ≥ h(m), then

|x(m,h)− ξ| ≤ |x(m)− ξ|+ 1

m. (6.20)

In view of this, we define m(·) : Nd → N as follows:

m(h) =

mr if h ∈ j ∈ Nd : j ≥ h(mr) = 1\j ∈ Nd : j ≥ h(mr+1),mr+1 if h ∈ j ∈ Nd : j ≥ h(mr+1)\j ∈ Nd : j ≥ h(mr+2),mr+2 if h ∈ j ∈ Nd : j ≥ h(mr+2)\j ∈ Nd : j ≥ h(mr+3),...

...

Clearly, m(h) → ∞ when h → ∞. Moreover, for all h ≥ h(mr+1) we have m(h) ∈ mr+1,mr+2, . . . and h ≥ h(m(h)). Hence,by (6.20), for all h ≥ h(mr+1) we have |x(m(h),h)− ξ| ≤ |x(m(h))− ξ|+ 1/m(h), which tends to 0 as h→∞.

Theorem 6.5. If a : [0, 1]d → C is Riemann-integrable and Dn(a)n is any matrix-sequence extracted from Dn(a)n∈Nd , thenDn(a)n ∼sLT a⊗ 1.

Proof. The proof is organized in two steps: we first show by induction on d that the thesis holds if a is continuous; then, by usingan approximation argument, we show that it holds even in the case where a is any Riemann-integrable function. As we shall see, theapproximation argument heavily relies on the Riemann-integrability of a.

1. We prove by induction on d that, if a ∈ C([0, 1]d), then

Dn(a) = LTmn (a, 1) +Rn,m +Nn,m, rank(Rn,m) ≤ N(n)

d∑i=1

mi

ni, ‖Nn,m‖ ≤

d∑i=1

ωa

( 1

mi+mi

ni

). (6.21)

Since ωa(δ) → 0 as δ → 0, eq. (6.21) implies that the thesis holds for any continuous function a ∈ C([0, 1]d); it suffices to choose, inDefinition 6.3, an index nm such that n ≥m2 for n ≥ nm, and to take c(m) =

∑di=1(1/mi), ω(m) =

∑di=1 ωa(2/mi).

In the case d = 1, LTmn (a, 1) is the n× n diagonal matrix given by

LTmn (a, 1) = Dm(a)⊗ Ibn/mc ⊕ Onmodm = a(1/m)Ibn/mc ⊕ a(2/m)Ibn/mc ⊕ · · · ⊕ a(1)Ibn/mc ⊕Onmodm.

For every i = 1, . . . ,mbn/mc, let j = j(i) be the index in 1, . . . ,m such that (j − 1)bn/mc+ 1 ≤ i ≤ jbn/mc. Then,

|[LTmn (a, 1)]ii − [Dn(a)]ii| = |a(j/m)− a(i/n)| ≤ ωa(1/m+m/n),

because ∣∣∣∣ jm − i

n

∣∣∣∣ ≤ j

m− (j − 1)bn/mc

n≤ j

m− (j − 1)(n/m− 1)

n=

1

m+j − 1

n≤ 1

m+m

n. (6.22)

Define the following n× n diagonal matrices:

Dn,m(a) = diagi=1,...,mbn/mc

a(i/n) ⊕ Onmodm, Dn,m(a) = Ombn/mc ⊕ diagi=mbn/mc+1,...,n

a(i/n).

Then,Dn(a)− LTmn (a, 1) = Dn,m(a) + Dn,m(a)− LTmn (a, 1) = Rn,m +Nn,m,

where Rn,m = Dn,m(a) and Nn,m = Dn,m(a)− LTmn (a, 1) satisfy

rank(Rn,m) ≤ nmodm ≤ m, ‖Nn,m‖ = maxi=1,...,mbn/mc

|[LTmn (a, 1)]ii − [Dn(a)]ii| ≤ ωa(1/m+m/n).

This shows that (6.21) holds for d = 1.In the case d > 1, LTmn (a, 1) is the N(n)×N(n) diagonal matrix given by

LTmn (a, 1) = diagj1=1,...,m1

Ibn1/m1c ⊗ LTm2,...,mdn2,...,nd

(a( j1m1

, ·), 1)⊕ O(n1 modm1)n2···nd , (6.23)

68

where, for any x1 ∈ [0, 1], a(x1, ·) : [0, 1]d−1 → C is the function (x2, . . . , xd) 7→ a(x1, x2, . . . , xd). For every j1 = 1, . . . ,m1 andevery i1 = (j1 − 1)bn1/m1c+ 1, . . . , j1bn1/m1c, by induction hypothesis we have

LTm2,...,mdn2,...,nd

(a( j1m1

, ·), 1)−Dn2,...,nd

(a( i1n1, ·))

=

[Dn2,...,nd

(a( j1m1

, ·))−Dn2,...,nd

(a( i1n1, ·))]

+R[j1/m1]n2,...,nd,m2,...,md

+N [j1/m1]n2,...,nd,m2,...,md

,

where

rank(R[j1/m1]n2,...,nd,m2,...,md

) ≤ n2 · · ·ndd∑k=2

mk

nk, ‖N [j1/m1]

n2,...,nd,m2,...,md‖ ≤

d∑k=2

ωa(j1/m1,·)

( 1

mk+mk

nk

)≤

d∑k=2

ωa

( 1

mk+mk

nk

).

Moreover, ∥∥∥∥Dn2,...,nd

(a( j1m1

, ·))−Dn2,...,nd

(a( i1n1, ·))∥∥∥∥ ≤ ωa( 1

m1+m1

n1

),

because one can show as in (6.22) that ∣∣∣∣ j1m1− i1n1

∣∣∣∣ ≤ 1

m1+m1

n1.

Thus,


(a( j1m1

, ·), 1)−Dn2,...,nd

(a( i1n1, ·))

= R[j1/m1]n2,...,nd,m2,...,md

+N [j1/m1, i1/n1]n,m

rank(R[j1/m1]n2,...,nd,m2,...,md

) ≤ n2 · · ·ndd∑k=2

mk

nk, ‖N [j1/m1, i1/n1]

n,m ‖ ≤d∑k=1

ωa

( 1

mk+mk

nk

). (6.24)

Now we observe that the diagonal matrices LTmn (a, 1) and Dn(a) can be written as

LTmn (a, 1) = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1cLTm2,...,md

n2,...,nd

(a( j1m1

, ·), 1)]⊕ O(n1 modm1)n2···nd ,

Dn(a) = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1cDn2,...,nd

(a( i1n1, ·))]

⊕ diagi1=m1bn1/m1c+1,...,n1

Dn2,...,nd

(a( i1n1, ·))

;

see (6.23) and (2.4). Hence,

Dn(a)− LTmn (a, 1) = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1c

[Dn2,...,nd

(a( i1n1, ·))− LTm2,...,md

n2,...,nd

(a( j1m1

, ·), 1)]]


Dn2,...,nd

(a( i1n1, ·))

= diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1c

[−R[j1/m1]

n2,...,nd,m2,...,md−N [j1/m1, i1/n1]

n,m

]]


Dn2,...,nd

(a( i1n1, ·))

= Rn,m +Nn,m,

where

Rn,m = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1c−R[j1/m1]

n2,...,nd,m2,...,md

]⊕ diag

i1=m1bn1/m1c+1,...,n1

Dn2,...,nd

(a( i1n1, ·)),

Nn,m = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1c−N [j1/m1, i1/n1]

n,m

]⊕ O(n1 modm1)n2···nd .

69

By (6.24) and (2.45), (2.47), we have

rank(Rn,m) ≤ m1

⌊n1

m1

⌋n2 · · ·nd

d∑k=2

mk

nk+ (n1 modm1)n2 · · ·nd ≤ n1n2 · · ·nd

d∑k=2

mk

nk+m1n2 · · ·nd = N(n)

d∑k=1

mk

nk,

‖Nn,m‖ ≤d∑k=1

ωa

( 1

mk+mk

nk

),


2. Let a : [0, 1]d → C be any Riemann-integrable function. Take any sequence of continuous functions am : [0, 1]d → C such thatam → a in L1([0, 1]d). Note that such a sequence exists because C([0, 1]d) is dense in L1([0, 1]d); see [81]. By the first part of the proof,Dn(am)n ∼sLT am ⊗ 1. Hence, for each m and each h ∈ Nd there is nm,h such that, for n ≥ nm,h,

Dn(am) = LThn (am, 1) +Rn,m,h +Nn,m,h,

rank(Rn,m,h) ≤ c(m,h)N(n), ‖Nn,m,h‖ ≤ ω(m,h),

wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, Dn(am)nm is an a.c.s. for Dn(a)n. Indeed,

‖Dn(a)−Dn(am)‖1 =

n∑j=1

∣∣∣∣a( jn)− am( jn)∣∣∣∣ = N(n)

1

N(n)

n∑j=1

∣∣∣∣a( jn)− am( jn)∣∣∣∣ .

By the Riemann-integrability of |a − am|, which follows from the Riemann-integrability of a and am, and by the fact that am → a inL1([0, 1]d), the quantity

ε(m,n) =1

N(n)

n∑j=1

∣∣∣∣a( jn)− am( jn)∣∣∣∣

satisfieslimm→∞

limn→∞

ε(m,n) = limm→∞

∫[0,1]d

|a(x)− am(x)|dx = limm→∞

‖a− am‖L1 = 0.

By Corollary 4.3, this implies that Dn(am)nm is an a.c.s. for Dn(a)n. Thus, for every m there exists nm such that, for n ≥ nm,

Dn(a) = Dn(am) +Rn,m +Nn,m,


wherelimm→∞

c(m) = limm→∞

ω(m) = 0.

It follows that, for every m, every h ∈ Nd and every n ≥ max(nm, nm,h),

Dn(a) = LThn (a, 1) +[LThn (am, 1)− LThn (a, 1)

]+ (Rn,m +Rn,m,h) + (Nn,m +Nn,m,h),

rank(Rn,m +Rn,m,h) ≤ (c(m) + c(m,h))N(n),

‖Nn,m +Nn,m,h‖ ≤ ω(m) + ω(m,h),∥∥LThn (am, 1)− LThn (a, 1)∥∥

1≤ N(n)

N(h)

h∑j=1

∣∣∣∣a( jh)− am( jh)∣∣∣∣ = ε(m,h)N(n),

where in the last inequality we used (6.8) and the identity Tbn/hc(1) = IN(bn/hc). For every h ∈ Nd, choosem(h) such thatm(h)→∞when h→∞ and

limh→∞

ε(m(h),h) = limh→∞

c(m(h),h) = limh→∞

ω(m(h),h) = 0.

70

An explicit construction of such a function m(h) has been given in Lemma 6.1 (apply the lemma with x(m,h) = ε(m,h) + c(m,h) +ω(m,h) and ξ = 0). Then, for every h ∈ Nd and every n ≥ max(nm(h), nm(h),h),

Dn(a) = LThn (a, 1) +[LThn (am(h), 1)− LThn (a, 1)

]+ (Rn,m(h) +Rn,m(h),h) + (Nn,m(h) +Nn,m(h),h),

rank(Rn,m(h) +Rn,m(h),h) ≤ (c(m(h)) + c(m(h),h))N(n),

‖Nn,m(h) +Nn,m(h),h‖ ≤ ω(m(h)) + ω(m(h),h),∥∥LThn (am(h), 1)− LThn (a, 1)∥∥

1≤ ε(m(h),h)N(n).

The application of Lemma 4.4 allows one to decompose LThn (am(h), 1) − LThn (a, 1) as the sum of a small-rank term Rn,h, with rankbounded by

√ε(m(h),h)N(n), plus a small-norm term Nn,h, with norm bounded by

√ε(m(h),h). This concludes the proof.

Theorem 6.6. If f ∈ L1([−π, π]d) and Tn(f)n is any matrix-sequence extracted from Tn(f)n∈Nd , then Tn(f)n ∼LT 1⊗ f .

Proof. The proof is organized in three steps: we first show by induction on d that the thesis holds if f is a separable d-variate trigonometricpolynomial; then, by linearity, we show that it also holds if f is an arbitrary d-variate trigonometric polynomial; finally, using anapproximation argument, we prove the theorem under the sole assumption that f ∈ L1([−π, π]d).

1. We show by induction on d that, if f is a separable d-variate trigonometric polynomial, say f = f1 ⊗ · · · ⊗ fd with f1, . . . , fdunivariate trigonometric polynomials, then

Tn(f) = LTmn (1, f) +Rn,m, rank(Rn,m) ≤ N(n)

d∑i=1

(2ri + 1)mi

ni, (6.25)

where ri is the degree of fi. From (6.25), it follows that the theorem holds for any separable trigonometric polynomial f ; it suffices tochoose, in Definition 6.3, an index nm such that n ≥m2 for n ≥ nm, and to take c(m) =

∑di=1(2ri + 1)/mi, ω(m) = 0.

In the case d = 1, let f(θ) =∑rj=−r fje

ijθ. Then,

LTmn (1, f) = Im ⊗ Tbn/mc(f) ⊕ Onmodm = Tbn/mc(f)⊕ · · · ⊕ Tbn/mc(f)⊕Onmodm.

Looking carefully at the structure of Tn(f) and LTmn (1, f), we see that the number of nonzero rows of the difference Tn(f)−LTmn (1, f)is at most 2rm− r + (nmodm). Hence,

Tn(f) = LTmn (1, f) +Rn,m, rank(Rn,m) ≤ 2rm− r + (nmodm) ≤ (2r + 1)m, (6.26)

and so (6.25) holds for d = 1.In the case d > 1, let f = f1 ⊗ · · · ⊗ fd with f1, . . . , fd univariate trigonometric polynomials of degrees r1, . . . , rd, respectively. By

induction hypothesis we have


(1, f2 ⊗ · · · ⊗ fd) = Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md ,

rank(Rn2,...,nd,m2,...,md) ≤ n2 · · ·ndd∑i=2

(2ri + 1)mi

ni.

From the definition of LTmn (1, f) and the properties of tensor products and direct sums (see Section 2.6.1), we obtain

LTmn (1, f) = diagj1=1,...,m1

Tbn1/m1c(f1)⊗ LTm2,...,mdn2,...,nd

(1, f2 ⊗ · · · ⊗ fd) ⊕ O(n1 modm1)n2···nd

=[

diagj1=1,...,m1

Tbn1/m1c(f1)]⊗[Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md

]⊕ O(n1 modm1)n2···nd

=[

diagj1=1,...,m1

Tbn1/m1c(f1) ⊕ On1 modm1

]⊗[Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md

]= LTm1

n1(1, f1)⊗

[Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md

]= LTm1

n1(1, f1)⊗ Tn2,...,nd(f2 ⊗ · · · ⊗ fd) + Rn1,...,nd,m1,...,md ,

where Rn1,...,nd,m1,...,md = LTm1n1

(1, f1)⊗Rn1,...,nd,m1,...,md satisfies

rank(Rn1,...,nd,m1,...,md) ≤ N(n)

d∑i=2

(2ri + 1)mi

ni.

71

Using (6.26), we can decompose LTm1n1

(1, f1) as the sum of Tn1(f1) plus a small-rank matrix Rn1,m1 , whose rank is bounded by(2r1 + 1)m1. Recalling Lemma 5.3, we arrive at

LTmn (1, f) = (Tn1(f1) +Rn1,m1)⊗ Tn2,...,nd(f2 ⊗ · · · ⊗ fd) + Rn1,...,nd,m1,...,md = Tn(f) +Rn,m,

where Rn,m = Rn1,m1⊗ Tn2,...,nd(f2 ⊗ · · · ⊗ fd) + Rn1,...,nd,m1,...,md satisfies

rank(Rn,m) ≤ (2r1 + 1)m1n2 · · ·nd +N(n)

d∑i=2

(2ri + 1)mi

ni= N(n)

d∑i=1

(2ri + 1)mi

ni.

This completes the proof of (6.25).

2. Let f be any d-variate trigonometric polynomial. By definition, f is a finite linear combination of the Fourier frequencieseij·θ, j ∈ Z, and so we can write f(θ) =

∑rj=−r fje

ij·θ for some separable trigonometric polynomials fjeij·θ. By linearity,

Tn(f) =

r∑j=−r

fjTn(eij·θ), LTmn (1, f) =

r∑j=−r

fjLTmn (1, eij·θ).

By the first part of the proof, Tn(eij·θ)n ∼LT 1⊗ eij·θ, hence LTmn (1, eij·θ)nm∈Nd is an a.c.s. for Tn(eij·θ)n. It follows thatLTmn (1, f)nm∈Nd is an a.c.s. for Tn(f)n; see Remark 4.6. Thus, Tn(f)n ∼LT 1⊗ f for every trigonometric polynomial f .

3. Let f ∈ L1([−π, π]d). Since the set of d-variate trigonometric polynomials is dense in L1([−π, π]d), there is a sequence fmof d-variate trigonometric polynomials such that fm → f in L1([−π, π]d). By the second part of the proof, Tn(fm)n ∼LT 1 ⊗ fm.Hence, for each m and each h ∈ Nd there is nm,h such that, for n ≥ nm,h,

Tn(fm) = LThn (1, fm) +Rn,m,h +Nn,m,h,


wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, by Theorem 5.2,‖Tn(f)− Tn(fm)‖1 = ‖Tn(f − fm)‖1 ≤ N(n)‖f − fm‖L1

and so Tn(fm)nm is an a.c.s. for Tn(f)n by Corollary 4.3: for every m there exists nm such that, for n ≥ nm,

Tn(f) = Tn(fm) +Rn,m +Nn,m,


wherelimm→∞

c(m) = limm→∞

ω(m) = 0.

It follows that, for every m, every h ∈ Nd and every n ≥ max(nm, nm,h),

Tn(f) = LThn (1, f) +[LThn (1, fm)− LThn (1, f)



‖Nn,m +Nn,m,h‖ ≤ ω(m) + ω(m,h),∥∥LThn (1, fm)− LThn (1, f)∥∥

1=∥∥LThn (1, fm − f)

∥∥1≤ N(n)‖f − fm‖L1 .

Choose, for every h ∈ Nd, a m(h) such that m(h)→∞ when h→∞ and

limh→∞


ω(m(h),h) = 0.

An explicit construction of such a function m(h) is given in Lemma 6.1 (apply the lemma with x(m,h) = c(m,h) + ω(m,h) andξ = 0). Then, for every h ∈ Nd and every n ≥ max(nm(h), nm(h),h),

Tn(f) = LThn (1, f) +[LThn (1, fm(h))− LThn (1, f)



‖Nn,m(h) +Nn,m(h),h‖ ≤ ω(m(h)) + ω(m(h),h),∥∥LThn (1, fm(h))− LThn (1, f)∥∥

1≤ ‖fm(h) − f‖L1N(n).

72

The application of Lemma 4.4 allows one to decompose LThn (1, fm(h)) − LThn (1, f) as the sum of a small-rank term Rn,h, withrank bounded by

√‖fm(h) − f‖L1 N(n), plus a small-norm term Nn,h, with norm bounded by

√‖fm(h) − f‖L1 . This concludes the

proof.

It follows from Theorem 6.6 that Tn(f)n ∼sLT 1⊗ f whenever f ∈ L1([−π, π]d) is separable.

6.4 Properties and characterizations of LT and sLT sequencesWe begin with a basic spectral result for LT sequences, which will be used in the proof of Theorem 6.7. Since it will be generalizedafterwards, in the more general context of GLT sequences, it is not necessary to remember it.

Lemma 6.2. If Ann ∼LT a⊗ f then Ann ∼σ a⊗ f.

Proof. Take any sequence of multi-indices m = m(m)m ⊆ Nd such thatm→∞ asm→∞. By Definition 6.3, LTmn (a, f)nmis an a.c.s. for Ann. By Theorem 6.2, we have LTmn (a, f)n ∼σ φm and φm → φ[|a⊗f |] pointwise over Cc(R), where we recallthat φ[g] is defined in (2.15). Therefore, Theorem 4.3 gives Ann ∼σ a⊗ f .

As a consequence of Lemma 6.2 and Proposition 4.3, every LT sequence is s.u. (see Definition 4.2). We now show, under mildassumptions, that the product of LT sequences is again a LT sequence whose symbol is given by the product of the symbols.

Theorem 6.7. Suppose thatAnn ∼LT a⊗ f, Ann ∼LT a⊗ f ,

where f ∈ Lp([−π, π]d), f ∈ Lq([−π, π]d), and p, q are conjugate exponents (1 ≤ p, q ≤ ∞). Then

AnAnn ∼LT aa⊗ ff .

Proof. As noted before the statement of the theorem, Lemma 6.2 and Proposition 4.3 imply that every LT sequence is s.u., so in particularAnn and Ann are s.u. Since LTmn (a, f)nm∈Nd is an a.c.s. for Ann and LTmn (a, f)nm∈Nd is an a.c.s. for Ann, theproduct LTmn (a, f)LTmn (a, f)nm∈Nd is an a.c.s. for AnAnn; see Remark 4.6. The thesis now follows from Definition 6.3 andProposition 6.3 (see in particular eq. (6.11) in Proposition 6.3).

As a consequence of Theorem 6.7 and Theorems 6.5–6.6, we immediately obtain the following result.

Theorem 6.8. Let a : [0, 1]d → C be Riemann-integrable, let f ∈ L1([−π, π]d), and consider the sequence of matrices Dn(a)Tn(f)n,where, of course, n→∞ as n→∞. Then Dn(a)Tn(f)n ∼LT a⊗ f.

Theorem 6.8 shows that, for any a, f as in Definition 6.3, there always exists a matrix-sequence Ann such that Ann ∼LT a⊗ f .Indeed, it suffices to take An = Dn(a)Tn(f). Theorem 6.9 shows that the sequences of the form Dn(a)Tn(f)n play a central rolein the world of LT sequences. Indeed, Ann ∼LT a ⊗ f if and only if An equals Dn(a)Tn(f) up to a small-rank plus small-normcorrection. In fact, any LT sequence Ann ∼LT a ⊗ f admits the fixed matrix-sequence Dn(a)Tn(f)n as an a.c.s., and, viceversa, any sequence Ann admitting Dn(a)Tn(f)n as an a.c.s. is a LT sequence with symbol a ⊗ f . Moreover, any LT sequenceAnn ∼LT a⊗ f also admits an a.c.s. of the form Dn(am)Tn(fm)nm, with am continuous and fm trigonometric polynomial; asone could guess, am will be chosen as an approximation of a, converging to a for m→∞, while fm will be chosen as an approximationof f , converging to f for m→∞.

Theorem 6.9 (characterizations of LT sequences). Let Ann be a matrix-sequence, let a : [0, 1]d → C be a Riemann-integrablefunction and let f ∈ L1([−π, π]d). Then, the following conditions are equivalent.

1. Ann ∼LT a⊗ f .

2. For all sequences amm, fmm, A(m)n nm with the following properties:

∗ am : [0, 1]d → C is Riemann-integrable and am → a in L1([0, 1]d);

∗ fm ∈ L1([−π, π]d) and fm → f in L1([−π, π]d);

∗ A(m)n n ∼LT am ⊗ fm;

it holds that A(m)n nm is an a.c.s. for Ann.

3. There exist sequences amm, fmm such that:

∗ am : [0, 1]d → C is continuous, ‖am‖∞ ≤ ‖a‖L∞ for all m and am → a a.e.;

73

∗ fm : [−π, π]d → C is a trigonometric polynomial and fm → f a.e. and in L1([−π, π]d);

∗ Dn(am)Tn(fm)nm is an a.c.s. for Ann.

4. There exist sequences amm, fmm, A(m)n nm such that:

∗ am : [0, 1]d → C is Riemann-integrable and am → a in L1([0, 1]d);

∗ fm ∈ L1([−π, π]d) and fm → f in L1([−π, π]d);

∗ A(m)n n ∼LT am ⊗ fm and A(m)

n nm is an a.c.s. for Ann.

5. Dn(a)Tn(f)nm is an a.c.s. for Ann.

6. For every n we have An = Dn(a)Tn(f) + Zn, where Znn is zero-distributed.

Proof. (1⇒ 2) Let am, fm, A(m)n nm be sequences with the properties specified in item 2. Since A(m)

n n ∼LT am ⊗ fm, foreach m and each h ∈ Nd there is nm,h such that, for n ≥ nm,h,

A(m)n = LThn (am, fm) +Rn,m,h +Nn,m,h,


wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, since Ann ∼LT a⊗ f , for every h ∈ Nd there is nh such that, for n ≥ nh,

An = LThn (a, f) +Rn,h +Nn,h,

rank(Rn,h) ≤ c(h)N(n), ‖Nn,h‖ ≤ ω(h),

wherelimh→∞

c(h) = limh→∞

ω(h) = 0.

Hence, for every m, every h ∈ Nd and every n ≥ max(nm,h, nh),

An = A(m)n +

[LThn (a, f)− LThn (am, fm)

]+ (Rn,h −Rn,m,h) + (Nn,h −Nn,m,h),

rank(Rn,h −Rn,m,h) ≤ (c(h) + c(m,h))N(n), ‖Nn,h −Nn,m,h‖ ≤ ω(h) + ω(m,h).(6.27)

Thanks to Propositions 6.1–6.2 and to Theorem 5.2, we have

‖LThn (a, f)− LThn (am, fm)‖1 ≤ ‖LThn (a, f − fm)‖1 + ‖LThn (a− am, fm)‖1

=

h∑j=1

∣∣∣a( jh

)∣∣∣‖Tbn/hc(f − fm)‖1 +

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣‖Tbn/hc(fm)‖1

≤ N(n)‖a‖∞‖f − fm‖L1 + ‖fm‖L1

N(n)

N(h)

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣≤

‖a‖∞‖f − fm‖L1 + supk‖fk‖L1

1

N(h)

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣N(n); (6.28)

note that ‖fk‖L1 is uniformly bounded with respect to k, because fk → f in L1([−π, π]d). By the Riemann-integrability of |a − am|,which follows from the Riemann-integrability of a and am, and by the fact that am → a in L1([0, 1]d) and fm → f in L1([−π, π]d), thequantity

ε(m,h) = ‖a‖∞‖f − fm‖L1 + supk‖fk‖L1

1

N(h)

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣ (6.29)

satisfies

limm→∞

limh→∞

ε(m,h) = limm→∞

(‖a‖∞‖f − fm‖L1 + sup

k‖fk‖L1

∫[0,1]d

|a(x)− am(x)|dx

)= 0. (6.30)

74

Choose any sequence of multi-indices h(m)m such that h(m)→∞ for m→∞ and

limm→∞

c(m,h(m)) = limm→∞

ω(m,h(m)) = limm→∞

ε(m,h(m)) = 0.

Then, by (6.27)–(6.28), for every m and every n ≥ max(nm,h(m), nh(m)),

An = A(m)n +

[LTh(m)

n (a, f)− LTh(m)n (am, fm)

]+ (Rn,h(m) −Rn,m,h(m)) + (Nn,h(m) −Nn,m,h(m)),

rank(Rn,h(m) −Rn,m,h(m)) ≤ [c(h(m)) + c(m,h(m))]N(n),

‖Nn,h(m) −Nn,m,h(m)‖ ≤ ω(h(m)) + ω(m,h(m)),∥∥LTh(m)n (a, f)− LTh(m)

n (am, fm)∥∥

1≤ ε(m,h(m))N(n).

Using Lemma 4.4, we can decompose LTh(m)n (a, f) − LTh(m)

n (am, fm) as the sum of a small-rank term Rn,m, with rank bounded by√ε(m,h(m))N(n), plus a small-norm term Nn,m, with norm bounded by

√ε(m,h(m)). This concludes the proof of the implication

1⇒ 2.(2⇒ 3) Since any Riemann-integrable function is bounded, we have a ∈ L∞([0, 1]d). Hence, by the Lusin theorem [81], there exists

a sequence of continuous functions am : [0, 1]d → C such that ‖am‖∞ ≤ ‖a‖L∞ for all m and am → a in measure. This implies thatam → a also in L1([0, 1]d), due to the uniform boundedness of ‖am‖∞. Thus, there exists a subsequence of am, say am, whichconverges to a a.e. in [0, 1]d. The sequence am satisfies all the properties required in item 3.

Since f ∈ L1([−π, π]d) and the set of d-variate trigonometric polynomials is dense in L1([−π, π]d), there exists a sequence fm ofd-variate trigonometric polynomials such that fm → f in L1([−π, π]d). Choosing a subsequence fm of fm which converges to fa.e., fm satisfies all the properties required in item 3.

By item 2 and Theorem 6.8, Dn(am)Tn(fm)nm is an a.c.s. for Ann, and the proof is finished.(3⇒ 4) Obvious. We just recall that, under the assumptions in item 3, am → a in L1([0, 1]d) by the dominated convergence theorem.

Moreover, Dn(am)Tn(fm)n ∼LT am ⊗ fm by Theorem 6.8.(4⇒ 1) Since A(m)

n n ∼LT am ⊗ fm, for each m and each h ∈ Nd there is nm,h such that, for n ≥ nm,h,

A(m)n = LThn (am, fm) +Rn,m,h +Nn,m,h,


wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, since A(m)n nm is an a.c.s. for Ann, for every m there exists nm such that, for n ≥ nm,

An = A(m)n +Rn,m +Nn,m,


wherelimm→∞

c(m) = limm→∞

ω(m) = 0.

Thus, for every m, every h ∈ Nd and every n ≥ max(nm, nm,h),

An = LThn (a, f) +[LThn (am, fm)− LThn (a, f)



‖Nn,m +Nn,m,h‖ ≤ ω(m) + ω(m,h),∥∥LThn (am, fm)− LThn (a, f)∥∥

1≤ ε(m,h)N(n),

where in the last inequalities we used (6.28); the quantity ε(m,h) is defined in (6.29) and satisfies (6.30). Choose, for every h ∈ Nd, am(h) such that m(h)→∞ when h→∞ and

limh→∞

ε(m(h),h) = limh→∞


ω(m(h),h) = 0.

75

A construction of such a function m(h) is provided in Lemma 6.1 (apply the lemma with x(m,h) = ε(m,h) + c(m,h) + ω(m,h) andξ = 0). Then, for every h ∈ Nd and every n ≥ max(nm(h), nm(h),h),

An = LThn (a, f) +[LThn (am(h), fm(h))− LThn (a, f)



‖Nn,m(h) +Nn,m(h),h‖ ≤ ω(m(h)) + ω(m(h),h),∥∥LThn (am(h), fm(h))− LThn (a, f)∥∥

1≤ ε(m(h),h)N(n).

The application of Lemma 4.4 allows one to decompose LThn (am(h), fm(h)) − LThn (a, f) as the sum of a small-rank term Rn,h, withrank bounded by

√ε(m(h),h)N(n), plus a small-norm term Nn,h, with norm bounded by

√ε(m(h),h). This concludes the proof of

the implication 4⇒ 1.(5⇔ 6) We note that Dn(a)Tn(f)n is a fixed matrix-sequence, independent of m. Item 5 is equivalent to saying that An −

Dn(a)Tn(f)nm is an a.c.s. of ON(n)n, which, by Theorem 6.4, is equivalent to saying that An − Dn(a)Tn(f)n is zero-distributed.

(2⇒ 5) Obvious (take am = a, fm = f and A(m)n = Dn(a)Tn(f)).

(5⇒ 4) Obvious (take am = a, fm = f and A(m)n = Dn(a)Tn(f)).

Remark 6.2. Theorem 6.9 continues to hold if f is assumed to be separable, item 1 is replaced by ‘ Ann ∼sLT a⊗ f ’, and we add initem 3 the requirement that each fm is separable. The proof is left as an exercise for the reader.

As an application of Theorem 6.9, we prove the following result about matrix-sequences of the form Dn(a) Tn(f)n, witha ∈ C([0, 1]d) and f a d-variate trigonometric polynomial; see Section 2.2 for the definition of Dn(a). The result will be used inChapter 8, when considering the applications of the theory of GLT sequences to the asymptotic spectral analysis of PDE discretizationmatrices.

Theorem 6.10. Let a : [0, 1]d → C be continuous and let f(θ) =∑rj=−r fje

ij·θ be a d-variate trigonometric polynomial. Then,

‖Dn(a) Tn(f)−Dn(a)Tn(f)‖ ≤ N(2r + 1)‖f‖∞ ωa

( d‖r‖∞min(n)

), n ∈ Nd. (6.31)

In particular, ‖Dn(a) Tn(f)‖ ≤ C for some constant C independent of n ∈ Nd, and Dn(a) Tn(f)n ∼LT a ⊗ f for all matrix-sequences Dn(a) Tn(f)n extracted from Dn(a) Tn(f)n∈Nd .

Proof. For all i, j = 1, . . . ,n:

• if ‖i− j‖∞ > ‖r‖∞, then the Fourier coefficient fi−j is zero and, consequently,

(Dn(a) Tn(f))ij = (Dn(a))ij(Tn(f))ij = a(i ∧ jn

)fi−j = 0,

(Dn(a)Tn(f))ij = (Dn(a))ii(Tn(f))ij = a( in

)fi−j = 0;

• if ‖i− j‖∞ ≤ ‖r‖∞, then, using (2.25) and (5.21), we obtain∣∣(Dn(a) Tn(f))ij − (Dn(a)Tn(f))ij∣∣ = |(Dn(a))ij(Tn(f))ij − (Dn(a))ii(Tn(f))ij |

=∣∣(Dn(a))ij − (Dn(a))ii

∣∣ |(Tn(f))ij |

≤∣∣∣∣a(i ∧ jn )

− a( in

)∣∣∣∣ ‖Tn(f)‖ ≤ ‖f‖∞ ωa

(∥∥∥i ∧ jn− i

n

∥∥∥).Since ‖i− j‖∞ ≤ ‖r‖∞, we have∥∥∥i ∧ j

n− i

n

∥∥∥ ≤ d∥∥∥i ∧ jn− i

n

∥∥∥∞≤ d‖j − i‖∞

min(n)≤ d‖r‖∞

min(n),

and so ∣∣(Dn(a) Tn(f))ij − (Dn(a)Tn(f))ij∣∣ ≤ ‖f‖∞ ωa

( d‖r‖∞min(n)

).

76

From the first item, it follows that the nonzero entries in each row and column of Dn(a) Tn(f)−Dn(a)Tn(f) are at most N(2r+ 1).Hence, we infer from the second item that the 1-norm and the ∞-norm of Dn(a) Tn(f) − Dn(a)Tn(f) are bounded by N(2r +

1)‖f‖∞ ωa(d‖r‖∞min(n) ). Thus, (2.26) yields (6.31). It follows from (6.31) that

‖Dn(a) Tn(f)‖ ≤ ‖Dn(a) Tn(f)−Dn(a)Tn(f)‖+ ‖Dn(a)‖ ‖Tn(f)‖ ≤ N(2r + 1)‖f‖∞ ωa

( d‖r‖∞min(n)

)+ ‖a‖∞‖f‖∞,

which implies that ‖Dn(a)Tn(f)‖ is uniformly bounded with respect to n ∈ Nd. Finally, if Dn(a)Tn(f)n is any matrix-sequenceextracted from Dn(a) Tn(f)n∈Nd , then Dn(a) Tn(f)−Dn(a)Tn(f)n is zero-distributed by (6.31) and Theorem 2.10, and so,by Theorem 6.9, Dn(a) Tn(f)n ∼LT a⊗ f .

We end with a proposition that provides a relation between LT and sLT sequences. This proposition will be used in the next sectionto show that any LT sequence is a GLT sequence, and, implicitly, that the definition of GLT sequences, originally formulated in [89, 90]in terms of sLT sequences, can be equivalently formulated in terms of LT sequences.

Proposition 6.4. Let Ann ∼LT a⊗ f . Then, for any m ∈ N there exist matrix-sequences A(i,m)n n ∼sLT a⊗ fi,m, i = 1, . . . , Nm,

such that∑Nmi=1 fi,m → f in L1([−π, π]d) when m→∞ and

∑Nmi=1A

(i,m)n nm is an a.c.s. for Ann.

Proof. Take any sequence of d-variate trigonometric polynomials fm such that fm → f in L1([−π, π]d). We recall that such a se-quence exists because the set of d-variate trigonometric polynomials is dense in L1([−π, π]d). By definition, any d-variate trigonometricpolynomial is a finite sum of separable d-variate trigonometric polynomials. Hence, we can write

fm =

Nm∑i=1

fi,m,

for some separable d-variate trigonometric polynomials fi,m, i = 1, . . . , Nm. Take arbitrary matrix-sequences A(i,m)n n ∼sLT

a ⊗ fi,m, i = 1, . . . , Nm. In view of Theorem 6.8, one can choose, for example, A(i,m)n = Dn(a)Tn(fi,m). By Exercise 6.1,

∑Nmi=1A

(i,m)n n ∼LT a⊗ (

∑Nmi=1 fi,m) = a⊗ fm. Hence,

∑Nmi=1A

(i,m)n nm is an a.c.s. for Ann by Theorem 6.9.

77

Chapter 7

GLT sequences

In this chapter, we develop the theory of Generalized Locally Toeplitz sequences. In particular, we prove all the statements contained initems GLT 1 – GLT 8 of Section 1.2.

7.1 DefinitionWe first report a ‘corrected’ version of the original definition of GLT sequences; cf. [89, Definition 2.3] and [90, Definition 1.5]. Thisdefinition is formulated in terms of a.c.s. parameterized by a positive ε→ 0 (see Definition 4.3).

Definition 7.1 (GLT sequence). Let Ann be a matrix-sequence, n ∈ Nd, and let κ : [0, 1]d× [−π, π]d → C be a measurable function.We say that Ann is a Generalized Locally Toeplitz (GLT) sequence with symbol κ, and we write Ann ∼GLT κ, if the followingcondition is met.

For every ε > 0 there exists a finite number of sLT sequences A(i,ε)n n ∼sLT ai,ε ⊗ fi,ε, i = 1, . . . , Nε, such that:

•∑Nεi=1 ai,ε ⊗ fi,ε → κ in measure over [0, 1]d × [−π, π]d when ε→ 0;

•∑Nεi=1A

(i,ε)n n

ε>0

is an a.c.s. of Ann for ε→ 0.

From now on, if we write Ann ∼GLT κ, it is understood that κ : [0, 1]d × [−π, π]d → C is measurable.It is clear that any sLT sequence is a GLT sequence. Indeed, if Ann ∼sLT a⊗ f then Ann ∼GLT a⊗ f . To see this, it suffices

to take, in Definition 7.1, Nε = 1, A(1,ε)n n = Ann, a1,ε = a and fi,ε = f , for all ε > 0. Proposition 6.4 and the first characterization

of GLT sequences (Proposition 7.1) imply that any LT sequence is a GLT sequence. More precisely,

Ann ∼LT a⊗ f ⇒ Ann ∼GLT a⊗ f. (7.1)

Proposition 7.1. We have Ann ∼GLT κ if and only if the following condition is met.

For every m varying in some infinite subset of N there exists a finite number of sLT sequences A(i,m)n n ∼sLT ai,m ⊗

fi,m, i = 1, . . . , Nm, such that:

•∑Nmi=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d when m→∞;

•∑Nmi=1A

(i,m)n n

m

is an a.c.s. for Ann.

Proof. If Ann ∼GLT κ, then the condition of the proposition holds with

ai,m = ai,ε(m), fi,m = fi,ε(m), A(i,m)n n = A(i,ε(m))

n n, Nm = Nε(m),

where ai,ε, fi,ε, A(i,ε)n n are as in Definition 7.1 and ε(m)m is any sequence of positive numbers such that ε(m)→ 0 as m→∞.

Conversely, suppose that the condition of the proposition holds. LetM ⊆ N be the infinite subset of N where m varies. Then, thecondition of Definition 7.1 holds with

ai,ε = ai,m(ε), fi,ε = fi,m(ε), A(i,ε)n n = A(i,m(ε))

n n, Nε = Nm(ε),

where m(ε)ε>0 ⊆M is any family of indices such that m(ε)→∞ when ε→ 0, and ai,m, fi,m, A(i,m)n n are as in the statement of

the proposition. Thus, Ann ∼GLT κ.

78

Proposition 7.1 is essentially the same as Definition 7.1, but it is easier to handle, because it is based on the standard notion of a.c.s.(Definition 4.1).

Corollary 7.1. If Ann ∼LT a⊗ f then Ann ∼GLT a⊗ f .

Proof. It follows directly from Proposition 7.1 and Proposition 6.4.

7.2 Singular value and eigenvalue distribution of GLT sequencesWe begin with a lemma concerning the singular value distribution of a finite sum of LT sequences. The lemma will be used in the proofof the singular value distribution result for GLT sequences (Theorem 7.1).

Lemma 7.1. Let A(i)n n ∼LT ai ⊗ fi, i = 1, . . . , p. Then

∑pi=1A

(i)n n ∼σ

∑pi=1 ai ⊗ fi.

Proof. Choose a sequence m = m(m)m ⊆ Nd such that m → ∞ when m → ∞. From the properties of a.c.s., see Proposition 4.1,and from the definition of LT sequences, we know that

∑pi=1 LT

mn (ai, fi)nm is an a.c.s. for

∑pi=1A

(i)n n. By Theorem 6.2,

∑pi=1 LT

mn (ai, fi)n ∼σ φm and φm → φ[ |

∑pi=1 ai⊗fi| ] pointwise over Cc(R), where we recall that the functional φ[g] is defined in

(2.15). Hence, by Theorem 4.3, ∑pi=1A

(i)n n ∼σ

∑pi=1 ai ⊗ fi.

Theorem 7.1. If Ann ∼GLT κ then Ann ∼σ κ.

Proof. By Proposition 7.1, there exist matrix-sequences A(i,m)n n ∼LT ai,m ⊗ fi,m, i = 1, . . . , Nm, such that

∑Nmi=1 ai,m ⊗ fi,m → κ

in measure and ∑Nmi=1A

(i,m)n nm is an a.c.s. for Ann. By Lemma 7.1, we have

∑Nmi=1A

(i,m)n n ∼σ

∑Nmi=1 ai,m ⊗ fi,m. Since∑Nm

i=1 ai,m ⊗ fi,m → κ in measure, all the assumptions of Corollary 4.1 are satisfied and so Ann ∼σ κ.

As a consequence of Theorem 7.1, every GLT sequence is s.u. in the sense of Definition 4.2 (see Proposition 4.3). Using Theorem 7.1,we show in Proposition 7.3 that the symbol of a GLT sequence is unique. For the proof of Proposition 7.3 we point out that any linearcombination of GLT sequences is again a GLT sequence whose symbol is given by the same linear combination of the symbols. Thisis one of the most elementary results in the world of the algebraic properties possessed by GLT sequences. These properties will beinvestigated in Section 7.5 and give rise to the so-called GLT algebra.

Proposition 7.2. Let Ann ∼GLT κ and Bnn ∼GLT ξ. Then, A∗nn ∼GLT κ and αAn+βBnn ∼GLT ακ+βξ for all α, β ∈ C.

The proof of Proposition 7.2 is easy: it suffices to write the meaning of Ann ∼GLT κ and Bnn ∼GLT ξ (using the characteri-zation of Proposition 7.1), and to apply Proposition 4.1, taking into account that Cnn ∼LT a ⊗ f implies C∗nn ∼LT a ⊗ f (by thedefinition of LT sequences, eq. (6.6) and Remark 4.3); the details are left to the reader.

Proposition 7.3. Assume that Ann ∼GLT κ and Ann ∼GLT ξ. Then κ = ξ a.e. in [0, 1]d × [−π, π]d.

Proof. By Proposition 7.2, ON(n)n ∼GLT κ− ξ. Therefore, by Theorem 7.1, for all test functions F ∈ Cc(R) we have

F (0) =1

(2π)d

∫[0,1]d×[−π,π]d

F (|κ(x,θ)− ξ(x,θ)|)dxdθ. (7.2)

This means that φ[|κ−ξ|] = φ[0] and so, by Remark 2.1, |κ− ξ| = 0 a.e.

Proposition 7.4. Let Ann ∼GLT κ and assume that the matrices An are Hermitian. Then κ ∈ R a.e.

Proof. Since the matrices An are Hermitian, by Proposition 7.2 we have Ann ∼GLT κ and Ann ∼GLT κ. Thus, by Proposition 7.3,κ = κ a.e., i.e., κ ∈ R a.e.

The next lemma deals with the spectral distribution of the real part of a finite sum of LT sequences. The lemma will be used in theproof of the eigenvalue distribution result for (Hermitian) GLT sequences (Theorem 7.2).

Lemma 7.2. Let A(i)n n ∼LT ai ⊗ fi, i = 1, . . . , p. Then <(

∑pi=1A

(i)n )n ∼λ <(

∑pi=1 ai ⊗ fi).

Proof. Choose a sequence m = m(m)m ⊆ Nd such that m → ∞ when m → ∞. From the properties of a.c.s., see Re-mark 4.3 and Proposition 4.1, and from the definition of LT sequences, <(

∑pi=1 LT

mn (ai, fi))nm is an a.c.s. for <(

∑pi=1A

(i)n )n.

By Theorem 6.3, <(∑pi=1 LT

mn (ai, fi))n ∼λ φm and φm → φ[<(

∑pi=1 ai⊗fi)]. Hence, by Theorem 4.5, <(

∑pi=1A

(i)n )n ∼λ

<(∑pi=1 ai ⊗ fi).

Theorem 7.2. If Ann ∼GLT κ and the matrices An are Hermitian, then Ann ∼λ κ.

79

Proof. By Proposition 7.1, there exist matrix-sequences A(i,m)n n ∼LT ai,m ⊗ fi,m, i = 1, . . . , Nm, such that

∑Nmi=1 ai,m ⊗ fi,m → κ

in measure and ∑Nmi=1A

(i,m)n nm is an a.c.s. for Ann. Since the matrices An are Hermitian, <(

∑Nmi=1A

(i,m)n )nm is another

a.c.s. for An = <(An)n, and it is formed by Hermitian matrices. By Lemma 7.2, <(∑Nmi=1A

(i,m)n )n ∼λ <(

∑Nmi=1 ai,m ⊗ fi,m).

The function κ is real a.e. by Proposition 7.4, and so from∑Nmi=1 ai,m ⊗ fi,m → κ (in measure) we get <(

∑Nmi=1 ai,m ⊗ fi,m) → κ (in

measure). All the assumptions of Corollary 4.2 are then satisfied, and it follows that Ann ∼λ κ.

Remark 7.1. By Proposition 7.2, Lemmas 6.2 and 7.1 are particular cases of Theorem 7.1, and Lemma 7.2 is a particular case ofTheorem 7.2.

We end this section with a spectral distribution result for GLT sequences formed by perturbed Hermitian matrices. This result is aconsequence of Theorem 7.2 and Corollary 3.1.

Theorem 7.3. Let An = Xn + Yn, with Ann ∼GLT κ or Xnn ∼GLT κ, and assume that:

1. every Xn is Hermitian;

2. ‖Xn‖, ‖Yn‖ ≤ C for all n, with C a constant independent of n;

3. ‖Yn‖1 = o(N(n)) as n→∞.

Then Ann ∼GLT κ and Ann ∼λ κ.

Proof. By Theorem 2.10, Ynn is a zero-distributed sequence, hence Ynn ∼GLT 0. Since An = Xn + Yn, it follows fromProposition 7.2 that Ann ∼GLT κ if and only if Xnn ∼GLT κ. Considering that at least one of these two conditions holds byhypothesis, we conclude that both the conditions hold. Moreover, Xnn ∼λ κ by Theorem 7.2 as the matrices Xn are Hermitian. Allthe assumptions of Corollary 3.1 are then satisfied and the thesis follows.

7.3 Approximation results for GLT sequencesTheorem 7.4 is the main approximation result for GLT sequences. It is the same as Corollaries 4.1–4.2 with ‘∼σ’ and ‘∼λ’ replaced by‘∼GLT’, and it is particularly useful to show that a given matrix-sequence Ann is a GLT sequence. Applications of Theorem 7.4 willbe seen in Section 7.4 and, especially, in Section 7.5.

Theorem 7.4. Let Ann be a matrix-sequence and let κ : [0, 1]d × [−π, π]d → C be a measurable function. Suppose that:

1. Bn,mnm is an a.c.s. for Ann;

2. Bn,mn ∼GLT κm for every m;

3. κm → κ in measure.

Then Ann ∼GLT κ.

Proof. Since Bn,mn ∼GLT κm, for every m and every h (varying in some infinite subset H ⊆ N) there exists a finite number ofmatrix-sequences A(i,h)

n,mn ∼sLT ai,h,m ⊗ fi,h,m, i = 1, . . . , Nh,m, such that:

•∑Nh,mi=1 ai,h,m ⊗ fi,h,m → κm in measure over [0, 1]d × [−π, π]d when h→∞;

•∑Nh,mi=1 A

(i,h)n,mn

h

is an a.c.s. for Bn,mn.

Hence, for every m and every h there exists nh,m such that, for n ≥ nh,m,

Bn,m =

Nh,m∑i=1

A(i,h)n,m +Rn,h,m +Nn,h,m,

rank(Rn,h,m) ≤ c(h,m)N(n), ‖Nn,h,m‖ ≤ ω(h,m),

where limh→∞ c(h,m) = limh→∞ ω(h,m) = 0. Let δmm be a sequence such that δm 0. Since∑Nh,mi=1 ai,h,m ⊗ fi,h,m → κm in

measure when h→∞, for every m we have

µ(m,h, δm) = µ2d

∣∣∣∣∣∣Nh,m∑i=1

ai,h,m ⊗ fi,h,m − κm

∣∣∣∣∣∣ ≥ δm→ 0 as h→∞.

80

Now we recall that Bn,mnm is an a.c.s. for Ann: for every m there exists nm such that, for n ≥ nm,



where limm→∞ c(m) = limm→∞ ω(m) = 0. It follows that, for every m, every h, and every n ≥ max(nm, nh,m),

An =

Nh,m∑i=1

A(i,h)n,m + (Rn,h,m +Rn,m) + (Nn,h,m +Nn,m),

rank(Rn,h,m +Rn,m) ≤ (c(h,m) + c(m))N(n), ‖Nn,h,m +Nn,m‖ ≤ ω(h,m) + ω(m).

Choose a sequence hmm ⊆ H such that hm ∞ and

limm→∞

c(hm,m) = limm→∞

ω(hm,m) = limm→∞

µ(m,hm, δm) = 0.

Then, for every m and every n ≥ max(nm, nhm,m),

An =

Nhm,m∑i=1

A(i,hm)n,m + (Rn,hm,m +Rn,m) + (Nn,hm,m +Nn,m),

rank(Rn,hm,m +Rn,m) ≤ (c(hm,m) + c(m))N(n), ‖Nn,hm,m +Nn,m‖ ≤ ω(hm,m) + ω(m).

It follows that ∑Nhm,mi=1 A

(i,hm)n,m nm is an a.c.s. for Ann. Moreover, A(i,hm)

n,m n ∼sLT ai,hm,m ⊗ fi,hm,m for all m and alli = 1, . . . , Nhm,m, and

∑Nhm,mi=1 ai,hm,m ⊗ fi,hm,m → κ in measure over [0, 1]d × [−π, π]d when m→∞. Indeed, for any δ > 0,

µ2d

∣∣∣∣∣∣Nhm,m∑i=1

ai,hm,m ⊗ fi,hm,m − κ

∣∣∣∣∣∣ ≥ δ ≤ µ2d

∣∣∣∣∣∣Nhm,m∑i=1

ai,hm,m ⊗ fi,hm,m − κm

∣∣∣∣∣∣ ≥ δ/2+ µ2d |κm − κ| ≥ δ/2 ,

µ2d |κm − κ| ≥ δ/2 → 0 by assumption (since κm → κ in measure), and

µ2d

∣∣∣∣∣∣Nhm,m∑i=1

ai,hm,m ⊗ fi,hm,m − κm

∣∣∣∣∣∣ ≥ δ/2 = µ(m,hm, δ/2)

tends to 0, because it is eventually less than µ(m,hm, δm). Thus, Ann ∼GLT κ by Proposition 7.1.

The approximation result for GLT sequences stated in Theorem 7.4 admits the following converse, which can be seen as anotherapproximation result for GLT sequences. In fact, Theorems 7.4–7.5 look like a characterization of GLT sequences in terms of a.c.s.

Theorem 7.5. Let Ann be a matrix-sequence and let Bn,mnm be a sequence of matrix-sequences. Suppose that:

1. Ann ∼GLT κ;

2. Bn,mn ∼GLT κm for every m.

Then, Bn,mnm is an a.c.s. for Ann if and only if κm → κ in measure.

Proof. (⇐) Assume that 1–2 hold and κm → κ in measure. By Proposition 7.2, An − Bn,mn ∼GLT κ − κm for each m. Hence, byTheorem 7.1, An −Bn,mn ∼σ κ− κm, with κ− κm tending to 0 in measure by hypothesis. Hence, by Corollary 4.4, Bn,mn is ana.c.s. for Ann.

(⇒) Assume that 1–2 hold and Bn,mnm is an a.c.s. for Ann. Then, An−Bn,mnm is an a.c.s. of ON(n)n. Moreover,An −Bn,mn ∼GLT κ− κm by Proposition 7.2, and so An −Bn,mn ∼σ κ− κm by Theorem 7.1. Hence, κm → κ in measure byProposition 4.5.

Corollary 7.2. Let Ann ∼GLT κ. Then, for all functions ai,m, fi,m, i = 1, . . . , Nm, with the following properties:

∗ ai,m : [0, 1]d → C is Riemann-integrable and fi,m ∈ L1([−π, π]d);

∗∑Nmi=1 ai,m ⊗ fi,m → κ in measure when m→∞;

81

it holds that ∑Nmi=1Dn(ai,m)Tn(fi,m)nm is an a.c.s. for Ann. In particular, Ann admits an a.c.s. of the form

Nm∑

j=−Nm

Dn(a(m)j )Tn(eij·θ)

n

m

, a(m)j ∈ C∞([0, 1]d), Nm ∈ Nd, (7.3)

where∑Nm

j=−Nma

(m)j (x)eij·θ → κ(x,θ) a.e.

Proof. Since Dn(ai,m)Tn(fi,m)n ∼GLT ai,m ⊗ fi,m, we have ∑Nmi=1Dn(ai,m)Tn(fi,m)n ∼GLT

∑Nmi=1 ai,m ⊗ fi,m by Proposi-

tion 7.2. Therefore, the thesis follows from Theorem 7.5 applied with Bn,m =∑Nmi=1Dn(ai,m)Tn(fi,m) and κm =

∑Nmi=1 ai,m ⊗ fi,m.

To obtain an a.c.s. for Ann of the form (7.3), it suffices to use the result of the corollary in combination with Lemma 2.7.

Remark 7.2. Let Ann, Bn,mn be matrix-sequences and let κ, κm : [0, 1]d × [−π, π]d → C be measurable functions. Consider thefollowing four conditions:

(1) Ann ∼GLT κ;

(2) Bn,mn ∼GLT κm for every m;

(3) Bn,mnm is an a.c.s. for Ann;

(4) κm → κ in measure.

Theorems 7.4–7.5 show that ‘(1)∧ (2)∧ (3)⇒ (4)’, ‘(1)∧ (2)∧ (4)⇒ (3)’ and ‘(2)∧ (3)∧ (4)⇒ (1)’.The implication ‘(1)∧ (3)∧ (4)⇒ (2)’, written in this way, is meaningless. However, a natural modification reads as follows:

‘(1)∧ (3)⇒ there exists a measurable function κm, tending to κ in measure, such that Bn,mn ∼GLT κm for all sufficiently largem’. This statement is false in general. As a counterexample, take An = On and Bn,m = (1 + (−1)n) 1

mIn, as in Remark 4.2. Since wehave seen in Remark 4.2 that the relation Bn,mn ∼σ φm cannot hold for any functional φm defined over Cc(R), in particular there isno κm such that Bn,mn ∼GLT κm.

As a first application of Theorem 7.4, we show in Theorem 7.6 that Dn(a)n ∼GLT a ⊗ 1 for all functions a : [0, 1]d → C thatare continuous a.e. in [0, 1]d. Since any Riemann-integrable function is continuous a.e. (see Section 2.5), Theorem 7.6 is an extension ofTheorem 6.5.

Theorem 7.6. If a : [0, 1]d → C is continuous a.e. and Dn(a)n is any matrix-sequence extracted from Dn(a)n∈Nd , thenDn(a)n ∼GLT a⊗ 1.

Proof. We first note that it suffices to prove the theorem under the additional assumption that a is real and nonnegative. Indeed, assumingwe have proved the theorem in this case, the general case follows from Proposition 7.2 and from the observations that

a = <(a) + i=(a) = <(a)+ −<(a)− + i=(a)+ − i=(a)−,

Dn(a) = Dn(<(a)+)−Dn(<(a)−) + iDn(=(a)+)− iDn(=(a)−),

where b+ = max(b, 0) and b− = −min(b, 0) are nonnegative a.e. continuous functions whenever b : [0, 1]d → R is a.e. continuous.Let a : [0, 1]d → [0,∞) be a nonnegative a.e. continuous function. Let am be the truncation of a at level m, i.e.,

am(x) =

a(x) if a(x) ≤ m,m if a(x) > m.

Since am is bounded and continuous a.e., am is Riemann-integrable. Hence Dn(am)n ∼GLT am ⊗ 1 by Theorem 6.5. Moreover, it isclear that am → a pointwise, so am ⊗ 1 → a⊗ 1 in measure. We show that Dn(am)nm is an a.c.s. of Dn(a)n, after which theapplication of Theorem 7.4 concludes the proof. To show that Dn(am)nm is an a.c.s. of Dn(a)n, we will prove that for every mthere exists nm such that, for n ≥ nm,

rank(Dn(a)−Dn(am)) = rank(Dn(a− am)) ≤ ε(m)N(n), (7.4)

where ε(m)→ 0 as m→∞. For every k, consider the partition of (0, 1]d given by

Ii,k =

(i− 1

2k,i

2k

]=

(i1 − 1

2k,i12k

]× · · · ×

(id − 1

2k,id2k

], i = 1, . . . , 2k1, (7.5)

and let

am,k =

2k1∑i=1

(sup

y∈Ii,kam(y)

)χIi,k .

82

For all x ∈ (0, 1]d and all m, k, we have

0 ≤ am(x) ≤ am,k+1(x) ≤ am,k(x) ≤ supy∈(0,1]d

am(y) ≤ m.

Recalling that a ≥ 0 on [0, 1]d, for all m,n, k we have

rank(Dn(a− am)) = #

j ∈ 1, . . . ,n : a

( jn

)− am

( jn

)6= 0

≤ #

j ∈ 1, . . . ,n : a

( jn

)≥ m

= #

j ∈ 1, . . . ,n : am

( jn

)= m

≤ #

j ∈ 1, . . . ,n : am,k

( jn

)= m

= #

j ∈ 1, . . . ,n :

j

n∈ am,k = m

= #

x ∈

jn

: j = 1, . . . ,n

: x ∈ am,k = m

= #

( jn

: j = 1, . . . ,n⋂

am,k = m)≤ µdam,k = mN(n+ 2k1)

N(n)N(n).

The last inequality is justified by the fact that am,k = m is a finite union of squares from (7.5) (say rm,k squares), and each of thesesquares cannot contain more than (n1/2

k + 1)(n2/2k + 1) · · · (nd/2k + 1) = N(n + 2k1)/2dk grid points of jn : j = 1, . . . ,n,

implying that am,k = m contains at most rm,kN(n+ 2k1)/2dk = µdam,k = mN(n+ 2k1) grid points. Now, am,k → am a.e. in[0, 1]d by Lemma 2.8, and the convergence is monotone, so

limk→∞

µdam,k = m = µd

( ∞⋂k=1

am,k = m)

= µdam = m = µda ≥ m.

For every m we choose km such that

µdam,km = m ≤ µda ≥ m+1

m.

Subsequently, we choose nm such that, for n ≥ nm,N(n+ 2km1)

N(n)≤ 2;

this choice is possible because

limn→∞

N(n+ 2km1)

N(n)= 1.

With the above choices, we see that (7.4) holds with ε(m) = 2(µda ≥ m+ 1

m

).

7.3.1 Topological closure of GLT sequencesTheorem 7.4 allows us to make an interesting observation concerning the topological closure of GLT sequences in the space of all matrix-sequences. This observation is connected with Section 4.1 and requires some background on general topology; the reader who is notinterested in topological issues may certainly decide to skip this part on first reading.

Fix a sequence of d-indices n = n(n)n ⊆ Nd such that n→∞ as n→∞. Let E be the set of all matrix-sequences,

E =Ann : Ann is a matrix-sequences

,

which is a topological (pseudometric) space with respect to the a.c.s. topology τa.c.s. generated by the a.c.s. distance da.c.s. introduced inSection 4.1. Let M be the set of all measurable functions defined on [0, 1]d × [−π, π]d,

M = κ : [0, 1]d × [−π, π]d → C : κ is measurable,

which is a topological (pseudometric) space with respect to the topology τµ associated with the convergence in measure (hereinaftercalled the ‘measure topology’). This topology is generated by the pseudometric

dµ(κ, ξ) = q(κ− ξ), q(ψ) = infµ2d|ψ| ≥ α+ α : α > 0

.1

Consider the product spaceE ×M = (Ann, κ) : Ann ∈ E , κ ∈M

1Note the formal analogy between the definitions of dµ and da.c.s. (see (4.6)), and the definitions of q(·) and p(·) (see (4.4)): it seems as if the a.c.s. topology is thecounterpart for matrix-sequences of the measure topology for measurable functions.

83

equipped with the product topology τa.c.s. × τµ; this is again a pseudometric space with respect to the ‘product distance’

(da.c.s. × dµ)((Ann, κ), (Bnn, ξ)) = da.c.s.(Ann, Bnn) + dµ(κ, ξ).

Now let A be the subset of E ×M consisting of all the ‘GLT pairs’, i.e.,

A = (Ann, κ) : Ann ∼GLT κ ⊆ E ×M.

Then, Theorem 7.4 can be summarized by saying that the set of GLT pairs A is a closed subset of E ×M. Indeed, as stated in Theorem7.4, if a sequence of GLT sequences Bn,mn ∼GLT κm converges in the a.c.s. topology to another matrix-sequence Ann and if thecorresponding sequence of symbols κm converges in measure to a measurable function κ (i.e., if a sequence of GLT pairs (Bn,mn, κm)converges to a pair (Ann, κ) in E ×M), then Ann ∼GLT κ (i.e., (Ann, κ) is a GLT pair).

7.4 Characterizations of GLT sequencesUsing Theorem 7.4, we show in Proposition 7.5 that GLT sequences could be defined in terms of LT sequences instead of sLT sequences.Proposition 7.5 is then a characterization of GLT sequences in terms of LT sequences and, specifically, it is the same as Proposition 7.1with ‘sLT’ replaced by ‘LT’.

Proposition 7.5. We have Ann ∼GLT κ if and only if the following condition is met.

For everym varying in some infinite subset of N there exists a finite number of LT sequences A(i,m)n n ∼LT ai,m⊗fi,m, i =

1, . . . , Nm, such that:

•∑Nmi=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d when m→∞;

•∑Nmi=1A

(i,m)n n

m

is an a.c.s. for Ann.

Proof. It is clear that, if Ann ∼GLT κ, then the condition holds by Proposition 7.1. Conversely, suppose that the condition holds.Then,

∑Nmi=1A

(i,m)n nm is an a.c.s. for Ann by hypothesis,

∑Nmi=1A

(i,m)n n ∼GLT

∑Nmi=1 ai,m ⊗ fi,m by Corollary 7.1 and

Proposition 7.2, and∑Nmi=1 ai,m ⊗ fi,m → κ in measure by hypothesis. Hence, the thesis follows from Theorem 7.4.

It is clear that Proposition 7.5, as well as Proposition 7.1, may be taken as the definition of GLT sequences.The next result is a characterization theorem for GLT sequences. All the characterizations provided in the theorem have already been

proved in the previous section, but it is anyway useful to collect them in a single statement.

Theorem 7.7 (characterizations of GLT sequences). Let Ann be a matrix-sequence and let κ : [0, 1]d × [−π, π]d → C be ameasurable function. Then, the following conditions are equivalent.

1. Ann ∼GLT κ.

2. For all sequences κmm, Bn,mnm with the following properties:

∗ Bn,mn ∼GLT κm for every m;

∗ κm → κ in measure;

it holds that Bn,mnm is an a.c.s. for Ann.

3. There exist functions ai,m, fi,m, i = 1, . . . , Nm, m ∈ N, such that:

∗ ai,m : [0, 1]d → C belongs to C∞([0, 1]d) and fi,m is a trigonometric monomial belonging to eij·θ : j ∈ Zd;∗∑Nmi=1 ai,m ⊗ fi,m → κ a.e.;

∗ ∑Nmi=1Dn(ai,m)Tn(fi,m)nm is an a.c.s. for Ann.

4. There exist sequences κmm, Bn,mnm such that:

∗ Bn,mn ∼GLT κm for every m;

∗ κm → κ in measure;

∗ Bn,mnm is an a.c.s. for Ann.

Proof. The implication (1⇒ 2) follows from Theorem 7.5. The implication (2⇒ 3) follows from the observation that, by Lemma 2.7,we can find functions ai,m, fi,m, i = 1, . . . , Nm, m ∈ N, with the first two properties specified in item 3. The implication (3⇒ 4) isobvious (take Bn,m =

∑Nmi=1Dn(ai,m)Tn(fi,m) and κm =

∑Nmi=1 ai,m ⊗ fi,m). Finally, the implication (4⇒ 1) is Theorem 7.4.

84

7.5 The GLT algebraWe investigate in this section the important algebraic properties possessed by GLT sequences, which give rise to the so-called GLT algebra.In short, these properties establish that, if A(1)

n n, . . . , A(r)n n are given GLT sequences with symbols κ1, . . . , κr, respectively, and if

An = ops(A(1)n , . . . , A

(r)n ) is obtained from A

(1)n , . . . , A

(r)n by means of certain operations ‘ops’, then Ann is a GLT sequence with

symbol κ = ops(κ1, . . . , κr) obtained by performing the same operations on the symbols κ1, . . . , κr.

Theorem 7.8. Suppose that Ann ∼GLT κ and Bnn ∼GLT ξ. Then:

1. A∗nn ∼GLT κ;

2. αAn + βBnn ∼GLT ακ+ βξ, for all α, β ∈ C;

3. AnBnn ∼GLT κξ.

Proof. The first two statements have already been settled before (see Proposition 7.2). We prove the third statement. By assumption andProposition 7.5, there exist LT sequences A(i,m)

n n ∼LT ai,m⊗fi,m, i = 1, . . . , Nm, and B(j,m)n n ∼LT bj,m⊗gj,m, j = 1, . . . ,Mm,

such that:

•∑Nmi=1 ai,m ⊗ fi,m → κ in measure and

∑Mm

j=1 bj,m ⊗ gj,m → ξ in measure;

• ∑Nmi=1A

(i,m)n nm is an a.c.s. for Ann and

∑Mm

j=1 B(j,m)n nm is an a.c.s. for Bnn.

Thanks to Theorem 7.7 (item 3), the functions fi,m, gj,m may be supposed to be in L∞([−π, π]d); actually, they might be supposed tobe trigonometric monomials, the functions ai,m, bj,m might be supposed to belong to C∞([0, 1]d), and A(i,m)

n n, B(j,m)n n might be

chosen of the form Dn(ai,m)Tn(fi,m)n, Dn(bj,m)Tn(gj,m)n. By Theorem 7.1, Ann ∼σ κ and Bnn ∼σ ξ, which implies,by Proposition 4.3, that Ann and Bnn are s.u. Thus, by Proposition 4.4,

(Nm∑i=1

A(i,m)n

)Mm∑j=1

B(j,m)n

n

m

=

Nm∑i=1

Mm∑j=1

A(i,m)n B(j,m)

n

n

m

is an a.c.s. for AnBnn. Since fi,m, gj,m ∈ L∞([−π, π]d), by Theorem 6.7 we have A(i,m)n B

(j,m)n n ∼LT ai,mbj,m ⊗ fi,mgj,m, i =

1, . . . , Nm, j = 1, . . . ,Mm. Finally,

Nm∑i=1

Mm∑j=1

ai,mbj,m ⊗ fi,mgj,m =

(Nm∑i=1

ai,m ⊗ fi,m

)Mm∑j=1

bj,m ⊗ gj,m

→ κξ

in measure by Lemma 2.4, and the proof is over.

Corollary 7.3. Let r, q1, . . . , qr ∈ N and, for i = 1, . . . , r and j = 1, . . . , qi, let A(i,j)n n ∼GLT κij . Then,

r∑i=1

qi∏j=1

A(i,j)n

n

∼GLT

r∑i=1

qi∏j=1

κij .

The results we have seen so far are enough to conclude that the set of GLT sequences is a *-algebra over the complex field C. Moreprecisely, fix any sequence of d-indices n = n(n)n ⊆ Nd such that n→∞ when n→∞; then,

A =Ann : Ann ∼GLT κ for some measurable function κ : [0, 1]d × [−π, π]d → C

(7.6)

is a *-algebra over C, with respect to the natural operations of Hermitian transposition, addition, scalar-multiplication and product ofmatrix-sequences (see (2.62)). We call A the GLT algebra. This algebra contains the algebra generated by zero-distributed sequences,by sequences of multilevel Toeplitz matrices, and by sequences of multilevel diagonal sampling matrices, because we have seen that allthese matrix-sequences fall in the class of GLT sequences. To be precise, let

B =

r∑i=1

qi∏j=1

X(i,j)n

: r, q1, . . . , qr ∈ N, X(i,j)

n n ∈ B0 for all i = 1, . . . , r and j = 1, . . . , qi

,

85

where

B0 =Tn(g)n : g ∈ L1([−π, π]d)

∪Dn(a)n : a : [0, 1]d → C is Riemann-integrable

∪ Znn : Znn ∼σ 0 .

Then, B is the algebra generated by B0 and B ⊆ A . We are going to see in Theorems 7.9–7.10 that the GLT algebra enjoys other niceproperties, in addition to those of Theorem 7.8, which make it look like a ‘big container’, closed under any type of ‘regular’ operation.

Theorem 7.9 provides a positive answer to a question raised in [91]. Incidentally, we recall that, as noted at the end of Section 4.3, inpaper [91] the authors proved an ‘a.c.s. result’ which is formally identical to the ‘GLT result’ contained in Theorem 7.9.

Theorem 7.9. Let Ann ∼GLT κ and suppose that the matrices An are Hermitian. Then

f(An)n ∼GLT f(κ)

for any continuous function f : R→ C.2

Proof. For each M > 0, let pm,Mm be a sequence of polynomials that converges uniformly to f over the compact interval [−M,M ]:

limm→∞

‖f − pm,M‖∞,[−M,M ] = 0.

For every M > 0 and every m,n, writef(An) = pm,M (An) + f(An)− pm,M (An). (7.7)

Since any GLT sequence is s.u. (by Theorem 7.1 and Proposition 4.3), the sequence Ann is s.u. Hence, by Proposition 4.2, for allM > 0 there exists nM such that, for n ≥ nM ,

An = An,M + An,M , rank(An,M ) ≤ r(M)N(n), ‖An,M‖ ≤M, (7.8)

where limM→∞ r(M) = 0. However, for the purpose of this proof we need a splitting of the form (7.8) such that g(An,M + An,M ) =

g(An,M ) + g(An,M ) for all functions g : R → C. Luckily, the matrices An are Hermitian and, consequently, such a splitting can beconstructed by following the same argument used in the proof of Proposition 4.2. For the reader’s convenience, we include the details ofthe construction. By definition, since Ann is s.u. and formed by Hermitian matrices, for every M > 0 there exists nM such that, forn ≥ nM ,

#i ∈ 1, . . . , N(n) : |λi(An)| > MN(n)

≤ r(M),

where limM→∞ r(M) = 0. Let An = UnΛnU∗n be a spectral decomposition of An. Let Λn,M be the matrix obtained from Λn by

setting to 0 all the eigenvalues of An whose absolute value is less than or equal toM , and let Λn,M = Λn− Λn,M be the matrix obtainedfrom Λn by setting to 0 all the eigenvalues of An whose absolute value is greater than M . Then, for M > 0 and n ≥ nM ,

An = UnΛnU∗n = UnΛn,MU

∗n + UnΛn,MU

∗n = An,M + An,M ,

where An,M = UnΛn,MU∗n and An,M = UnΛn,MU

∗n. The matrices An,M , An,M constructed in this way are Hermitian, satisfy the

properties in (7.8) and, moreover,

g(An,M + An,M ) = g(An,M ) + g(An,M ) = Un g(Λn,M )U∗n + Un g(Λn,M )U∗n

for all functions g : R→ C.Going back to (7.7), for every M > 0, every m and every n ≥ nM we can write

f(An) = pm,M (An) + f(An,M ) + f(An,M )− pm,M (An,M )− pm,M (An,M )

= pm,M (An) + (f − pm,M )(An,M ) + (f − pm,M )(An,M ). (7.9)

The term (f − pm,M )(An,M ) can be split in the sum of two terms Rn,m,M + N ′n,m,M : Rn,m,M is obtained from (f − pm,M )(An,M )

by setting to 0 all the eigenvalues that are equal to (f − pm,Mm)(0), so that rank(Rn,m,M ) = rank(An,M ); while N ′n,m,M is obtained

from (f − pm,M )(An,M ) by setting to 0 all the eigenvalues that are different from (f − pm,Mm)(0). Let N ′′n,m,M = (f − pm,M )(An,M )and Nn,m,M = N ′n,m,M +N ′′n,m,M . From (7.9), for every M > 0, every m and every n ≥ nM we have

f(An) = pm,M (An) +Rn,m,M +Nn,m,M , (7.10)

2Recall from Proposition 7.4 that κ ∈ R a.e., because every An is Hermitian. Hence, f(κ) is well-defined.

86

and, by our construction,

rank(Rn,m,M ) = rank(An,M ) ≤ r(M)N(n),

‖Nn,m,M‖ ≤ |f(0)− pm,M (0)|+ ‖f − pm,M‖∞,[−M,M ] ≤ 2‖f − pm,M‖∞,[−M,M ].(7.11)

Choose a sequence Mmm such that, when m→∞,

Mm →∞, ‖f − pm,Mm‖∞,[−Mm,Mm] → 0. (7.12)

Then, for every m and every n ≥ nMm,

f(An) = pm,Mm(An) +Rn,m,Mm

+Nn,m,Mm,

rank(Rn,m,Mm) ≤ r(Mm)N(n), ‖Nn,m,Mm‖ ≤ 2‖f − pm,Mm‖∞,[−Mm,Mm],

which implies that pm,Mm(An)nm is an a.c.s. for f(An)n. Moreover, pm,Mm(An)n ∼GLT pm,Mm(κ) by Theorem 7.8.Finally, pm,Mm(κ)→ f(κ) a.e. in [0, 1]d × [−π, π]d, due to (7.12). In conclusion, all the hypotheses of Theorem 7.4 are satisfied and sof(An)n ∼GLT f(κ).

The last issue we are interested in is to know if A−1n n ∼GLT κ−1 in the case where Ann ∼GLT κ, each An is invertible, and

κ 6= 0 a.e. (so that κ−1 is a well-defined measurable function). More in general, we may ask if A†nn ∼GLT κ−1 when An ∼GLT κand κ 6= 0 a.e. The answer to both the previous questions is affirmative, but some work is needed to bring out the related proofs. Notethat these results cannot be inferred from Theorem 7.9, because the matrices An may fail to be Hermitian and, moreover, f(x) = x−1 isnot continuous on R. We begin by introducing the concept of sparsely vanishing matrix-sequences.

Definition 7.2 (sparsely vanishing matrix-sequence). Let Ann be a matrix-sequence. We say that Ann is sparsely vanishing (s.v.)if for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

≤ r(M),


It is clear from Definition 7.2 that if Ann is s.v. then A†nn is s.u.; it suffices to recall that the singular values of A† are1/σ1(A), . . . , 1/σr(A), 0, . . . , 0, where σ1(A) . . . σr(A) are the nonzero singular values of A (r = rank(A)).

Remark 7.3. Let Ann be a matrix-sequence. Then, Ann is s.v. if and only if

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

= 0. (7.13)

The proof of this equivalence is easy and follows the same line as the proof of the equivalence (1⇔ 2) in Proposition 4.2; the details areleft to the reader. Note that (7.13) can be rewritten as

limM→∞

lim supn→∞

1

N(n)

N(n)∑i=1

χ[0,1/M)(σi(An)) = 0.

Proposition 7.6. If Ann ∼σ f then Ann is s.v. if and only if f 6= 0 a.e.

Proof. Let D ⊂ Rk be the domain of the function f . Fix M > 0 and take FM ∈ Cc(R) such that FM = 1 over [0, 1/M ], FM = 0 over[2/M,∞] and 0 ≤ FM ≤ 1 over R. Note that FM ≥ χ[0,1/M) over [0,∞). Then,

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

=1

N(n)

N(n)∑i=1

χ[0,1/M)(σi(An))

≤ 1

N(n)

N(n)∑i=1

FM (σi(An))n→∞−→ 1

µk(D)

∫D

FM (|f(x)|)dx

and

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

≤ 1

µk(D)

∫D

FM (|f(x)|)dx.

87

Since FM (|f(x)|)→ χf=0(x) a.e. and |F (|f(x)|)| ≤ 1, by the dominated convergence theorem we get

limM→∞

∫D

FM (|f(x)|)dx =µkf 6= 0µk(D)

.

Thus,

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

= 0

if and only if f = 0 a.e. By Remark 7.3, this means that Ann is s.v. if and only if f = 0 a.e.

We are now ready to state and prove the last main theorem about the GLT algebra. This theorem is the counterpart for GLT sequencesof the ‘a.c.s. result’ mentioned at the end of Section 4.3.

Theorem 7.10. Let Ann ∼GLT κ with κ 6= 0 a.e., then A†nn ∼GLT κ−1.

Proof. Take a sequence of matrix-sequences Bn,mnm such that Bn,mm ∼GLT ξm for every m and ξm → κ−1 in measure. Notethat a sequence Bn,mnm with these properties exists. Indeed, by Lemma 2.7 there exists a sequence ξmm, with ξm of the form

ξm(x,θ) =

Nm∑j=−Nm

a(m)j (x)eij·θ, a

(m)j ∈ C∞([0, 1]d), Nm ∈ Nd,

such that ξm → κ−1 a.e. (and hence also in measure). Therefore, it suffices to take Bn,m =∑Nm

j=−NmDn(a

(m)j )Tn(eij·θ) and to

observe that Bn,mn ∼GLT ξm (see Theorem 6.8, Corollary 7.1 and Theorem 7.8).By Theorem 7.8, we have Bn,mAn − IN(n)n ∼GLT ξmκ − 1 for every m, and ξmκ − 1 → 0 a.e. (and hence also in measure).

Therefore, by Theorem 4.7, for every m there exists nm such that, for n ≥ nm,

Bn,mAn = IN(n) +Rn,m +Nn,m,


where limm→∞ c(m) = limm→∞ ω(m) = 0. Multiplying (7.14) by A†n, we obtain that, for every m and every n ≥ nm,

Bn,mAnA†n = A†n + (Rn,m +Nn,m)A†n. (7.15)

Since κ 6= 0 a.e. by hypothesis, Ann is s.v. (by Theorem 7.1 and Proposition 7.6). It follows that A†nn is s.u. and so, byProposition 4.2, for all M > 0 there is nM such that, for n ≥ nM ,

A†n = A†n,M + A†n,M ,

rank(A†n,M ) ≤ r(M)N(n), ‖A†n,M‖ ≤M,

where limM→∞ r(M) = 0. Choosing Mm = [ω(m)]−1/2, from (7.15) we see that, for every m and every n ≥ max(nm, nMm),

Bn,mAnA†n = A†n +R′n,m +N ′n,m,

rank(R′n,m) ≤(c(m) + r(Mm)

)N(n), ‖N ′n,m‖ ≤ [ω(m)]1/2,

(7.16)

where we have set R′n,m = Rn,mA†n +Nn,mA

†n,Mm

and N ′n,m = Nn,mA†n,Mm

.If the matrices An were invertible, then A†n = A−1

n and (7.16) would imply that Bn,mnm is an a.c.s. for A−1n n; this, in

combination with the approximation result for GLT sequences (Theorem 7.4), would conclude the proof. In the general case where thematrices An are not invertible, the thesis will follow again from (7.16) and Theorem 7.4 as soon as we have proved the following: forevery m there exists nm such that, for n ≥ nm,

AnA†n = IN(n) + Sn, rank(Sn) ≤ θ(m)N(n),

where limm→∞ θ(m) = 0. This is easy, because, by definition of A†n, the rank of the matrix Sn = AnA†n − IN(n) is given by

rank(Sn) = #i ∈ 1, . . . , N(n) : σi(An) = 0. Hence, the previous claim follows directly from the fact that Ann is s.v. (seeDefinition 7.2).

88

7.6 Algebraic-topological definition of GLT sequencesBefore concluding the theory of GLT sequences, it is interesting to give an abstract definition of GLT sequences based on the algebraic-topological results obtained in Sections 7.3 and 7.5. This definition can, however, be omitted on first reading.

Fix a sequence of d-indices n = n(n)n ⊆ Nd such that n→∞ as n→∞. Let E be the set of all matrix-sequences,

E =Ann : Ann is a matrix-sequences

,

which is a *-algebra with respect to the natural operations (2.62) and is also a topological (pseudometric) space with respect to the a.c.s.topology τa.c.s. generated by the a.c.s. distance da.c.s. introduced in Section 4.1. Let M be the set of all measurable functions defined on[0, 1]d × [−π, π]d,

M = κ : [0, 1]d × [−π, π]d → C : κ is measurable,

which is a *-algebra with respect to the natural operations and is also a topological (pseudometric) space with respect to the topology τµassociated with the convergence in measure (hereinafter called the ‘measure topology’). This topology is generated by the pseudometric

dµ(κ, ξ) = q(κ− ξ), q(ψ) = infµd|ψ| ≥ α+ α : α > 0

;

see, e.g., [17]. Consider the product space

E ×M = (Ann, κ) : Ann ∈ E , κ ∈M

equipped with the product topology τa.c.s. × τµ. Since E ×M is a product of two *-algebras, it is again a *-algebra with respect to thenatural pointwise operations

(Ann, κ)∗ = (An∗n, κ), (Ann, κ) + (Bnn, ξ) = (Ann + Bnn, κ+ ξ),

α(Ann, κ) = (αAnn, ακ), (Ann, κ)(Bnn, ξ) = (AnnBnn, κξ).

Moreover, E ×M is also a pseudometric space with respect to the ‘product distance’

(da.c.s. × dµ)((Ann, κ), (Bnn, ξ)) = da.c.s.(Ann, Bnn) + dµ(κ, ξ).

Now let A be the subset of E ×M consisting of all the ‘GLT pairs’, i.e.,

A = (Ann, κ) : Ann ∼GLT κ ⊆ E ×M.

Then, the following result holds.

Proposition 7.7. The set of GLT pairs A is the closed subalgebra of E ×M generated by the pairs(Tn(eij·θ)n, 1⊗ eij·θ) : j ∈ Zd

∪

(Dn(a)n, a⊗ 1) : a ∈ C∞([0, 1]d). (7.17)

In other words, A is the smallest closed subalgebra of E ×M containing the pairs (7.17).

Proof. The set of GLT pairs A contains the pairs (7.17), and it is a closed subalgebra of E ×M by the discussion in Section 7.3.1 andTheorem 7.8. Hence, A contains the closed subalgebra of E ×M generated by the pairs (7.17). Conversely, let Ann ∼GLT κ. ByCorollary 7.2, Ann admits an a.c.s. of the form

Nm∑

j=−Nm


n

m

,

where a(m)j ∈ C∞([0, 1]d),Nm ∈ Nd, and

∑Nm

j=−Nma

(m)j (x) eij·θ → κ(x,θ) a.e. (and hence also in measure). This means that

Nm∑j=−Nm


n

,

Nm∑j=−Nm

a(m)j (x) eij·θ

−→ (Ann, κ) in E ×M. (7.18)

Now, it is clear that the pairs in the left-hand side of (7.18) belong to the algebra generated by the pairs in (7.17). Hence, (Ann, κ)is the limit of paris belonging to the algebra generated by (7.17), and so (Ann, κ) belongs to the closure of the algebra generated by(7.17). Therefore, (Ann, κ) belongs to the closed subalgebra of E ×M generated by the pairs (7.17). This concludes the proof.

89

Proposition 7.7 allows us to give the following algebraic-topological definition of GLT sequences. Note that one could also decide tostart the theory of GLT sequences from this definition rather than Definition 7.1.

Definition 7.3 (algebraic-topological definition of GLT sequences). Let Ann be a matrix-sequence, n ∈ Nd, and let κ : [0, 1]d ×[−π, π]d → C be a measurable function. We say that Ann is a GLT sequence with symbol κ, and we write Ann ∼GLT κ, if(Ann, κ) belongs to the smallest closed subalgebra of E ×M generated by the pairs (7.17).

Exercise 7.1. Show that A is the closure in E ×M of the algebra generated by the pairs (7.17), i.e.,

A = Algebra(T ∪ D),

whereT =

(Tn(eij·θ)n, 1⊗ eij·θ) : j ∈ Zd

, D =

(Dn(a)n, a⊗ 1) : a ∈ C∞([0, 1]d)

.

7.7 Summary of the theory of GLT sequencesAfter developing the theory of GLT sequences, we note at this point that we have proved all the items GLT 1 – GLT 8 of Section 1.2. Inparticular, items GLT 1 – GLT 2 were proved in Theorems 7.1, 7.2, 7.3, item GLT 3 was proved in Theorems 6.4, 6.5, 6.6, 6.10, itemsGLT 4 – GLT 7 were proved in Theorems 7.8, 7.9, 7.10, and item GLT 8 is contained in Theorem 7.7.

We note however that in Section 1.2 the precise notion of matrix-sequence had not been introduced yet, and we adopted for simplicitya slightly wrong notation, using ‘n’ instead of ‘n’. For the reader’s convenience, we report again items GLT 1 – GLT 8 in the correctnotation. We also simplify a little bit their statements and add further details.

GLT 1. If Ann ∼GLT κ, then Ann ∼σ κ. If moreover the matrices An are Hermitian, then Ann ∼λ κ.

GLT 2. If An = Xn + Yn, with Ann ∼GLT κ or Xnn ∼GLT κ, and:

• every Xn is Hermitian;

• ‖Xn‖, ‖Yn‖ are uniformly bounded with respect to n;

• ‖Yn‖1 = o(N(n));

then Ann ∼λ κ.

GLT 3. The following properties hold.

• Tn(f)n ∼GLT 1⊗ f if f ∈ L1([−π, π]d).

• Dn(a)n ∼GLT a⊗ 1 if a : [0, 1]d → C is continuous a.e.

• Znn ∼GLT 0 if and only if Znn ∼σ 0. Moreover:

– Znn ∼σ 0 if and only if Zn = Rn +Nn with limn→∞

rank(Rn)

N(n)= limn→∞

‖Nn‖ = 0;

– Znn ∼σ 0 if limn→∞

‖Zn‖pN(n)1/p

= 0 for some p ∈ [1,∞].

• Dn(a) Tn(f)n ∼GLT a⊗ f if a : [0, 1]d → C is continuous and f is a d-variate trigonometric polynomial. Moreover:

– ‖Dn(a) Tn(f)−Dn(a)Tn(f)‖ ≤ C ωa(

1min(n)

)for some constant C independent of n;

– ‖Dn(a) Tn(f)‖ is uniformly bounded with respect to n.

GLT 4. If Ann ∼GLT κ then A∗nn ∼GLT κ.

GLT 5. If An =∑ri=1 αi

∏qij=1A

(i,j)n , where r, q1, . . . , qr ∈ N, α1, . . . , αr ∈ C, and A(i,j)

n n ∼GLT κij , then Ann ∼GLT κ =∑ri=1 αi

∏qij=1 κij .

GLT 6. If Ann ∼GLT κ and κ 6= 0 a.e., then A†nn ∼GLT κ−1.

GLT 7. If Ann ∼GLT κ and each An is Hermitian, then f(An)n ∼GLT f(κ) for all continuous functions f : R→ C.

GLT 8. Ann ∼GLT κ if and only if there exist GLT sequences Bn,mn ∼GLT κm such that Bn,mnma.c.s.−→ Ann and κm → κ

in measure.

90

Chapter 8

Applications

In this section, we present some applications of the theory of GLT sequences. This is done to give a flavor of the applicative interest ofthe theory. For more applications, we refer the reader to Section 1.1, where specific pointers to the available literature are provided.

8.1 The algebra generated by Toeplitz sequencesFix a sequence of d-indices n = n(n)n such that n → ∞ as n → ∞. In this section, we talk about the algebra T over the complexfield C generated by the Toeplitz sequences of the form Tn(g)n, g ∈ L1([−π, π]d). It is not difficult to see that

T =

r∑i=1

qi∏j=1

Tn(gij)n

: r, q1, . . . , qr ∈ N, gij ∈ L1([−π, π]d) for all i = 1, . . . , r and j = 1, . . . , qi

. (8.1)

Clearly, T is a sub-algebra of the GLT algebra A defined in (7.6). Indeed, according to GLT 3 and GLT 5,

r∑i=1

qi∏j=1

Tn(gij)n∼GLT

r∑i=1

qi∏j=1

1⊗ gij = 1⊗r∑i=1

qi∏j=1

gij .

Since∫

[0,1]d×[−π,π]d(1⊗ g) =

∫[−π,π]d

g, by GLT 1 we have

r∑i=1

qi∏j=1

Tn(gij)n∼σ

r∑i=1

qi∏j=1

gij . (8.2)

Moreover, if the matrices∑ri=1

∏qij=1 Tn(gij) are Hermitian, GLT 1 gives

r∑i=1

qi∏j=1

Tn(gij)n∼λ

r∑i=1

qi∏j=1

gij . (8.3)

The result in (8.2) was originally obtained in [87]. In the case where d = 1 and gij ∈ L∞([−π, π]) for all i, j, both (8.2) and (8.3) alreadyappeared in [24, Section 5.7]. Clearly, the distribution relations (8.2)–(8.3) extend Theorem 5.4.

The extension of the spectral distribution relation (8.3) to the case where the matrices∑ri=1

∏qij=1 Tn(gij) are not Hermitian has been

the subject of recent researches [37, 92]. Note that, if we remove the hypothesis of Hermitianess, then we necessarily have to add someadditional assumption. Indeed, (8.3) does not hold in general; a counterexample is provided, e.g., by the sequence of (1-level) Toeplitzmatrices Tn(eijθ)n. The hypothesis added in [37, 92] is a topological assumption on the range of the functions gij . A completelyanalogous hypothesis has been used in [36, 38] and, especially, in the pioneering work by Tilli [102], in order to extend the spectraldistribution relation expressed in Theorem 5.4 to the case where the generating function f is not real (and hence the related multilevelToeplitz matrices Tn(f) are not Hermitian).

We conclude this section by observing that the theory of GLT sequences allows us to manage matrix-sequences obtained from morecomplicated operations on Toeplitz sequences than sums and products. For example, in a context of Toeplitz preconditioning, one isinterested in the singular value and eigenvalue distribution of a sequence of preconditioned matrices of the form Tn(g)−1Tn(f)n.If g 6= 0 a.e., then GLT 3 and GLT 5 – GLT 6 give Tn(g)−1Tn(f)n ∼GLT 1 ⊗ g−1f . As a consequence, by GLT 1 we haveTn(g)−1Tn(f)n ∼σ g−1f and, if the matrices Tn(g)−1Tn(f) are Hermitian, Tn(g)−1Tn(f)n ∼λ g−1f . In the case of classicalCG preconditioning, the functions f, g are nonnegative and the matrices Tn(f), Tn(g) are Hermitian positive definite. In this situation, the

91

spectral distribution Tn(g)−1Tn(f)n ∼λ g−1f holds even if the preconditioned matrices Tn(g)−1Tn(f) are not Hermitian. Indeed,Tn(g)−1Tn(f) is similar to the Hermitian matrix Tn(g)−1/2Tn(f)Tn(g)−1/2. Assuming as before that g 6= 0 a.e., by GLT 3 andGLT 5 – GLT 7 we obtain 1

Tn(g)−1/2Tn(f)Tn(g)−1/2n ∼GLT g−1/2fg−1/2 = g−1f.

Hence, GLT 1 yields Tn(g)−1/2Tn(f)Tn(g)−1/2n ∼λ g−1f , and, by similarity, Tn(g)−1Tn(f)n ∼λ g−1f .

8.2 Variable-coefficient Toeplitz matricesLet L1 be the space of functions a : [0, 1]2 × [−π, π] → C such that a(x, y, ·) ∈ L1([−π, π]) for all (x, y) ∈ [0, 1]2. Every a ∈ L1 canbe formally represented by its Fourier series in the last variable,

a(x, y, θ) =∑k∈Z

ak(x, y)eikθ,

where

ak(x, y) =1

2π

∫ π

−πa(x, y, θ)e−ikθdθ, k ∈ Z,

are the Fourier coefficients of a(x, y, ·). For n ≥ 2, the n× n variable-coefficient Toeplitz matrix associated with a is defined as

An(a) =

[ai−j

( i− 1

n− 1,j − 1

n− 1

)]ni,j=1

.

Note that An(a) = Tn(a) whenever a is independent of x and y. The matrix-sequence An(a)n is referred to as the variable-coefficientToeplitz sequence associated with a, which in turn is called the generating function of An(a)n. Variable-coefficient Toeplitz matricesare also known as generalized convolutions and appear in many different contexts. As testified by the literature, this kind of matrices hasreceived a certain attention in the last years; see, e.g., [21, 22, 40, 73, 84, 97, 98, 106, 107]. We also refer the reader to [41, 69, 70] fora numerical-oriented literature about orthogonal polynomials with varying recurrence coefficients: the associated Jacobi matrices can beinterpreted as tridiagonal symmetric Toeplitz matrices with variable coefficients.

Following the analysis in [53], in this section we show that, under suitable assumptions on a, An(a)n is a GLT sequence withsymbol a(x, x, θ). This property, in combination with the theory of GLT sequences, allows one to derive a lot of singular value andeigenvalue distribution results for various matrix-sequences, including those obtained from algebraic and non-algebraic operations onvariable-coefficient Toeplitz sequences. Let us formulate the main result of this section in a precise way. For any ε > 0, define the strip

Sε = (x, y) ∈ [0, 1]2 : |x− y| ≤ ε

and set

W =a ∈ L1 :

∑k∈Z

supx,y∈[0,1]2

|ak(x, y)| <∞,

for all k ∈ Z there is ε(k) > 0 such that ak(·, ·) is continuous on Sε(k)

. (8.4)

Note thatW contains every continuous function a ∈ C([0, 1]2 × [−π, π]) satisfying the Wiener-type condition∑k∈Z

supx,y∈[0,1]2

|ak(x, y)| <∞. (8.5)

The main result of this section is the following.

Theorem 8.1. If a ∈ W then An(a)n ∼GLT a(x, x, θ).

Some work is necessary to prove Theorem 8.1. If α : [0, 1]2 → C, n ≥ 2 and k ∈ Z, we define the n× n diagonal matrix

Dn,k(α) = diagh=1,...,n

α( (h− 1 + k) modn

n− 1,h− 1

n− 1

).

Lemma 8.1. For every a ∈ L1 and n ≥ 2, we have

An(a) =

n−1∑k=−(n−1)

Tn(eikθ)Dn,k(ak).

1GLT 7 is applied with the function t 7→ |t|1/2.

92

Proof. For n ≥ 2 and k ∈ Z, we have

(Tn(eikθ))ij = δi−j,k =

1 if i− j = k,0 otherwise,

i, j = 1, . . . , n.

Hence, for all i, j = 1, . . . , n, n−1∑k=−(n−1)

Tn(eikθ)Dn,k(ak)

ij

=

n−1∑k=−(n−1)

δi−j,k ak

( (j − 1 + k) modn

n− 1,j − 1

n− 1

)= ai−j

( i− 1

n− 1,j − 1

n− 1

)= (An(a))ij ,

and the lemma is proved.

Lemma 8.2. If α : [0, 1]2 → C is continuous on the strip Sε for some ε > 0 and k ∈ Z, then

Dn,k(α)n ∼GLT α(x, x).

Proof. We show that, for all n ≥ 2 and k ∈ Z,

Dn,k(α) = Dn(α(x, x)) +Rn,k +Nn,k, (8.6)

where

limn→∞

rank(Rn,k)

n= limn→∞

‖Nn,k‖ = 0. (8.7)

This implies that the sequence Zn,k = Rn,k +Nn,kn is zero-distributed (see GLT 3), and the thesis follows from (8.6) in combinationwith GLT 3 and GLT 5.

Let ωα,ε(·) be the modulus of continuity of α over the strip Sε,

ωα,ε(δ) = max(x,y),(x′,y′)∈Sε|x−x′|,|y−y′|≤δ

|α(x, y)− α(x′, y′)|, δ > 0.

If k ≥ 0 and n > kε + 1, for h = 1, . . . , n− k we have

|(Dn,k(α))hh − (Dn(α(x, x)))hh| =∣∣∣∣α(h− 1 + k

n− 1,h− 1

n− 1

)− α

(hn,h

n

)∣∣∣∣ ≤ ωα,ε(k + 1

n− 1

), (8.8)

which tends to 0 as n→∞. WriteDn,k(α)−Dn(α(x, x)) = Nn,k +Rn,k,

where Nn,k (resp., Rn,k) is the matrix obtained from Dn,k(α)−Dn(α(x, x)) by setting to 0 all the diagonal elements corresponding toindices h > n−k (resp., h ≤ n−k). The decomposition (8.6)–(8.7) follows from (8.8) and from the (obvious) inequality rank(Rn,k) ≤ k.

if k < 0 and n > |k|ε + 1, for h = |k|+ 1, . . . , n we have

|(Dn,k(α))hh − (Dn(α(x, x)))hh| =∣∣∣∣α(h− 1 + k

n− 1,h− 1

n− 1

)− α

(hn,h

n

)∣∣∣∣ ≤ ωα,ε( |k|+ 1

n− 1

), (8.9)

which tends to 0 as n→∞. WriteDn,k(α)−Dn(α(x, x)) = Nn,k +Rn,k,

where Nn,k (resp., Rn,k) is the matrix obtained from Dn,k(α) − Dn(α(x, x)) by setting to 0 all the diagonal elements correspondingto indices h < |k| + 1 (resp., h ≥ |k| + 1). The decomposition (8.6)–(8.7) follows from (8.9) and from the (obvious) inequalityrank(Rn,k) ≤ k.

We are now ready to prove Theorem 8.1.

Proof of Theorem 8.1. For n ≥ 2 and m ∈ N, consider the matrix

An,m(a) =

m∑k=−m

Tn(eikθ)Dn,k(ak). (8.10)

Note that An,m(a) “resembles” An(a) because, by Lemma 8.1, the only difference between these two matrices is the range of indiceswhere k varies. By Lemma 8.1 we also have An,m(a) = An(am) for n > m, where

am(x, y, θ) =

m∑k=−m

ak(x, y)eikθ (8.11)

is the m-th Fuorier sum of a(x, y, θ). We are going to show that:

93

i. An,m(a)n ∼GLT am(x, x, θ) for every m ∈ N;

ii. am(x, x, θ)→ a(x, x, θ) in measure over [0, 1]× [−π, π];

iii. An,m(a)na.c.s.−→ An(a)n.

Once this is done, the thesis follows directly from GLT 8. By the hypothesis on a, each ak is continuous on Sε(k) for some ε(k) > 0, andso Dn,k(ak)n ∼GLT ak(x, x) by Lemma 8.2. Thus, by GLT 3 and GLT 5,

An,m(a)n ∼GLT

m∑k=−m

eikθak(x, x) = am(x, x, θ),

and item i is proved. Since a satisfies the Wiener-type condition (8.5), am(x, y, θ) → a(x, y, θ) uniformly on [−π, π] for each fixed(x, y) ∈ [0, 1]2. In particular, the sequence of continuous functions am(x, x, θ) converges pointwise to a(x, x, θ) over [0, 1] × [−π, π].Since the pointwise convergence on a set of finite measure implies the convergence in measure, item ii is proved. Finally, by Lemma 8.1and by the equality

‖Tn(eikθ)‖ = 1, |k| < n,

for every n > m we have

‖An(a)−An,m(a)‖ =

∥∥∥∥∥∥∑

n>|k|>m

Tn(eikθ)Dn,k(ak)

∥∥∥∥∥∥≤

∑n>|k|>m

‖Tn(eikθ)‖ ‖Dn,k(ak)‖

≤∑

n>|k|>m

supx,y∈[0,1]

|ak(x, y)|

= ε(m,n).

Recalling that a satisfies the Wiener-type condition (8.5), we have

limm→∞

lim supn→∞

ε(m,n) = 0,

and Corollary 4.3 implies that An,m(a)na.c.s.−→ An(a)n. We then conclude that item iii holds.

8.2.1 Consequences of Theorem 8.1Consequences of Theorem 8.1 (and of the theory of GLT sequences) are all the results that can be deduced from the list of propertiesGLT 1 – GLT 8 in which GLT 3 is extended to include the following additional item:

• An(a)n ∼GLT κ(x, θ) = a(x, x, θ) if a ∈ W .

For the sake of clarity, we use a different notation for the extended version of GLT 3 and we give it the label GLT 3. In other words,GLT 3 is the property obtained by adding to GLT 3 the above additional item, which is the content of Theorem 8.1. In this section wediscuss some consequences of GLT 1 – GLT 2, GLT 3, GLT 4 – GLT 8.

Singular value and eigenvalue distribution results on the algebra generated by variable-coefficient Toeplitz sequences Let Cdenote the *-algebra generated by the variable-coefficient Toeplitz sequences An(a)n with a ∈ W . It is not difficult to see that

C =

r∑i=1

qi∏j=1

An(aij)

n

: r, q1, . . . , qr ∈ N, aij ∈ W for all i = 1, . . . , r and j = 1, . . . , qi

.

By GLT 3 and GLT 5, C is a subalgebra of the GLT algebra A , and for the generic element of C we haver∑i=1

qi∏j=1

An(aij)

n

∼GLT

r∑i=1

qi∏j=1

aij(x, x, θ).

94

Hence, by GLT 1, r∑i=1

qi∏j=1

An(aij)

n

∼σr∑i=1

qi∏j=1

aij(x, x, θ) (8.12)

and, if the matrices∑ri=1

∏qij=1An(aij) are Hermitian,

r∑i=1

qi∏j=1

An(aij)

n

∼λr∑i=1

qi∏j=1

aij(x, x, θ). (8.13)

A result analogous to (8.12)–(8.13) was obtained by Silbermann and Zabroda in [97, Theorem 7.2].

Singular value and eigenvalue distribution results beyond the algebra generated by variable-coefficient Toeplitz sequences It isclear that the relations (8.12)–(8.13) are far from exhausting the singular value and eigenvalue distribution results that can be derivedfrom GLT 1 – GLT 2, GLT 3, GLT 4 – GLT 8. In particular, GLT 1, GLT 3 and GLT 5 – GLT 7 allow one to compute the singular valueand eigenvalue distribution of matrix-sequences that are obtained not only through sums and products of variable-coefficient Toeplitzsequences, but also through more complex operations involving all the GLT sequences listed in GLT 3. For example, let a ∈ W . Ifa(x, x, θ) 6= 0 a.e., then GLT 1, GLT 3, GLT 6 yield

An(a)†n ∼GLT1

a(x, x, θ),

An(a)†n ∼σ1

a(x, x, θ).

If in addition the matrices An(a) are Hermitian for all n, then GLT 1, GLT 3, GLT 6 also yield

An(a)†n ∼λ1

a(x, x, θ).

If the matrices An(a) are Hermitian for all n, then GLT 1, GLT 3, GLT 7 give

sinAn(a)n ∼GLT sin(a(x, x, θ)),

sinAn(a)n ∼σ sin(a(x, x, θ)),

sinAn(a)n ∼λ sin(a(x, x, θ)).

If a(x, x, θ) 6= 0 a.e. and the matrices An(a) are Hermitian for all n, then GLT 1, GLT 3, GLT 5 – GLT 7 yield

Tn(|θ|−1/2)An(a)†eAn(a)An(a)†Tn(|θ|−1/2) + 8Dn(log x)An(a)†Dn(log x)n ∼GLTea(x,x,θ)

|θ|a(x, x, θ)2+

8 log2 x

a(x, x, θ),

Tn(|θ|−1/2)An(a)†eAn(a)An(a)†Tn(|θ|−1/2) + 8Dn(log x)An(a)†Dn(log x)n ∼σea(x,x,θ)

|θ|a(x, x, θ)2+

8 log2 x

a(x, x, θ),

Tn(|θ|−1/2)An(a)†eAn(a)An(a)†Tn(|θ|−1/2) + 8Dn(log x)An(a)†Dn(log x)n ∼λea(x,x,θ)

|θ|a(x, x, θ)2+

8 log2 x

a(x, x, θ).

We could continue with this game indefinitely...

8.2.2 Possible extensions of Theorem 8.1Let us outline in this section two possible ways to extend Theorem 8.1, i.e., to prove the relation An(a)n ∼GLT a(x, x, θ) for a spaceof functions a larger thanW .

1. The first way originates from the observation that, when proving items i–iii under the assumption a ∈ W , we actually proved morethan necessary. We emphasize in particular the following two aspects.

• When proving item ii, we showed that am(x, x, θ)→ a(x, x, θ) pointwise. On the other hand, it would have been enough toshow that am(x, x, θ)→ a(x, x, θ) in measure.

95

• When proving item iii, we showed that‖An(a)−An,m(a)‖ ≤ ε(m,n) (8.14)

for some ε(m,n) satisfyinglimm→∞

lim supn→∞

ε(m,n) = 0. (8.15)

On the other hand, it would have been enough to show that An,m(a)na.c.s.−→ An(a)n, by proving for example that

‖An(a)−An,m(a)‖p ≤ ε(m,n)n1/p (8.16)

for some p ∈ [1,∞) and some ε(m,n) satisfying (8.15); see Corollary 4.3. Note that (8.16) is a condition weaker than (8.14),because for all p ∈ [1,∞) we have

‖A‖pn1/p

≤ ‖A‖,

with the equality holding if and only if all the singular values of A are equal.

In view of these considerations, there is room to refine the arguments used for the proof of Theorem 8.1, so as to weaken thehypotheses on a and, consequently, to enlarge the space of functions a such that An(a)n ∼GLT a(x, x, θ).

2. Suppose that, for all functions a belonging to a certain class K we can construct a sequence amm such that: the relationAn(am)n ∼GLT am(x, x, θ) holds for all m, am converges to a in some sense that guarantees the convergence in measuream(x, x, θ) → a(x, x, θ), and An(am)n

a.c.s.−→ An(a)n. Then An(a)n ∼GLT a(x, x, θ) for all a ∈ K (by GLT 8). Thistechnique is the second possible way to extend Theorem 8.1, and it was already used in the proof of Theorem 8.1 with K = W .Note that Theorem 8.1 allows us to take any sequence of functions inW as the sequence amm.

Exercise 8.1. Let

U =

q∑r=1

αr(x, y)βr(θ) : αr ∈ C([0, 1]2) and βr ∈ L2([−π, π]) for all r = 1, . . . , q

.

Show that An(a)n ∼GLT a(x, x, θ) for all a ∈ U .

Exercise 8.2. Let

V =

q∑r=1

αr(x)βr(y)γr(θ) : αr, βr are bounded measurable functions on [0, 1] and γr ∈ L1([−π, π]) for all r = 1, . . . , q

.

Show that An(a)n ∼GLT a(x, x, θ) for all a ∈ V .

8.3 Geometric means of matricesEveryone knows that the geometric mean of two positive numbers a, b is G(a, b) = (ab)1/2. But what is the geometric mean G(A,B)of two HPD matrices A,B ∈ Cn×n? An appropriate definition was proposed in a remarkable paper by Ando, Li and Mathias [2]. Theapproach of these authors was axiomatic: denoting by Pn the set of HPD matrices of size n, a function G : Pn ×Pn →Pn is said tobe a geometric mean if it satisfies a suitable list of properties that any geometric mean worthy of the name should satisfy. Ando, Li andMathias proposed a list of ten properties, which are referred to as the ALM axioms. Let us mention three of them.

1. Permutation invariance: G(A,B) = G(B,A) for all A,B ∈Pn.

2. Congruence invariance: G(M∗AM,M∗BM) = M∗G(A,B)M for all A,B ∈Pn and all invertible matrices M .

3. Consistency with scalars: G(A,B) = (AB)1/2 for all commuting matrices A,B ∈Pn.

It may be proved [11, Chapter 4] that the unique function G : Pn ×Pn →Pn satisfying both consistency with scalars and congruenceinvariance is G(A,B) = A(A−1B)1/2, which, moreover, satisfies all the ALM axioms. Using the general identity f(M−1AM) =M−1f(A)M (see [60, Theorem 1.13]) and the permutation invariance equation G(A,B) = G(B,A), we obtain

(A−1B)1/2 = (A−1/2A−1/2BA−1/2A1/2)1/2 = A−1/2(A−1/2BA−1/2)1/2A1/2, A,B ∈Pn,

andG(A,B) = A1/2(A−1/2BA−1/2)1/2A1/2 = B1/2(B−1/2AB−1/2)1/2B1/2, A,B ∈Pn. (8.17)

96

Suppose now that Ann ∼GLT κ and Bnn ∼GLT ξ, where An, Bn ∈ PN(n). Due to GLT 1, the positive definiteness of thematrices An, Bn, and Theorem 2.8, the essential ranges ER(κ), ER(ξ) are contained in [0,∞). Hence, κ, ξ ≥ 0 a.e. We assume thatat least one between κ and ξ is nonzero a.e. Under this assumption, we show that the sequence of geometric means G(An, Bn)n is aGLT sequence whose symbol is given by the geometric mean of the symbols κ, ξ. In other words, we prove the following relation:

G(An, Bn)n ∼GLT (κξ)1/2. (8.18)

Let f : R → R be any continuous function such that f(x) = x1/2 for all x ≥ 0. If κ 6= 0 a.e., then κ > 0 a.e., and so, using the firstexpression of G(An, Bn) in (8.17), we see that

G(An, Bn) = A1/2n (A−1/2

n BnA−1/2n )1/2A1/2

n = f(An)f(f(An)−1Bnf(An)

−1)f(An).

By GLT 5, GLT 6, GLT 7, it follows that

G(An, Bn)n ∼GLT f(κ)f(f(κ)−1ξf(κ)

−1)f(κ).

Since κ > 0 a.e., we have

f(κ)f(f(κ)−1ξf(κ)

−1)f(κ) = κ1/2(κ−1/2ξκ−1/2)1/2κ1/2 = (κξ)1/2 a.e.,

and the relation (8.18) is proved. If ξ 6= 0 a.e., then the proof of (8.18) follows the same pattern as in the case κ 6= 0 a.e., with theonly difference that now we use the second expression of G(An, Bn) in (8.17) instead of the first one. Noting that G(An, Bn) is HPDwhenever An, Bn are HPD, from (8.18) and GLT 1 we get

G(An, Bn)n ∼σ, λ (κξ)1/2. (8.19)

While it is easy to generalize the concept of geometric mean to the case where the numbers to be averaged are k > 2, the same isnot true for HPD matrices. In particular, the axiomatic approach by Ando, Li and Mathias is not satisfying for k > 2, because the tenALM axioms do not lead to a unique definition [16, 75]. The path to the right definition was different, involving a little bit of differentialgeometry. In fact, the geometric mean (or Karcher mean) of k matricesA(1), . . . , A(k) ∈Pn was defined as the barycenter of the matriceswith respect to a certain Riemannian distance; see [12] and [11, Chapter 6]. More precisely, the Karcher mean G(A(1), . . . , A(k)) is theunique minimizer over Pn of the functional

D(· ;A(1), . . . , A(k)) : Pn → R, D(X;A(1), . . . , A(k)) =

k∑i=1

[δ(X,A(i))

]2, (8.20)

where δ(A,B) is the distance given by the Riemannian structure,

δ(A,B) = ‖ log(A−1/2BA−1/2)‖2 =

(n∑`=1

log2(λ`(A−1B))

)1/2

. (8.21)

It was proved with some effort [13, 62, 72] that the Karcher mean satisfies all the ALM axioms and some further properties, and thusnow everyone agrees that the Karcher mean has the right to be called the geometric mean of matrices. Besides the mathematical interest,the Karcher mean has been used in several applications (see [67, Section 7] and the references in [15]), and suitable algorithms for itscomputation have been designed [15, 68].

Suppose now that A(i)n n ∼GLT κi for i = 1, . . . , k, where A(1)

n , . . . , A(k)n ∈ PN(n). By GLT 1, the positive definiteness of

A(i)n , and Theorem 2.8, each κi is nonnegative a.e. In this situation, we have reason to believe that the sequence of Karcher meansG(A

(1)n , . . . , A

(k)n )n is a GLT sequence with symbol (κ1 · · ·κk)1/k. The formal proof of this result, which might be achieved via

GLT 8, is certainly an interesting subject for future research, also considering that geometric means of Toeplitz matrices are of interest inpractical applications. For example, in a radar application one is interested in computing a geometric mean of HPD matrices which areToeplitz or block Toeplitz with Toeplitz blocks (i.e., 2-level Toeplitz); see [5, 6, 71].

8.4 PDE discretizations: the 1-dimensional caseThe main application of the theory of GLT sequences was already described in Section 1.1. It consists in the computation of the spectraldistribution of the sequences of discretization matrices arising from the approximation of PDEs by numerical methods. In fact, thesematrix-sequences are often GLT sequences. In this section and in the next one, we present the GLT analysis of several PDEs approximatedby various numerical methods. The idea is to show that items GLT 1 – GLT 8 are a powerful tool for computing the asymptotic spectraldistribution of PDE discretization matrices.

97

8.4.1 FD discretization of diffusion equationsConsider the following second-order differential problem:

−(a(x)u′(x))′ = f(x), x ∈ (0, 1),

u(0) = α, u(1) = β,(8.22)

where a : [0, 1]→ R is continuous. To ensure that problem (8.22) is well-posed, at least in its weak formulation, one should require thata(x) satisfy some other properties; for example, a ∈ C1([0, 1]) and a(x) > 0 for every x ∈ [0, 1], so that problem (8.22) is elliptic (see[27, Chapter 8, especially the Sturm-Liouville problem on p. 223]). However, for the GLT analysis presented in this section it suffices torequire that a ∈ C([0, 1]). We consider the discretization of (8.22) by using the classical second-order central FD scheme. In the casewhere a(x) is constant, this is also known as the (−1, 2,−1) FD scheme. Let us describe it shortly; for more details on FD methods, werefer the reader to the available literature (see, e.g., [99] or any good book on FDs). We choose a discretization parameter n ∈ N, weset h = 1

n+1 and xj = jh for all j ∈ [0, n + 1], and we note that, for j = 1, . . . , n, we can approximate −(a(x)u′(x))′|x=xj by thefollowing FD formula:

−(a(x)u′(x))′|x=xj ≈ −a(xj+ 1

2)u′(xj+ 1

2)− a(xj− 1

2)u′(xj− 1

2)

h≈ −

a(xj+ 12)u(xj+1)− u(xj)

h− a(xj− 1

2)u(xj)− u(xj−1)

hh

=−a(xj+ 1

2)u(xj+1) +

(a(xj+ 1

2) + a(xj− 1

2))u(xj)− a(xj− 1

2)u(xj−1)

h2. (8.23)

Then, we approximate the solution of (8.22) by the piecewise linear function that takes the value uj in xj for j = 0, . . . , n + 1, whereu0 = α, un+1 = β, and u = (u1, . . . , un)T is the solution of the linear system

−a(xj+ 12)uj+1 +

(a(xj+ 1

2) + a(xj− 1

2))uj − a(xj− 1

2)uj−1 = h2f(xj), j = 1, . . . , n. (8.24)

The matrix of the linear system (8.24) is the tridiagonal symmetric matrix given by

An =

a(x 12) + a(x 3

2) −a(x 3

2)

−a(x 32) a(x 3

2) + a(x 5

2) −a(x 5

2)

−a(x 52) a(x 5

2) + a(x 7

2) −a(x 7

2)

−a(x 72)

. . . . . .

. . . . . . −a(xn− 12)

−a(xn− 12) a(xn− 1

2) + a(xn+ 1

2)

. (8.25)

In this example, we will see that the theory of GLT sequences allows us to compute the singular value and eigenvalue distribution of thesequence of discretization matrices Ann. Actually, this is the fundamental example that led to the birth of the theory of LT sequences[101] and, subsequently, of GLT sequences [50, 51, 52, 89, 90]. Given the importance, we will compute the singular value and eigenvaluedistribution of Ann by two different methods, both of them being instructive.

Method 1. Suppose first that a(x) is constant, say a(x) = 1 identically. In this case, the matrix An becomes

An =

2 −1−1 2 −1

−1 2 −1

−1. . . . . .. . . . . . −1

−1 2

.

Therefore, An = Tn(2− 2 cos θ) is simply the Toeplitz matrix generated by the function 2− 2 cos θ; see (5.7)–(5.8). Using Theorem 5.4,we have Ann ∼σ, λ 2− 2 cos θ. Since the function 2− 2 cos θ is even, the relations Ann ∼σ, λ 2− 2 cos θ continue to hold even ifwe consider [0, π] as the domain of 2 − 2 cos θ instead of [−π, π]. According to the informal meaning behind the definition of spectraland singular value distribution, see Remark 2.3, we may conclude that the eigenvalues and the singular values of An are approximatelya uniform sampling over [0, π] of the nonnegative function 2 − 2 cos θ. However, this result is known also analytically. Indeed, since

98

n 202 402 602 802 1002 1202

‖ςn − εn‖∞ 0.4074 0.1981 0.1268 0.0926 0.0733 0.0594

Table 8.1: computation of ‖ςn − εn‖∞ for a(x) = 2 + cos(3x) and for increasing values of n.

Tn(2 − 2 cos θ) is SPD, the eigenvalues coincide with the singular values and are given explicitly by 2 − 2 cos jπn+1 , j = 1, . . . , n; see

[20, p. 35] or [99, p. 154].Now let us turn to the case where a(x) is not constant. In this case, the expression of An is given by (8.25) and the Toeplitzness

seems to be completely lost. In reality, we find it again ‘in an approximated sense’ and ‘at a local scale’. Indeed, we note that a(x)varies smoothly from a(0) to a(1), because it is uniformly continuous. Therefore, assuming that n is large with respect to k, any k × kleading principal submatrix of An shows an approximate Toeplitz structure. Let us be more quantitative. Fix a large m ∈ N and assumen > m. Then, n is large with respect to bn/mc, and so, according to the previous reasoning, any bn/mc × bn/mc leading principalsubmatrix of An shows an approximate Toeplitz structure. In fact, the evaluations of a(x) appearing in the first bn/mc × bn/mc leadingprincipal submatrix are approximately equal to a( 1

m ); the evaluations of a(x) appearing in the second bn/mc× bn/mc leading principalsubmatrix are approximately equal to a( 2

m ); and so on until the evaluations of a(x) appearing in the m-th bn/mc × bn/mc leadingprincipal submatrix, which are approximately equal to a(1). If, for all j = 1, . . . ,m, we replace by a( jm ) the evaluations of a(x) in thej-th bn/mc × bn/mc leading principal submatrix, this submatrix becomes a( jm )Tbn/mc(2 − 2 cos θ). In conclusion, the matrix An isapproximated by the Locally Toeplitz operator

LTmn (a(x), 2− 2 cos θ) =

a( 1m )Tbn/mc(2− 2 cos θ)

a( 2m )Tbn/mc(2− 2 cos θ)

. . .

a(1)Tbn/mc(2− 2 cos θ)

Onmodm

.

In fact, LTmn (a(x), 2− 2 cos θ)nm is an a.c.s. for Ann, because it can be shown that

An = LTmn (a(x), 2− 2 cos θ) +Rn,m +Nn,m

rank(Rn,m) ≤ 3m, ‖Nn,m‖ ≤ ωa( 1

m+m+ 1

n+ 1

).

Thus, by definition,Ann ∼LT a(x)(2− 2 cos θ), (8.26)

and soAnn ∼σ, λ a(x)(2− 2 cos θ). (8.27)

Since a(x)(2 − 2 cos θ) is symmetric with respect to the Fourier variable θ, the relations (8.27) continue to hold even if we consider[0, 1] × [0, π] as the domain of a(x)(2 − 2 cos θ) instead of [0, 1] × [−π, π]; this follows directly from the definition of spectral andsingular value distribution. According to the informal meaning of (8.27), see Remark 2.3, if n = `2 is large enough, the eigenvalues ofAn are approximately given by the uniform sampling a( i` )(2− 2 cos jπ

`+1 ), i, j = 1, . . . , `. This is confirmed by Figure 8.1 and Table 8.1for the case a(x) = 2+cos(3x). In Figure 8.1, we plotted the spectrum ofAn together with the values a( i` )(2−2 cos jπ

`+1 ), i, j = 1, . . . , `,for n = `2 = 400. Both the eigenvalues of An and the samplings of a(x)(2− 2 cos θ) are depicted in non-increasing order. In Table 8.1,we computed, for increasing values of n = `2, the∞-norm of the difference ςn − εn, where

• εn is the vector of the eigenvalues of An;

• ςn is the vector of the samples a( i` )(2− 2 cos jπ`+1 ), i, j = 1, . . . , `.

Both ςn and εn are arranged in non-increasing order. As shown in Table 8.1, the norm ‖ςn − εn‖∞ converges to 0 when n → ∞,although the convergence is very slow.

Method 2. As already pointed out, the example we are dealing with led to the birth of the theory of GLT sequences. In particular, theprocedure followed in Method 1 to obtain (8.27) motivated the definition of Locally Toeplitz sequences, as well as the introduction of theLocally Toeplitz operator. However, now that we have developed the theory of GLT sequences, we should say that Method 1 is probablynot the most effective way to obtain (8.27). The method we are going to see now seems to be much more effective.

99

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

Λ(A

n)

uniform sampling of a(x)(2−2cosθ)

Figure 8.1: spectrum of An and samplings a( i` )(2− 2 cos jπ`+1 ), i, j = 1, . . . , `, for n = `2 = 400, a(x) = 2 + cos(3x).

Consider the matrix

Dn(a)Tn(2− 2 cos θ) =

2a( 1n ) −a( 1

n )

−a( 2n ) 2a( 2

n ) −a( 2n )

−a( 3n ) 2a( 3

n ) −a( 3n )

−a( 4n )

. . . . . .

. . . . . . −a(n−1n )

−a(1) 2a(1)

. (8.28)

In view of the inequalities∣∣xj − j

n

∣∣ ≤ 1n+1 = h, j = 1, . . . , n, a direct comparison between (8.28) and (8.25) shows that the modulus

of each diagonal entry of the matrix An −Dn(a)Tn(2− 2 cos θ) is bounded by 2ωa(3h/2), and the modulus of each off-diagonal entryof An − Dn(a)Tn(2 − 2 cos θ) is bounded by ωa(3h/2). Therefore, the 1-norm and the ∞-norm of An − Dn(a)Tn(2 − 2 cos θ) arebounded by 4ωa(3h/2), and so, by (2.26),

‖An −Dn(a)Tn(2− 2 cos θ)‖ ≤ 4ωa(3h/2).

In particular, ‖An − Dn(a)Tn(2 − 2 cos θ)‖ → 0 as n → ∞. Setting Zn = An − Dn(a)Tn(2 − 2 cos θ), we have Znn ∼σ 0 byGLT 3. Since

An = Dn(a)Tn(2− 2 cos θ) + Zn, (8.29)

by GLT 3 and GLT 5 we haveAnn ∼GLT a(x)(2− 2 cos θ), (8.30)

and (8.27) follows from GLT 1.

Remark 8.1. From a formal viewpoint (i.e., disregarding the regularity of a(x) and u(x)), problem (8.22) can be rewritten in the form−a(x)u′′(x)− a′(x)u′(x) = f(x), x ∈ (0, 1),

u(0) = α, u(1) = β.(8.31)

From this reformulation, it appears more clearly that the symbol a(x)(2− 2 cos θ) consists of two ‘ingredients’:

• the coefficient a(x) associated with the higher-order differential operator of (8.31), namely −a(x)u′′(x) (this is the principalsymbol in the Hörmander theory [61]);

• the trigonometric polynomial 2− 2 cos θ = −eiθ + 2− e−iθ associated with the FD formula (−1, 2,−1) used to approximate thehigher-order derivative −u′′(x).

100

In particular, the term −a′(x)u′(x), which only depends on lower-order derivatives of u(x), does not enter the expression of the symbol.We also note that the trigonometric polynomial 2 − 2 cos θ is nonnegative on [−π, π] and it has a unique zero of order 2 at θ = 0; thisreflects the fact that the associated FD formula (−1, 2,−1) approximates −u′′(x), which is a differential operator of order 2.

8.4.2 FD discretization of convection-diffusion-reaction equations (part I)Suppose that we add to the diffusion equation (8.22) a convenction and a reaction term. In this way, we obtain the following convection-diffusion-reaction PDE in divergence form with Dirichlet boundary conditions:

−(a(x)u′(x))′ + b(x)u′(x) + c(x)u(x) = f(x), x ∈ (0, 1),

u(0) = α, u(1) = β,(8.32)

where we assume that b, c : [0, 1]→ R are bounded. Based on the discussion in Remark 8.1, we expect that the term b(x)u′(x)+c(x)u(x),which only involves lower-order derivatives of u(x), does not enter the expression of the symbol. In other words, if we discretize thehigher-order term −(a(x)u′(x))′ as in (8.23), the symbol of the FD discretization matrices Bn resulting from (8.32) should be againa(x)(2− 2 cos θ). This is in fact the case. Let us provide the details of the proof.

Consider the discretization of (8.32) by the FD scheme defined as follows:

• to approximate the higher-order term −(a(x)u′(x))′, use again the FD formula (8.23);

• to approximate the first-order term b(x)u′(x), use any (consistent) FD formula; to fix the ideas, we assume to use the centralformula

b(x)u′(x)|x=xj ≈ b(xj)u(xj+1)− u(xj−1)

2h; (8.33)

• to approximate the reaction term c(x)u(x), use the obvious equation

c(x)u(x)|x=xj = c(xj)u(xj). (8.34)

Let Bn be the resulting discretization matrix. Then,Bn = An + Zn, (8.35)

where Zn is the matrix coming from the approximation of the term b(x)u′(x) + c(x)u(x). Since b(x)u′(x) + c(x)u(x) only involveslower-order derivatives of u(x), it can be shown that ‖Zn‖ ≤ C/n for some constant C independent of n. Indeed, we have

Zn =h

2

0 b(x1)

−b(x2) 0 b(x2)

. . . . . . . . .

−b(xn−1) 0 b(xn−1)

−b(xn) 0

+ h2

c(x1)

c(x2)

. . .

c(xn−1)

c(xn)

, (8.36)

and it follows from (2.26) that ‖Zn‖ ≤ h‖b‖∞+h2‖c‖∞. Hence Znn ∼σ 0, and it follows from (8.30), (8.35) and GLT 3, GLT 5 that

Bnn ∼GLT a(x)(2− 2 cos θ). (8.37)

Now, if the convection term is not present, i.e. b(x) = 0 identically, then Bn is symmetric and so, by (8.37) and GLT 1,

Bnn ∼σ, λ a(x)(2− 2 cos θ). (8.38)

If b(x) is not identically 0, Bn is not symmetric and thus (8.37) and GLT 1 only imply that Bnn ∼σ a(x)(2 − 2 cos θ). However,in view of the decomposition (8.35), since ‖Zn‖ ≤ C/n, ‖Zn‖1 = O(1) by (2.27), and ‖An‖ ≤ 4‖a‖∞ by (2.26), the relationBnn ∼λ a(x)(2 − 2 cos θ) holds even if b(x) is an arbitrary (bounded) function, by GLT 2. In short, the relations (8.38) are alwayssatisfied.

In Figure 8.2, we fixed a(x) = 2 + cos(3x), b(x) = 0 and c(x) = 30000, and we plotted the spectrum of Bn together with thevalues a( i` )(2 − 2 cos jπ

`+1 ), i, j = 1, . . . , `, for n = `2 = 400. Both the eigenvalues of Bn and the samplings of a(x)(2 − 2 cos θ)are depicted in non-increasing order. Note that the samplings of a(x)(2 − 2 cos θ) are exactly the same as in Figure 8.1. We see fromFigure 8.2 that the approximation of Λ(Bn) provided by the symbol a(x)(2− 2 cos θ) is not as good as in Figure 8.1. This is due to thepresence of a reaction term c which is quite large with respect to 1/h2, being h2 the coefficient appearing in front of the reaction matrix

101

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

Λ(B

n)


Figure 8.2: spectrum of Bn and samplings a( i` )(2− 2 cos jπ`+1 ), i, j = 1, . . . , `, for n = `2 = 400, a(x) = 2 + cos(3x), b(x) = 0 and

c(x) = 30000.

c = 3

n 202 402 602 802 1002 1202

‖ςn − εn‖∞ 0.4074 0.1981 0.1268 0.0926 0.0733 0.0594

c = 300

n 202 402 602 802 1002 1202

‖ςn − εn‖∞ 0.4093 0.1982 0.1268 0.0926 0.0733 0.0594

c = 30000

n 202 402 602 802 1002 1202

‖ςn − εn‖∞ 0.5940 0.2098 0.1291 0.0934 0.0736 0.0595

Table 8.2: computation of ‖ςn − εn‖∞ for a(x) = 2 + cos(3x), b(x) = 0, c(x) = c = 3, 300, 30000, and for increasing values of n.

diagj=1,...,n c(xj) (see (8.36)). Actually, c is approximately of the same order as 1/h2 = (n + 1)2 = 160801. The reduced accuracy inthe symbol-to-spectrum approximation could be expected, because the symbol does not depend on c and therefore it is unlikely that ityields a good approximation of Λ(Bn) for any c and n. However, as soon as h2c becomes small, the approximation of Λ(Bn) providedby the symbol a(x)(2− 2 cos θ) becomes accurate. This is shown in Table 8.2, in which we fixed a(x) = 2 + cos(3x) and b(x) = 0, andwe computed, for c(x) = c = 3, 300, 30000 and for increasing values of n = `2, the∞-norm of the difference ςn − εn, where:

• εn is the vector of the eigenvalues of Bn;

• ςn is the vector of the samples a( i` )(2− 2 cos jπ`+1 ), i, j = 1, . . . , `.

Both ςn and εn are arranged in non-increasing order. Table 8.2 shows that the norm ‖ςn − εn‖∞ converges to 0 when n → ∞ withthe same asymptotic speed as in the absence of the reaction term; cf. Table 8.2 and Table 8.1. The presence of a large reaction term likec = 30000 affects the norm ‖ςn − εn‖∞ only when n is small (more precisely, when the quantity h2c is not negligible), whereas for nlarge enough the value of ‖ςn − εn‖∞ provided by Table 8.2 is essentially the same as the corresponding value provided by Table 8.1.

Remark 8.2. In the considered numerical example, we addressed the case of a nonzero reaction term c(x). Similar considerations alsohold in the presence of a nonzero convection term b(x). In particular, a bad symbol-to-spectrum approximation could be expected if bis large with respect to h, being h the coefficient appearing in front of the convection matrix in (8.36). However, as soon as the quantityhb becomes negligible, the approximation of Λ(Bn) provided by the symbol a(x)(2 − 2 cos θ) becomes accurate. In other words, theinfluence of the convection/reaction term desappears when n is large enough, because in the limit where n → ∞ we have ‖Zn‖ ≈ 0,Bn ≈ An by (8.35), and, consequently, Λ(Bn) ≈ Λ(An). In this respect, we recall that the notion of spectral symbol is asymptotic: forsmall n, the spectral symbol could be far from approximating the spectrum.

102

Effect of boundary conditions So far, we only considered PDEs with Dirichlet boundary conditions. A natural question is the follow-ing: if we change the boundary conditions in (8.32), does the expression of the symbol change? The answer is ‘no’: boundary conditionsdo not affect the singular value and eigenvalue distribution because they only produce a small-rank perturbation in the resulting dis-cretization matrices. To understand better this point, consider, for example, problem (8.32) with Neumann boundary conditions:

−(a(x)u′(x))′ + b(x)u′(x) + c(x)u(x) = f(x), x ∈ (0, 1),

u′(0) = α, u′(1) = β.(8.39)

We discretize (8.39) by the same FD scheme considered above, which is defined by (8.23), (8.33) and (8.34). In this way, we arrive at thelinear system

− a(xj+ 12)uj+1 +

(a(xj+ 1

2) + a(xj− 1

2))uj − a(xj− 1

2)uj−1 +

h

2

(b(xj)uj+1 − b(xj)uj−1

)+ h2c(xj)uj = h2f(xj),

j = 1, . . . , n, (8.40)

which is formed by n equations in the n+ 2 unknowns u0, u1, . . . , un, un+1. However, as it is common in the FD context, u0 and un+1

are expressed in terms of u1, . . . , un, by exploiting the boundary conditions. The simplest choice is to express u0 and un+1 as a functionof u1 and un, respectively, by imposing the conditions

u1 − u0

h= α,

un+1 − unh

= β.

From these relations we get u0 = u1 − αh and un+1 = un + βh. Substituting into (8.40), we obtain a linear system with n equationsand n unknowns u1, . . . , un. Its coefficient matrix is

Cn = Bn +Rn = An + Zn +Rn, (8.41)

where Bn, An, Zn are given, respectively, by (8.35), (8.25), (8.36), and

Rn =

−a(x 1

2)− h

2 b(x1)

−a(xn+ 12) + h

2 b(xn)

is a small-rank correction coming from the discretization of the boundary conditions. Clearly ‖Rn‖ ≤ ‖a‖∞ + h

2 ‖b‖∞ is uniformlybounded with respect to n and ‖Rn‖1 ≤ 2‖Rn‖ = o(n). Thus, Rnn is zero-distributed like Znn, and so (8.41) together with (8.30)and GLT 3, GLT 5 gives

Cnn ∼GLT a(x)(2− 2 cos θ). (8.42)

If the matrices Cn are symmetric (this happens when b(x) = 0), from (8.42) we get

Cnn ∼σ, λ a(x)(2− 2 cos θ). (8.43)

If the matrices Cn are not symmetric, from (8.42) we only obtain the singular value distribution Cnn ∼σ a(x)(2− 2 cos θ). However,in view of (8.41), since ‖Rn +Zn‖1 = o(n) and ‖Rn +Zn‖, ‖An‖ are uniformly bounded with respect to n, the eigenvalue distributionCnn ∼λ a(x)(2− 2 cos θ) holds by GLT 2 even if the matrices Cn are not symmetric. Thus, the relations (8.43) are always satisfied.

8.4.3 FD discretization of convection-diffusion-reaction equations (part II)Consider the following convection-diffusion-reaction problem in non-divergence form:

−a(x)u′′(x) + b(x)u′(x) + c(x)u(x) = f(x), x ∈ (0, 1),

u(0) = α, u(1) = β,(8.44)

where a : [0, 1] → R is continuous and b, c : [0, 1] → R are bounded. We choose n ∈ N, we set h = 1n+1 and xj = jh for all

j = 0, . . . , n + 1, and we discretize again (8.44) by the central second-order FD scheme, which in this case is defined by the followingformulas:

−a(x)u′′(x)|x=xj ≈ a(xj)−u(xj+1) + 2u(xj)− u(xj−1)

h2, j = 1, . . . , n,

b(x)u′(x)|x=xj ≈ b(xj)u(xj+1)− u(xj−1)

2h, j = 1, . . . , n,

c(x)u(x)|x=xj = c(xj)u(xj), j = 1, . . . , n.

103


a(xj)(−uj+1 + 2uj − uj−1

)+h

2b(xj)

(uj+1 − uj−1

)+ h2c(xj)uj = h2f(xj), j = 1, . . . , n.

The matrix An of this linear system can be decomposed according to the diffusion, convection and reaction term, as follows:

An = Kn + Zn, (8.45)

where Zn is the sum of the convection and reaction matrix, and is given by (8.36), while

Kn =

2 a(x1) −a(x1)

−a(x2) 2 a(x2) −a(x2)

. . . . . . . . .

−a(xn−1) 2 a(xn−1) −a(xn−1)

−a(xn) 2 a(xn)

. (8.46)

We are going to show thatAnn ∼GLT a(x)(2− 2 cos θ) (8.47)

andAnn ∼σ, λ a(x)(2− 2 cos θ). (8.48)

On the basis of Remark 8.1, the relations (8.47)–(8.48) are not unexpected, because we used again the FD scheme (−1, 2,−1) toapproximate the higer-order derivative −u′′(x). However, contrary to Sections 8.4.1–8.4.2, this time the diffusion matrix Kn is notsymmetric and the proof of the eigenvalue distribution in (8.48) will require a little more work.

The relation (8.47) is proved by the following observations.

• Znn ∼GLT 0 by GLT 3, because Znn ∼σ 0 (indeed, ‖Zn‖ ≤ h‖b‖∞ + h2‖c‖∞ → 0 as n→∞).

• Knn ∼GLT a(x)(2− 2 cos θ), because:

– by comparing (8.28) and (8.46), we see that the 1-norm and the ∞-norm of Kn − Dn(a)Tn(2 − 2 cos θ) are bounded by4ωa(h), hence ‖Kn −Dn(a)Tn(2− 2 cos θ)‖ ≤ 4ωa(h)→ 0 as n→∞;

– Dn(a)Tn(2− 2 cos θ)n ∼GLT a(x)(2− 2 cos θ) by GLT 3 and GLT 5.

From (8.47) and GLT 1, we get the singular value distribution in (8.48). To prove the spectral distribution, the idea is to exploit the fact thatKn is ‘almost’ symmetric, because a(x) varies continuously when x ranges in [0, 1], and thus a(xj) ≈ a(xj+1) for all j = 1, . . . , n− 1

(when n is large enough). Therefore, by replacing Kn with one of its symmetric approximations Kn, we can write

An = Kn + (Kn − Kn) + Zn, (8.49)

and then we will want to obtain the relation Ann ∼λ a(x)(2−2 cos θ) from GLT 2 applied withXn = Kn and Yn = (Kn−Kn)+Zn.Let

Kn =

2 a(x1) −a(x1)

−a(x1) 2 a(x2) −a(x2)

. . . . . . . . .

−a(xn−2) 2 a(xn−1) −a(xn−1)

−a(xn−1) 2 a(xn)

. (8.50)

Since ‖Zn‖ → 0, ‖Kn‖ ≤√|Kn|1|Kn|∞ ≤ 4‖a‖∞, and ‖Kn − Kn‖ ≤

√|Kn − Kn|1 |Kn − Kn|∞ ≤ ωa(h) → 0 as n → ∞, it

follows from GLT 2 that Ann ∼λ a(x)(2− 2 cos θ).

104

Remark 8.3. We could also choose

Kn = Dn(a) Tn(2− 2 cos θ) =

2 a( 1n ) −a( 1

n )

−a( 1n ) 2 a( 2

n ) −a( 2n )

. . . . . . . . .

−a(n−2n ) 2 a(n−1

n ) −a(n−1n )

−a(n−1n ) 2 a(1)

.

With this choice of Kn, nothing changes except for the bound of ‖Kn − Kn‖, which is replaced by ‖Kn − Kn‖ ≤ 4ωa(h).

8.4.4 FD discretization of convection-diffusion-reaction equations (part III)Based on the discussion in Remark 8.1, if we change the FD scheme to approximate (8.44), the symbol should become a(x)p(θ), wherep(θ) is the trigonometric polynomial associated with the new FD formula used to approximate the second derivative −u′′(x) (the higher-order differential operator). In this section, we show through an example that this is indeed the case.

Consider again the convection-diffusion-reaction problem (8.44). Instead of the second-order central FD scheme (−1, 2,−1), thistime we use the fourth-order central FD scheme 1

12 (1,−16, 30,−16, 1) to approximate the second derivative−u′′(x). In other words, forj = 2, . . . , n− 1 we approximate the higher-order term −a(x)u′′(x) by the FD formula

−a(x)u′′(x)|x=xj ≈ a(xj)u(xj+2)− 16u(xj+1) + 30u(xj)− 16u(xj−1) + u(xj−2)

12h2, j = 2, . . . , n− 1,

while for j = 1, n we use again the FD scheme (−1, 2,−1),

−a(x)u′′(x)|x=xj ≈ a(xj)−u(xj+1) + 2u(xj)− u(xj−1)

h2, j = 1, n.

As already observed in the previous sections, the FD schemes used to approximate the lower-order terms b(x)u′(x) and c(x)u(x) do notaffect the singular value and eigenvalue distribution of the resulting sequence of discretization matrices. In this example, we assume toapproximate b(x)u′(x) and c(x)u(x) by the following ‘strange’ FD formulas:

b(x)u′(x)|x=xj ≈ b(xj)u(xj)− u(xj−1)

h, j = 1, . . . , n,

c(x)u(x)|x=xj ≈ c(xj)u(xj+1) + u(xj) + u(xj−1)

3, j = 1, . . . , n.

The resulting (normalized) discretization matrix An can be decomposed according to the diffusion, convection and reaction term, asfollows:

An = Kn + Zn, (8.51)

where Zn is the sum of the convection and reaction matrix, and is given by

Zn = h

b(x1)

−b(x2) b(x2)

. . . . . .

−b(xn−1) b(xn−1)

−b(xn) b(xn)

+h2

3

c(x1) c(x1)

c(x2) c(x2) c(x2)

. . . . . . . . .

c(xn−1) c(xn−1) c(xn−1)

c(xn) c(xn)

,

105

while

Kn =1

12

24 a(x1) −12 a(x1)

−16 a(x2) 30 a(x2) −16 a(x2) a(x2)

a(x3) −16 a(x3) 30 a(x3) −16 a(x3) a(x3)

. . . . . . . . . . . . . . .

a(xn−2) −16 a(xn−2) 30 a(xn−2) −16 a(xn−2) a(xn−2)

a(xn−1) −16 a(xn−1) 30 a(xn−1) −16 a(xn−1)

−12 a(xn) 24 a(xn)

. (8.52)

The trigonometric polynomial associated with the FD formula used to approximate the second derivative −u′′(x) is in this case p(θ) =112 (e−2iθ − 16e−iθ + 30− 16eiθ + e2iθ) = 1

12 (30− 32 cos θ + 2 cos(2θ)); and, indeed, we are going to show that

Ann ∼GLT a(x)p(θ) (8.53)

andAnn ∼σ, λ a(x)p(θ). (8.54)

To obtain simultaneously (8.53) and (8.54), we consider the following decomposition of An,

An = Kn + (Kn − Kn) + Zn,

where Kn is the symmetric approximation of Kn given by

Kn = Dn(a) Tn(p).

We show that:

(a) Knn ∼GLT a(x)p(θ);

(b) ‖Kn‖, ‖Kn‖ are uniformly bounded with respect to n and ‖Zn‖ → 0;

(c) ‖Kn − Kn‖1 = o(n).

Note that (b)–(c) imply that (Kn−Kn)+Znn ∼σ 0. Once we have proved (a)–(c), the relation (8.53) follows from GLT 5, the singularvalue distribution in (8.54) follows from (8.53), and the spectral distribution in (8.54) follows from GLT 2 applied with Xn = Kn andYn = (Kn − Kn) + Zn.

Proof of (a). The result is contained in GLT 3.

Proof of (b). ‖Zn‖ ≤ 2h‖b‖∞ + h2‖c‖∞ → 0, ‖Kn‖ ≤√|Kn|1 |Kn|∞ ≤ 64

12‖a‖∞, and ‖Kn‖ ≤√|Kn|1 |Kn|∞ ≤ 64

12‖a‖∞.

Note that the uniform boundedness of ‖Kn‖ was already known from GLT 3.

Proof of (c). A direct comparison between Kn and Kn shows that

Kn = Kn +Rn +Nn,

where

Rn =1

12

−30 a( 1

n ) + 24 a(x1) 16 a( 1n )− 12 a(x1)

16 a(n−1n )− 12 a(xn) −30 a(1) + 24 a(xn)

.Since ‖Rn‖ ≤ 82

12‖a‖∞, rank(Rn) ≤ 2, and ‖Nn‖ ≤ 6412 ωa( 2

n ), we have ‖Kn − Kn‖1 ≤ 826 ‖a‖∞ + 64

12 ωa( 2n )n = o(n).

Remark 8.4. Despite we have changed the FD scheme to approximate the second derivative −u′′(x), the resulting trigonometric poly-nomial p(θ) shares some properties of 2 − 2 cos θ. In particular, p(θ) is nonnegative over [−π, π] and it has a unique zero of order 2 atθ = 0, because

limθ→0

p(θ)

θ2= 1 = lim

θ→0

2− 2 cos θ

θ2.

This reflects the fact the the associated FD formula 112 (1,−16, 30,−16, 1) approximates −u′′(x) (a differential operator of order 2); cf.

Remark 8.1.

106

8.4.5 FD discretization of higher-order PDEsIn this section, we would like to investigate what happens if we approximate by FDs an higher-order differential equation. We will focuson the following simple fourth-order problem with homogeneous Dirichlet-Neumann boundary conditions:

a(x)u(4)(x) = f(x), x ∈ (0, 1),u(0) = u′(0) = 0, u(1) = u′(1) = 0,

(8.55)

where a : [0, 1] → R is continuous. We do not consider more complicated boundary conditions, and we do not include terms withlower-order derivatives, because we know from the previous sections that both these ‘ingredients’ only serve to complicate things, butultimately they do not affect the singular value and eigenvalue distribution of the resulting discretizations matrices. To approximate thefourth derivative u(4)(x), we use the second-order central FD scheme (1,−4, 6,−4, 1), which yields the approximation

a(x)u(4)(x)|x=xj ≈ a(xj)u(xj+2)− 4u(xj+1) + 6u(xj)− 4u(xj−1) + u(xj−2)

h4, j = 2, . . . , n+ 1;

here, xj = jh, j = 0, . . . , n+ 3, and h = 1n+3 . Taking into account the homogeneous boundary conditions, we approximate the solution

of (8.55) by the piecewise linear function that takes the value uj in xj for j = 0, . . . , n + 3, where u0 = u1 = un+2 = un+3 = 0 andu = (u2, . . . , un+1)T is the solution of the linear system

a(xj)(uj+2 − 4uj+1 + 6uj − 4uj−1 + uj−2

)= h4f(xj), j = 2, . . . , n+ 1.

The matrix An of this linear system is given by

An =

6 a(x2) −4 a(x2) a(x2)

−4 a(x3) 6 a(x3) −4 a(x3) a(x3)

a(x4) −4 a(x4) 6 a(x4) −4 a(x4) a(x4)

. . . . . . . . . . . . . . .

a(xn−1) −4 a(xn−1) 6 a(xn−1) −4 a(xn−1) a(xn−1)

a(xn) −4 a(xn) 6 a(xn) −4 a(xn)

a(xn+1) −4 a(xn+1) 6 a(xn+1)

.

Let q(θ) = e−2iθ − 4e−iθ + 6− 4eiθ + e2iθ = 6− 8 cos θ + 2 cos(2θ). We prove that

Ann ∼GLT a(x)q(θ) (8.56)

andAnn ∼σ, λ a(x)q(θ), (8.57)

by showing that‖An − Dn(a) Tn(q)‖ → 0. (8.58)

Once (8.58) is proved, since Dn(a)Tn(q)n ∼GLT a(x)q(θ) and ‖Dn(a)Tn(q)‖ is uniformly bounded with respect to n (by GLT 3),and since ‖An‖ ≤ 16‖a‖∞ by (2.26), it follows from the decomposition

An = Dn(a) Tn(q) + (An − Dn(a) Tn(q))

and from GLT 5, GLT 1, GLT 2 that (8.56)–(8.57) hold.Let us prove (8.58). The matrices An and Dn(a) Tn(q) are banded (pentadiagonal), and, for all i, j = 1, . . . , n with |i− j| ≤ 2, a

crude estimates gives∣∣(An)ij − (Dn(a) Tn(q))ij∣∣ =

∣∣∣a(xi+1)(Tn(q))ij − a(min(i, j)

n

)(Tn(q))ij

∣∣∣ =∣∣∣a( i+ 1

n+ 3

)− a(min(i, j)

n

)∣∣∣ |(Tn(q))ij |

≤ 6ωa

( 6

n

).

Hence, by (2.26), ‖An − Dn(a) Tn(q)‖ ≤ 5 · 6ωa( 6n ).

Remark 8.5. The polynomial q(θ) is nonnegative over [−π, π] and, since the associated FD formula (1,−4, 6,−4, 1) approximatesu(4)(x), it is no surprise that

limθ→0

q(θ)

θ4= 1.

That is, q(θ) has a (unique) zero of order 4 at θ = 0; see also Remarks 8.1 and 8.4.

107

8.4.6 Non-uniform FD discretizationsAll the FD discretizations considered so far were based on uniform grids. It is natural to ask whether the theory of GLT sequences findsapplications also in the context of FD discretizations based on non-uniform grids. The answer to this question is affirmative, at least inthe case where the non-uniform grid is obtained as the mapping of a uniform grid through a fixed function G, independent of the meshsize. Let us illustrate this claim by means of a specific example.

Consider the simple diffusion equation (8.22) with a ∈ C([0, 1]). Choose a discretization parameter n ∈ N, and set h = 1n+1 and

xj = jh for all j = 0, . . . , n+ 1. Let G : [0, 1]→ [0, 1] be a fixed (invertible and increasing) map that generates the non-uniform meshwe intend to use. This means that the mesh consists of the points xj = G(xj), j = 0, . . . , n + 1, and of the stepsizes hj = xj − xj−1,j = 1, . . . , n+ 1. For j = 1, . . . , n, we approximate (a(x)u′(x))′|x=xj by the FD formula

−(a(x)u′(x))′|x=xj ≈ −a(xj +

hj+1

2 )u′(xj +hj+1

2 )− a(xj − hj2 )u′(xj − hj

2 )hj+1

2 +hj2

≈ −a(xj +

hj+1

2 )u(xj+1)− u(xj)

hj+1− a(xj − hj

2 )u(xj)− u(xj−1)

hjhj+1

2 +hj2

=2

hj + hj+1

[−a(xj − hj

2 )

hju(xj−1) +

(a(xj − hj

2 )

hj+a(xj +

hj+1

2 )

hj+1

)u(xj)−

a(xj +hj+1

2 )

hj+1u(xj+1)

].


−a(xj − hj

2 )

hjuj−1 +

(a(xj − hj

2 )

hj+a(xj +

hj+1

2 )

hj+1

)uj −

a(xj +hj+1

2 )

hj+1uj+1 =

hj + hj+1

2f(xj), j = 1, . . . , n. (8.59)

The matrix of this system is the n× n tridiagonal symmetric matrix given by

AG,n = tridiagn

[−a(xj − hj

2 )

hj

a(xj − hj2 )

hj+a(xj +

hj+1

2 )

hj+1−a(xj +

hj+1

2 )

hj+1

]. (8.60)

We perform the GLT analysis of the normalized matrix-sequence 1n+1AG,nn in the following two cases:

(i) G is regular, i.e., G ∈ C1([0, 1]) and G′(x) 6= 0 for all x ∈ [0, 1];

(ii) G ∈ C1([0, 1]) and there exists a finite number of points x where G′(x) = 0.

Note that G′(x) ≥ 0 for all x ∈ [0, 1], because G is assumed to be increasing and invertible (bijective). In particular, the assumptionG′(x) 6= 0 in (i) is equivalent to G′(x) > 0. In the case (ii), the map G is said to be singular and each point where G′ vanishes is referredto as a singularity point. Actually, case (ii) contains (i) as a special subcase, which occurs when the number of singularity points is 0.However, it is better to first address case (i) before passing to the more complicated case (ii). Under the assumptions in (i) and (ii), wewill show that 1

n+ 1AG,n

n∼GLT

a(G′(x))

G′(x)(2− 2 cos θ) (8.61)

and 1

n+ 1AG,n

n∼σ,λ

a(G′(x))

G′(x)(2− 2 cos θ). (8.62)

We begin by observing that, since G ∈ C1([0, 1]), for all j = 1, . . . , n+ 1 there exists ξj ∈ [xj−1, xj ] = [(j − 1)h, jh] such that

hj = G(xj)−G(xj−1) = G′(ξj)h = h(G′(xj) + εj), |εj | = |G′(ξj)−G′(xj)| ≤ ωG′(h) ≤ 2‖G′‖∞.

Hence, for all j = 1, . . . , n,

a(xj −

hj2

)= a

(G(xj)−

h

2(G′(xj) + εj)

)= a(G(xj)) + δj ,

a(xj +

hj+1

2

)= a

(G(xj) +

h

2(G′(xj+1) + εj+1)

)= a(G(xj)) + ηj ,

108

where δj = a(G(xj)− h2 (G′(xj) + εj))− a(G(xj)) and ηj = a(G(xj) + h

2 (G′(xj+1) + εj+1))− a(G(xj)) satisfy

|δj |, |ηj | ≤ ωa(3‖G′‖∞h

2

)≤ Cωa(h),

and C is a constant independent of j, n. We then arrive at the following expression:

1

n+ 1AG,n = hAG,n = tridiagn

[−a(G(xj)) + δj

G′(xj) + εj

a(G(xj)) + δjG′(xj) + εj

+a(G(xj)) + ηjG′(xj+1) + εj+1

− a(G(xj)) + ηjG′(xj+1) + εj+1

].

Consider the matrix

Dn

(a(G(x))

G′(x)

)Tn(2− 2 cos θ) = tridiagn

[−a(G(xj))

G′(xj)2a(G(xj))

G′(xj)− a(G(xj))

G′(xj)

].

Note that the function a(G(x))/G′(x) is either continuous (in case (i)) or continuous a.e. (in case (ii)), so in both cases we haveDn

(a(G(x))

G′(x)

)Tn(2− 2 cos θ)

n

∼GLTa(G(x))

G′(x)(2− 2 cos θ)

by GLT 3 and GLT 5. We will show that the fixed matrix-sequenceDn

(a(G(x))/G′(x)

)Tn(2−2 cos θ)

n

is an a.c.s. for 1n+1AG,nn

in both the cases (i) and (ii) (in case (i) we will directly show that the matrix-sequence Znn = 1n+1AG,n−Dn

(a(G(x))/G′(x)

)Tn(2−

2 cos θ)n is zero-distributed). Once this is proved, (8.61)–(8.62) follow from GLT 8 and GLT 1, taking into account that AG,n issymmetric.

Case (i). Let

Zn =1

n+ 1AG,n −Dn

(a(G(x))

G′(x)

)Tn(2− 2 cos θ). (8.63)

Zn is tridiagonal and a direct computation shows that all its components are bounded in modulus by a quantity that depends only onn,G, a and that and that converges to 0 as n→∞. For example, for all j = 2, . . . , n we have

|(Zn)j,j−1| =∣∣∣∣a(G(xj)) + δjG′(xj) + εj

− a(G(xj))

G′(xj)

∣∣∣∣ ≤ ∣∣∣∣a(G(xj)) + δjG′(xj) + εj

− a(G(xj))

G′(xj) + εj

∣∣∣∣+

∣∣∣∣ a(G(xj))

G′(xj) + εj− a(G(xj))

G′(xj)

∣∣∣∣=

∣∣∣∣ δjG′(xj) + εj

∣∣∣∣+

∣∣∣∣ a(G(xj))εjG′(xj)(G′(xj) + εj)

∣∣∣∣≤ Cωa(h)

minx∈[0,1]G′(x)− ωG′(h)+

‖a‖∞ωG′(h)

minx∈[0,1]G′(x)(minx∈[0,1]G′(x)− ωG′(h)), (8.64)

which tends to 0 as n → ∞ (recall that minx∈[0,1]G′(x) > 0 because in case (i) it is assumed that G is regular). Thus, ‖Zn‖ → 0 as

n→∞, and Znn ∼σ 0.

Case (ii). In this case, G is singular at a finite number of points, and the previous argument does not work because minx∈[0,1]G′(x) = 0.

However, we can still show thatDn

(a(G(x))/G′(x)

)Tn(2 − 2 cos θ)

n

is an a.c.s. for 1n+1AG,nn in the following way. Let

x(1), . . . , x(r) be the singularity points in which G′ vanishes, and consider the balls (intervals) B(x(k), 1m ) = x ∈ [0, 1] : |x− x(k)| <

1m. The function G′ is continuous and positive on the complement of the union

⋃rk=1B(x(k), 1

m ), so

minx∈[0,1]\

⋃rk=1 B(x(k), 1

m )G′(x) > 0.

For all indices j = 1, . . . , n such that xj ∈ [0, 1]\⋃rk=1B(x(k), 1

m ), the components in the j-th row of the matrix (8.63) are bounded inmodulus by a quantity that depends only on n,m,G, a and that converges to 0 as n → ∞. This becomes immediately clear if we notethat, for such indices j, the inequality (8.64) holds unchanged with minx∈[0,1]G

′(x) replaced by minx∈[0,1]\⋃rk=1 B(x(k), 1

m )G′(x). The

remaining rows of Zn, i.e., the rows corresponding to indices j such that xj ∈⋃rk=1B(x(k), 1

m ), are at most 2r(n+ 1)/m+ r; indeed,each interval B(x(k), 1

m ) has length 2/m (at most) and can contain at most a number of grid points xj equal to 2(n + 1)/m + 1. Thus,for every n,m we can split the matrix Zn into the sum of two terms, i.e.,

Zn = Rn,m +Nn,m, (8.65)

where Nn,m is obtained from Zn by setting to zero all the rows corresponding to indices j such that xj ∈⋃rk=1B(x(k), 1

m ) andRn,m = Zn−Nn,m is obtained fromZn by setting to zero all the rows corresponding to indices j such that xj ∈ [0, 1]\

⋃rk=1B(x(k), 1

m ).By the above discussion, we have

limn→∞

‖Nn,m‖ = 0 ∀m,

109

rank(Rn,m) ≤ 2r(n+ 1)

m+ r ∀m,n.

Since for every m we can choose nm such that, for n ≥ nm, ‖Nn,m‖ ≤ 1/m and rank(Rn,m) ≤ 3rn/m, the fixed matrix-sequenceDn

(a(G(x))/G′(x)

)Tn(2− 2 cos θ)

n

is an a.c.s. for 1n+1AG,nn.

Before concluding this section, it is worth emphasizing that the choice of a map G with one or more singularity points corresponds toadopting a local refinement strategy, according to which the grid points rapidly accumulate at the G-images of the singularity points asthe mesh size increases. For example, if

G(x) = xq, q > 1, (8.66)

then 0 is a singularity point of G (because G′(0) = 0) and the grid points

G(xj) = G( j

n+ 1

)=( j

n+ 1

)q, i = 0, . . . , n+ 1,

rapidly accumulate at the G-image of 0 (i.e., G(0) = 0) as n → ∞. Let us then answer to the following question: why should onebe interested in discretizing the diffusion equation (8.22) with a grid that rapidly accumulates at a point? The answer is that this localrefinement is necessary in some situations where the coefficient a(x) is strongly anisotropic. If a uniform discretization were used,the associated discretization step should be chosen very small, and this would result in a linear system with extremely large size: thecomputational cost to solve it would be unsustainable. For this reason, one adopts a local refinement, so that a coarse grid is used in thesubregions of the domain where a(x) is sufficiently smooth, and a finer grid is used only in the subregions where a(x) is, say, ‘not well-behaved’ (e.g., remarkably oscillatory). For example, the map G in (8.66) is a way to better approximate the solution in a neighborhoodof 0, and this is necessary if a(x) has a ‘wild behavior’ near 0, like for instance in the case where a(x) = 1

x+ε sin( 1x+ε ), with ε ≈ 0.

Exercise 8.3. Consider the matrix

BG,n = tridiagn

[− 2

hj + hj+1

a(xj − hj2 )

hj

2

hj + hj+1

(a(xj − hj

2 )

hj+a(xj +

hj+1

2 )

hj+1

)− 2

hj + hj+1

a(xj +hj+1

2 )

hj+1

].

(8.67)This would be the matrix of the linear system (8.59) if the factor hj+hj+1

2 in the right-hand side were moved to the left-hand side. Assumethat G is regular as in case (i) above, and show that 1

(n+ 1)2BG,n

n∼GLT

a(G(x))

(G′(x))2(2− 2 cos θ) (8.68)

and 1

(n+ 1)2BG,n

n∼σ,λ

a(G(x))

(G′(x))2(2− 2 cos θ). (8.69)

8.4.7 FE approximation of convection-diffusion-reaction equationsConsider the following second-order differential problem:

−(a(x)u′(x))′ + b(x)u′(x) + c(x)u(x) = f(x), x ∈ (0, 1),u(0) = u(1) = 0,

(8.70)

where f ∈ L2([0, 1]) and a, b, c : [0, 1]→ R are only assumed to be in L∞([0, 1]). We consider the approximation of (8.70) by classicallinear FEs on the uniform mesh in [0, 1] with stepsize h = 1

n+1 . Let us briefly describe this approximation technique; for more details,we refer the reader to [78, Chapter 4]. The weak form of (8.70) reads as follows [27, Chapter 8]: find u ∈ H1

0 ([0, 1]) such that, for allw ∈ H1

0 ([0, 1]),a(u,w) = f(w), (8.71)

where

a(u,w) =

∫ 1

0

a(x)u′(x)w′(x)dx+

∫ 1

0

b(x)u′(x)w(x)dx+

∫ 1

0

c(x)u(x)w(x)dx, f(w) =

∫ 1

0

f(x)w(x)dx.

Set xj = jh, j = 0, . . . , n + 1, and fix the subspace Wn = span(ϕ1, . . . , ϕn) ⊂ H10 ([0, 1]), where ϕ1, . . . , ϕn are the so-called

hat-functions. The function ϕi is given explicitly by

ϕi(x) =x− xi−1

xi − xi−1χ[xi−1,xi)(x) +

xi+1 − xxi+1 − xi

χ[xi,xi+1)(x), i = 1, . . . , n. (8.72)

110

Note that Wn is the space of piecewise linear functions corresponding to the sequence of points 0 = x0 < x1 < . . . < xn+1 = 1, i.e.,

Wn =s : [0, 1]→ R : s|[ i

n+1 ,i+1n+1 ) ∈ P1, i = 0, . . . , n

,

where P1 is the space of polynomials of degree less than or equal to 1. In the linear FE approach, we look for an approximation uWnof

u by solving the following (Galerkin) problem: find uWn∈ Wn such that, for all w ∈ Wn,

a(uWn , w) = f(w). (8.73)

Since ϕ1, . . . , ϕn is a basis of Wn, we can write uWn =∑nj=1 ujϕj for a unique vector u = (u1, . . . , un)T . By linearity, the

computation of uWn(i.e., of u) reduces to solving the linear system

Anu = f , (8.74)

where f = (f(ϕ1), . . . , f(ϕn))T and An is the stiffness matrix,

An = [a(ϕj , ϕi)]ni,j=1. (8.75)

Note that An admits the following decomposition:An = Kn + Zn, (8.76)

where

Kn =

[∫ 1

0

a(x)ϕ′j(x)ϕ′i(x)dx

]ni,j=1

(8.77)

is the (symmetric) diffusion matrix, and

Zn =

[∫ 1

0

b(x)ϕ′j(x)ϕi(x)dx

]ni,j=1

+

[∫ 1

0

c(x)ϕj(x)ϕi(x)dx

]ni,j=1

(8.78)

is the sum of the convection and reaction matrix. In the following, we compute the spectral and singular value distribution of the sequenceof normalized stiffness matrices 1

n+1Ann using the theory of GLT sequences. More precisely, we prove that 1

n+ 1An

n∼GLT a(x)(2− 2 cos θ) (8.79)

and 1

n+ 1An

n∼σ, λ a(x)(2− 2 cos θ). (8.80)

The proof consists of the following steps.

Step 1. We show that ∥∥∥ 1

n+ 1Kn

∥∥∥ ≤ C (8.81)

for some constant C independent of n, and ∥∥∥ 1

n+ 1Zn

∥∥∥ = O(n−1). (8.82)

Intuitively, (8.82) follows from (8.81) and from the fact that Zn is the matrix resulting from the discretization of the terms in (8.70) withlower-order derivatives: since 1

n+1 is the ‘correct’ normalization factor, which keeps the spectral norm of the diffusion matrix 1n+1Kn

bounded away from∞ (and from 0), we should expect that ‖ 1n+1Zn‖ → 0, or, more precisely, that ‖ 1

n+1Zn‖ = O(n−1).To prove (8.81), we note that Kn is a tridiagonal matrix, due to the local support property supp(ϕi) = [xi−1, xi+1], i = 1, . . . , n.

Moreover, by the inequality |ϕ′i(x)| ≤ n+ 1, for all i, j = 1, . . . , n we have

|(Kn)ij | =∣∣∣∣∫ 1

0


∣∣∣∣ =

∣∣∣∣∣∫ xi+1

xi−1


∣∣∣∣∣ ≤ (n+ 1)2‖a‖L∞∫ xi+1

xi−1

dx = 2(n+ 1)‖a‖L∞ .

Thus, the components of the tridiagonal matrix 1n+1Kn are bounded (in modulus) by 2‖a‖L∞ , and (8.81) follows from (2.26).

To prove (8.82), we follow the same argument as for the proof of (8.81). Due to the local support property of the hat-functions, Zn istridiagonal. Moreover, by the inequalities |ϕi(x)| ≤ 1 and |ϕ′i(x)| ≤ n+ 1, for all i, j = 1, . . . , n we have

|(Zn)ij | =

∣∣∣∣∣∫ xi+1

xi−1

b(x)ϕ′j(x)ϕi(x)dx+

∫ xi+1

xi−1

c(x)ϕj(x)ϕi(x)dx

∣∣∣∣∣ ≤ 2‖b‖L∞ +2‖c‖L∞n+ 1

,

111

and (8.82) follows from (2.26).

Step 2. We define the linear operator Ln(·) : L1([0, 1])→ Rn×n,

Ln(h) =

[∫ 1

0

h(x)ϕ′j(x)ϕ′i(x)dx

]ni,j=1

. (8.83)

By (8.77), we have Kn = Ln(a). The next three steps are devoted to show that 1

n+ 1Ln(h)

n∼GLT h(x)(2− 2 cos θ) ∀h ∈ L1([0, 1]). (8.84)

This implies in particular that 1n+1Knn ∼GLT a(x)(2− 2 cos θ).

Step 3. We first show that (8.84) holds when h(x) = 1 identically. Taking h(x) = 1 in (8.83), a direct computation based on (8.72)shows that Ln(1) = (n+ 1)Tn(2− 2 cos θ). Hence, by GLT 3, 1

n+ 1Ln(1)

n∼GLT 2− 2 cos θ. (8.85)

It is precisely the analysis of the constant-coefficient case considered in this step that allows one to realize what is the correct normalizationfactor. In our case, the correct factor is 1

n+1 , which removes the n + 1 from Ln(1) and yields a normalized matrix 1n+1Ln(1) =

Tn(2− 2 cos θ) whose components are bounded away from 0 and∞ (actually, in this case they are even constant).

Step 4. Next, we show that (8.84) holds if h ∈ C([0, 1]). We first illustrate the idea and then we go into the details. The proof isbased on the fact that the hat-functions (8.72) are ‘locally supported’. Indeed, the support [xi−1, xi+1] of the i-th hat-function ϕi(x) islocalized near the point i

n ∈ [xi, xi+1], and its width tends to 0 when n → ∞. Since h(x) varies continuously over [0, 1], the (i, j)-thentry of Ln(h) can be approximated as follows, for all i, j = 1, . . . , n:

(Ln(h))ij =

∫ 1

0

h(x)ϕ′j(x)ϕ′i(x)dx =

∫ xi+1

xi−1

h(x)ϕ′j(x)ϕ′i(x)dx

≈ h( in

)∫ xi+1

xi−1

ϕ′j(x)ϕ′i(x)dx = h( in

)∫ 1

0

ϕ′j(x)ϕ′i(x)dx = h( in

)(Ln(1))ij . (8.86)

After normalization, we can rewrite (8.86) in matrix form,

1

n+ 1Ln(h) ≈ 1

n+ 1Dn(h)Ln(1). (8.87)

As we shall see, the approximation (8.87) implies 1n+1Ln(h) − 1

n+1Dn(h)Ln(1)n ∼σ 0, and so (8.84) follows from the relation 1n+1Ln(1)n ∼GLT 2− 2 cos θ (see Step 3) and from GLT 3, GLT 5.

Now let us go into the details. Since |ϕ′i(x)| ≤ n+ 1, for all i, j = 1, . . . , n we have

|(Ln(h))ij − (Dn(h)Ln(1))ij | =∣∣∣∣∫ 1

0

[h(x)− h

( in

)]ϕ′j(x)ϕ′i(x)dx

∣∣∣∣ ≤ (n+ 1)2

∫ xi+1

xi−1

∣∣∣h(x)− h( in

)∣∣∣dx≤ 2(n+ 1)ωh

( 2

n+ 1

).

It follows that each component of the matrix 1n+1Ln(h) − 1

n+1Dn(h)Ln(1) is bounded (in modulus) by 2ωh( 2n+1 ). Moreover,

1n+1Ln(h)− 1

n+1Dn(h)Ln(1) is tridiagonal, because of the local support property of the hat-functions. Thus, both the 1-norm and the∞-norm of 1

n+1Ln(h)− 1n+1Dn(h)Ln(1) are bounded by 6ωh( 2

n+1 ), and (2.26) yields ‖ 1n+1Ln(h)− 1

n+1Dn(h)Ln(1)‖ ≤ 6ωh( 2n+1 ).

Hence, 1n+1Ln(h)− 1

n+1Dn(h)Ln(1)n ∼σ 0, and this implies (8.84) by 1n+1Ln(1)n ∼GLT 2− 2 cos θ and GLT 3, GLT 5.

Step 5. Finally, we show that (8.84) holds in the general case where h ∈ L1([0, 1]). By the density of C([0, 1]) in L1([0, 1]), thereexists a sequence of continuous functions hm ∈ C([0, 1]) such that hm → h in L1([0, 1]). Clearly hm → h in measure over [0, 1]. ByStep 4, 1

n+1Ln(hm)n ∼GLT hm(x)(2 − 2 cos θ). In addition, 1n+1Ln(hm)nm is an a.c.s. for 1

n+1Ln(h)n. Indeed, using(2.29) and observing that

∑ni=1 |ϕ′i(x)| ≤ 2(n+ 1) for all x ∈ [0, 1], we obtain

‖Ln(h)−Ln(hm)‖1 ≤n∑

i,j=1

|(Ln(h))ij − (Ln(hm))ij | =n∑

i,j=1

∣∣∣∣∫ 1

0

[h(x)− hm(x)

]ϕ′j(x)ϕ′i(x)dx

∣∣∣∣≤∫ 1

0

|h(x)− hm(x)|n∑

i,j=1

|ϕ′j(x)| |ϕ′i(x)|dx ≤ 4(n+ 1)2

∫ 1

0

|h(x)− hm(x)|dx,

112

100 200 300 400 500 600 700 800 900−8

−6

−4

−2

0

2

4

6

Λ((n+1)−1An)


Figure 8.3: spectrum of 1n+1An and samplings a( i` )(2 − 2 cos jπ` ), i, j = 1, . . . , `, for n = `2 = 900, a(x) = ex sin(4x) and b(x) =

c(x) = 0.

and ∥∥∥ 1

n+ 1Ln(h)− 1

n+ 1Ln(hm)

∥∥∥1≤ Cn‖h− hm‖L1

for some constant C independent of n and m. 1n+1Ln(hm)nm is then an a.c.s. for 1

n+1Ln(h)n by Corollary 4.3, and (8.84)follows from GLT 8.

Step 6. As already noted in Step 2, from (8.84) we get 1n+1Knn ∼GLT a(x)(2 − 2 cos θ). Since we have seen in Step 1 that

1n+1Znn is zero-distributed, (8.79) follows from the decomposition

1

n+ 1An =

1

n+ 1Kn +

1

n+ 1Zn (8.88)

(cf. (8.76)) and from GLT 3, GLT 5; and the singular value distribution in (8.80) follows from GLT 1. If b(x) = 0 identically, 1n+1An is

symmetric and also the spectral distribution in (8.80) follows from GLT 1. If b(x) is not identically 0, the spectral distribution in (8.80)follows from GLT 2 applied to the decomposition (8.88), taking into account what we have seen in Step 1.

Figure 8.3 shows, for n = `2 = 900, the spectrum of 1n+1An and the uniform samplings a( i` )(2 − 2 cos jπ` ), i, j = 1, . . . , `, in the

case where a(x) = ex sin(4x) and b(x) = c(x) = 0. Note that in this case An is just Kn. We see from the figure that the symbol-to-spectrum approximation is quite accurate. The∞-norm of the difference between the vector of the eigenvalues and the vector of thesamplings, both sorted in non-increasing order as in Figure 8.3, is about 0.8197.

Exercise 8.4. Consider problem (8.70) and assume that b(x) = 0 identically. Note that the matrices An are symmetric in this case. Showthat (8.79)–(8.80) continue to hold even if the PDE coefficients a(x) and c(x) are only assumed to be in L1([0, 1]).

8.4.8 Schur complements of matrices arising from the FE approximation of a system of PDEsWe consider in this section the linear FE approximation of a system of PDEs. The resulting discretization matrices show up in saddlepoint form [9], and we will see the way to compute the asymptotic spectral distribution of their Schur complements using the theoryof GLT sequences. We recall that the Schur complement is a key tool for the numerical treatment of the related linear systems; see [9,Section 5]. The analysis of this section is the same as the analysis in [39, Section 2], with the only difference that the discretizationtechnique considered herein is a pure FE approximation, whereas in [39, Section 2] the authors adopted a mixed FD/FE technique. Foran extension of the analysis of this section to the 2-dimensional setting, as well as for the GLT analysis of the linear elasticity equations,we refer the reader to [39].

Consider the system of PDEs −(a(x)u′(x))′ + v′(x) = f(x), x ∈ (0, 1),

−u′(x)− ρ v(x) = g(x), x ∈ (0, 1),(8.89)

113

with homogeneous Dirichlet boundary conditions: u(0) = u(1) = 0 and v(0) = v(1) = 0. In (8.89), a ∈ L∞([0, 1]) and ρ is a constant.As in Section 8.4.7, we consider the approximation of (8.89) by linear FEs on the uniform mesh in [0, 1] with stepsize h = 1

n+1 . Theweak form of (8.89) reads as follows: find u, v ∈ H1

0 ([0, 1]) such that, for all w ∈ H10 ([0, 1]),

∫ 1

0a(x)u′(x)w′(x)dx+

∫ 1

0v′(x)w(x)dx =

∫ 1

0f(x)w(x)dx,

−∫ 1

0u′(x)w(x)dx− ρ

∫ 1

0v(x)w(x)dx =

∫ 1

0g(x)w(x)dx.

(8.90)

Set xj = jh, j = 0, . . . , n + 1, and fix the subspace Wn = span(ϕ1, . . . , ϕn) ⊂ H10 ([0, 1]), where ϕ1, . . . , ϕn are the hat-functions in

(8.72). Then, we look for approximations uWn, vWn

of u, v by solving the following (Galerkin) problem: find uWn, vWn

∈ Wn such that,for all w ∈ Wn,

∫ 1

0a(x)u′Wn

(x)w′(x)dx+∫ 1

0v′Wn

(x)w(x)dx =∫ 1

0f(x)w(x)dx,

−∫ 1

0u′Wn

(x)w(x)dx− ρ∫ 1

0vWn(x)w(x)dx =

∫ 1

0g(x)w(x)dx.

(8.91)

Since ϕ1, . . . , ϕn is a basis of Wn, we can write uWn=∑nj=1 ujϕj and vWn

=∑nj=1 vjϕj for unique vectors u = (u1, . . . , un)T

and v = (v1, . . . , vn)T . By linearity, the computation of uWn, vWn

(i.e., of u,v) reduces to solving the linear system

A2n

[uv

]=

[fg

], (8.92)

where f =[∫ 1

0f(x)ϕi(x)dx

]ni=1

, g =[∫ 1

0g(x)ϕi(x)dx

]ni=1

, and A2n is the stiffness matrix, which admits the following saddle pointstructure:

A2n =

[Kn Hn

HTn −ρMn

].

Here, the blocks Kn, Hn,Mn are square matrices of size n, and precisely

Kn =

[∫ 1

0


]ni,j=1

, (8.93)

Hn =

[∫ 1

0

ϕ′j(x)ϕi(x)dx

]ni,j=1

=1

2

0 1−1 0 1

. . . . . . . . .−1 0 1

−1 0

= −iTn(sin θ), (8.94)

Mn =

[∫ 1

0

ϕj(x)ϕi(x)dx

]ni,j=1

=h

6

4 11 4 1

. . . . . . . . .1 4 1

1 4

=h

3Tn(2 + cos θ). (8.95)

Note that Kn is exactly the matrix appearing in (8.77). The matrices Kn, Mn are symmetric, while Hn is skew-symmetric. In particular,HTn = −Hn = iTn(sin θ). In the case where a(x) is constant, say a(x) = a0 identically, the matrix Kn becomes

Kn = a0

[∫ 1

0

ϕ′j(x)ϕ′i(x)dx

]ni,j=1

= a01

h

2 −1−1 2 −1

. . . . . . . . .−1 2 −1

−1 2

= a01

hTn(2− 2 cos θ). (8.96)

From now on we assume that the matrices Kn are invertible.2 The (negative) Schur complement of A2n is the symmetric matrix given by

Sn = ρMn +HnK−1n HT

n =ρh

3Tn(2 + cos θ) + Tn(sin θ)K−1

n Tn(sin θ). (8.97)

In the following, we perform the GLT analysis of the sequence of normalized Schur complements (n + 1)Snn, and we compute itsasymptotic spectral and singular value distribution.

2This is satisfied, for example, if a(x) = a0 a.e. with a0 6= 0, or if a > 0 a.e., in which case the matrices Kn are positive definite.

114

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

Λ((n+1)S

n)

graph of ς(θ)

Figure 8.4: graph of ς(θ) and spectrum of (n+ 1)Sn for n = 50, with a0 = 1, ρ = 1.2.

We first consider the case where a(x) is constant, a(x) = a0 identically (a0 6= 0). In this case, Sn can be expressed in terms ofToeplitz matrices as follows:

Sn =ρh

3Tn(2 + cos θ) +

h

a0Tn(sin θ)

(Tn(2− 2 cos θ)

)−1Tn(sin θ). (8.98)

According to GLT 3, we have Tn(2 + cos θ)n ∼GLT 2 + cos θ, Tn(sin θ)n ∼GLT sin θ and Tn(2− 2 cos θ)n ∼GLT 2− 2 cos θ.By GLT 5 – GLT 6, taking into account that 2− 2 cos θ 6= 0 a.e. and (n+ 1)h = 1, we get

(n+ 1)Snn ∼GLT ς(θ) =ρ

3(2 + cos θ) +

1

a0

sin2 θ

2− 2 cos θ. (8.99)

Since Sn is symmetric, (8.99) and GLT 1 imply that Snn ∼σ, λ ς(θ). Considering that ς(θ) is even, the relations Snn ∼σ, λ ς(θ)continue to hold if we consider [0, π] as the domain of ς(θ) instead of [0, 1] × [−π, π]. In particular, according to Remark 2.3, for largen the eigenvalues of (n + 1)Sn are approximately given by ς( jπ

n+1 ), j = 1, . . . , n. This theoretical forecast is confirmed by Figure 8.4,where we fixed a0 = 1 and ρ = 1.2, and we plotted the graph of ς(θ) together with the spectrum of (n + 1)Sn for n = 50. Notethat, in the figure, the spectrum of (n + 1)Sn is represented by the pairs ( jπ

n+1 , λj((n + 1)Sn)), j = 1, . . . , n, where the eigenvaluesλj((n+ 1)Sn) are labeled in decreasing order.

Now we consider the general case where a(x) is an arbitrary function in L∞([0, 1]). In this case, eq. (8.96) does not hold and Kn isno longer a Toeplitz matrix. However, as proved in Section 8.4.7 (see Step 2), 1

n+1Knn ∼GLT a(x)(2− 2 cos θ). Therefore, assumingthat a(x) 6= 0 a.e., from the expression (8.97) and GLT 3, GLT 5 – GLT 6 we obtain

(n+ 1)Snn ∼GLT ς(x, θ) =ρ

3(2 + cos θ) +

sin2 θ

a(x)(2− 2 cos θ). (8.100)

As a consequence, by GLT 1 we get(n+ 1)Snn ∼σ, λ ς(x, θ), (8.101)

and these relations continue to hold even if we consider [0, 1]× [0, π] as the domain of ς(x, θ) instead of [0, 1]× [−π, π], because ς(x, θ)is symmetric with respect to the Fourier variable θ. According to Remark 2.3, if n = `2 is large enough, the eigenvalues of (n+ 1)Sn areapproximately given by the uniform sampling ς( i` ,

jπ`+1 ), i, j = 1, . . . , `. This is confirmed by Figure 8.5, where we fixed a(x) = 1 + x

and ρ = 0.9, and we plotted the spectrum of (n+ 1)Sn together with the values ς( i` ,jπ`+1 ), i, j = 1, . . . , `, for n = `2 = 100. Note that,

in the figure, the eigenvalues of (n+ 1)Sn, as well as the samplings of the symbol ς(x, θ), are depicted in non-increasing order.

Remark 8.6. In view of Exercise 8.4, the relation 1n+1Knn ∼GLT a(x)(2 − 2 cos θ) and, consequently, (8.100)–(8.101) continue to

hold even if a(x) is only assumed to be in L1([0, 1]). The reason for which we assumed a(x) to be in L∞([0, 1]) was to ensure that theweak form (8.91) is well-defined. This shows that the theory of GLT sequences allows one to obtain spectral distribution results for PDEdiscretization matrices under minimal assumptions on the PDE coefficients. In the present case, such assumptions do not even guaranteethe well-posedness of the differential problem under consideration.

115

0 10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Λ((n+1)S

n)

uniform sampling of ς(x,θ)

Figure 8.5: spectrum of (n+ 1)Sn and samplings ς( i` ,jπ`+1 ), i, j = 1, . . . , `, for n = `2 = 100, a(x) = 1 + x, ρ = 0.9.

8.4.9 B-spline IgA collocation approximation of convection-diffusion-reaction equationsIn this section, we focus on the differential problem

−(a(x)u′(x))′ + b(x)u′(x) + c(x)u(x) = f(x), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,

(8.102)

where Ω is a bounded open interval in R, a : Ω → R is a function in C1(Ω), and b, c, f : Ω → R are functions in C(Ω). We considerthe isogeometric collocation approximation of (8.102) based on uniform B-splines of degree p ≥ 2. Since this approximation techniqueis not as known as FDs or FEs, we describe it below in some detail.

It is worth highlighting the following aspect. In its original formulation [28, 63], IgA employs Galerkin discretizations rather thancollocation discretizations. In the Galerkin framework an efficient implementation requires special numerical quadrature rules whenconstructing the system of equations; see, e.g., [66]. To avoid this issue, isogeometric collocation methods have been recently introducedin [3, 83]. The latter methods are the subject of this section, whereas isogeometric Galerkin methods will be considered in Section 8.4.10.

Isogeometric collocation approximation Problem (8.102) can be reformulated as follows:−a(x)u′′(x) + s(x)u′(x) + c(x)u(x) = f(x), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,

(8.103)

where s(x) = b(x)−a′(x). In the standard collocation method, we choose a finite dimensional vector space W , consisting of sufficientlysmooth functions defined on Ω and vanishing on the boundary ∂Ω. We call W the approximation space. Then, we introduce a set ofN = dimW collocation points τ1, . . . , τN ⊂ Ω, and we look for a function uW ∈ W such that

−a(τi)u′′W (τi) + s(τi)u

′W (τi) + c(τi)uW (τi) = f(τi), i = 1, . . . , N. (8.104)

The function uW is taken as an approximation to the solution of (8.103). If ϕ1, . . . , ϕN is a basis of W , then uW =∑Ni=1 uiϕi for a

unique vector u = (u1, . . . , uN )T , and, by linearity, the computation of uW (i.e., of u) reduces to solving the linear system

Au = f , (8.105)

where

A =[−a(τi)ϕ

′′j (τi) + s(τi)ϕ

′j(τi) + c(τi)ϕj(τi)

]Ni,j=1

=(

diagi=1,...,N

a(τi))[−ϕ′′j (τi)

]Ni,j=1

+(

diagi=1,...,N

s(τi))[ϕ′j(τi)

]Ni,j=1

+(

diagi=1,...,N

c(τi))[ϕj(τi)

]Ni,j=1

(8.106)

is the collocation matrix and f =[f(τi)

]Ni=1

.Now, suppose that the physical domain Ω can be described by a global geometry function G : [0, 1] → Ω, which is invertible and

satisfies G(∂([0, 1])) = ∂Ω. Letϕ1, . . . , ϕN (8.107)

116

be a set of basis functions defined on the parametric (or reference) domain [0, 1] and vanishing on the boundary ∂([0, 1]). Let

τ1, . . . , τN (8.108)

be a set of N collocation points in the parametric domain [0, 1]. In the isogeometric collocation approach, we find an approximation uW

of u by using the standard collocation method described above, in which:

• the approximation space is chosen as W = span(ϕi, i = 1, . . . , N), with

ϕi(x) = ϕi(G−1(x)) = ϕi(x), x = G(x); (8.109)

• the collocation points in the physical domain Ω are defined as follows:

τi = G(τi), i = 1, . . . , N. (8.110)

The resulting collocation matrix A is given by (8.106), with the basis functions ϕi and the collocation points τi defined as in (8.109)–(8.110).

Assuming that G and ϕi, i = 1, . . . , N , are sufficiently regular, we can apply standard differential calculus to express A in terms ofG and ϕi, τi, i = 1, . . . , N . Let us work out this expression. For any u : Ω → R, consider its corresponding function defined on theparametric domain by

u : [0, 1]→ R, u(x) = u(x), x = G(x). (8.111)

In other words, u(x) = u(G(x)).3 Then, u satisfies (8.103) if and only if u satisfies the corresponding transformed problem−aG(x)u′′(x) + sG(x)u′(x) + cG(x)u(x) = f(G(x)), x ∈ (0, 1),u(x) = 0, x ∈ ∂((0, 1)),

(8.112)

where aG, sG, cG are, respectively, the transformed diffusion, convection, reaction coefficient of the PDE (8.103). They are given by

aG(x) =a(G(x))

(G′(x))2, sG(x) =

a(G(x))G′′(x)

(G′(x))3+s(G(x))

G′(x), cG(x) = c(G(x)), x ∈ [0, 1]. (8.113)

The collocation matrix A in (8.106) can be expressed in terms of G and ϕi, τi, i = 1, . . . , N , as follows:

A =[−aG(τi)ϕ

′′j (τi) + sG(τi)ϕ

′j(τi) + cG(τi)ϕj(τi)

]Ni,j=1

=(

diagi=1,...,N

aG(τi))[−ϕ′′j (τi)

]Ni,j=1

+(

diagi=1,...,N

sG(τi))[ϕ′j(τi)

]Ni,j=1

+(

diagi=1,...,N

cG(τi))[ϕj(τi)

]Ni,j=1

. (8.114)

In the context of IgA, the geometry map G is expressed in terms of the functions ϕi, in accordance with the isoparametric approach[28, Section 3.1]. Moreover, the functions ϕi themselves are usually B-splines or their rational versions, the so-called NURBS. In thissection, the role of the ϕi will be played by B-splines over uniform knot sequences (for the case of NURBS, see [49]). Furthermore,we do not limit ourselves to the isoparametric approach, but we allow the geometry map G to be any sufficiently regular function from[0, 1] to Ω, not necessarily expressed in terms of B-splines. Finally, following [3], the collocation points τi will be chosen as the Grevilleabscissae corresponding to the B-splines ϕi.

B-splines and Greville abscissae For p, n ≥ 1, consider the uniform knot sequence

t1 = · · · = tp+1 = 0 < tp+2 < · · · < tp+n < 1 = tp+n+1 = · · · = t2p+n+1, (8.115)

whereti+p+1 =

i

n, i = 0, . . . , n. (8.116)

The B-splines of degree p on this knot sequence are denoted by

Ni,[p] : [0, 1]→ R, i = 1, . . . , n+ p, (8.117)

and are defined recursively as follows [31]: for 1 ≤ i ≤ n+ 2p,

Ni,[0](t) = χ[ti,ti+1)(t), t ∈ [0, 1]; (8.118)

3Note that ϕi(x) = ϕi(G(x)) for i = 1, . . . , N , so ϕ1, . . . , ϕN are obtained from ϕ1, . . . , ϕN by the rule (8.111). Moreover, the equation τi = G(τi) is the sameas the relation x = G(x) in (8.111).

117

for 1 ≤ k ≤ p and 1 ≤ i ≤ n+ 2p− k,

Ni,[k](t) =t− ti

ti+k − tiNi,[k−1](t) +

ti+k+1 − tti+k+1 − ti+1

Ni+1,[k−1](t), t ∈ [0, 1], (8.119)

where we assume that a fraction with zero denominator is zero. The Greville abscissa ξi,[p] associated with the B-spline Ni,[p] is definedby

ξi,[p] =ti+1 + . . .+ ti+p

p, i = 1, . . . , n+ p. (8.120)

We know from [31] that the functions N1,[p], . . . , Nn+p,[p] form a basis for the spline spaces ∈ Cp−1([0, 1]) : s|[ in , i+1

n ) ∈ Pp, i = 0, . . . , n− 1,

where Pp is the space of polynomials of degree less than or equal to p. Moreover, N1,[p], . . . , Nn+p,[p] possess the following properties.

• Local support property:supp(Ni,[p]) = [ti, ti+p+1], i = 1, . . . , n+ p. (8.121)

• Vanishing on the boundary:Ni,[p](0) = Ni,[p](1) = 0, i = 2, . . . , n+ p− 1. (8.122)

• Nonnegative partition of unity:

Ni,[p](t) ≥ 0, t ∈ [0, 1], i = 1, . . . , n+ p, (8.123)n+p∑i=1

Ni,[p](t) = 1, t ∈ [0, 1]. (8.124)

• Bounds for derivatives:

n+p∑i=1

|N ′i,[p](t)| ≤ 2pn,

n+p∑i=1

|N ′′i,[p](t)| ≤ 4p(p− 1)n2, t ∈ [0, 1]. (8.125)

Note that the derivatives N ′1,[p](t), . . . , N′n+p,[p](t) (resp., N ′′1,[p](t), . . . , N

′′n+p,[p](t)) may not be defined at some of the points

1n , . . . ,

n−1n when p = 1 (resp., p = 1, 2). In (8.125), it is understood that the undefined values are counted as 0 in the summations.

Let φ[q] be the cardinal B-spline of degree q ≥ 0 over the uniform knot sequence 0, 1, . . . , q + 1, which is defined recursively asfollows [31]:

φ[0](t) = χ[0,1)(t), t ∈ R, (8.126)

φ[q](t) =t

qφ[q−1](t) +

q + 1− tq

φ[q−1](t− 1), t ∈ R, q ≥ 1. (8.127)

It is known that [30, 31]supp(φ[q]) = [0, q + 1]. (8.128)

Moreover, the following symmetry property holds [46, Lemma 3] (see also [30, p. 86]):

φ(r)[q]

(q + 1

2+ t)

= (−1)rφ(r)[q]

(q + 1

2− t), t ∈ R, r, q ≥ 0, (8.129)

where φ(r)[q] is the r-th derivative of φ[q]. Note that φ(r)

[q] (t) is defined for all t ∈ R if r < q, and for all t ∈ R\0, 1, . . . , q + 1 ifr ≥ q. Nevertheless, (8.129) holds for all t ∈ R, because when the left-hand side is not defined, the right-hand side is not defined as well.Concerning the L2 inner products of derivatives of cardinal B-splines, it was proved in [46, Lemma 4] that∫

Rφ

(r1)[q1] (t)φ

(r2)[q2] (t+τ)dt = (−1)r1φ

(r1+r2)[q1+q2+1](q1 +1+τ) = (−1)r2φ

(r1+r2)[q1+q2+1](q2 +1−τ), τ ∈ R, q1, q2, r1, r2 ≥ 0. (8.130)

Eq. (8.130) generalizes the result given in [30, p. 89]. Cardinal B-splines are of interest herein, because the so-called central basisfunctionsNi,[p], i = p+1, . . . , n, are uniformly shifted and scaled versions of the cardinal B-spline φ[p]. This is illustrated in Figures 8.6–8.7 for p = 3. More precisely, we have

Ni,[p](t) = φ[p](nt− i+ p+ 1), t ∈ [0, 1], i = p+ 1, . . . , n, (8.131)

118

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Figure 8.6: graph of the B-splines Ni,[p], i = 1, . . . , n+ p, for p = 3 and n = 10; the central basis functions Ni,[p], i = p+ 1, . . . , n, aredepicted in blue.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

Figure 8.7: graph of the cubic cardinal B-spline φ[3].

and, consequently,

N ′i,[p](t) = nφ′[p](nt− i+ p+ 1), t ∈ [0, 1], i = p+ 1, . . . , n, (8.132)

N ′′i,[p](t) = n2φ′′[p](nt− i+ p+ 1), t ∈ [0, 1], i = p+ 1, . . . , n. (8.133)

In view of (8.120) and (8.121), it is clear that the Greville abscissa ξi,[p] lies in the support of Ni,[p],

ξi,[p] ∈ supp(Ni,[p]) = [ti, ti+p+1], i = 1, . . . , n+ p. (8.134)

The central Greville abscissae ξi,[p], i = p+1, . . . , n, which are the Greville abscissae associated with the central basis functions (8.131),simplify to

ξi,[p] =i

n− p+ 1

2n, i = p+ 1, . . . , n. (8.135)

The Greville abscissae are somehow equivalent, in an asymptotic sense, to the uniform knots in [0, 1]. More precisely,∣∣∣ξi,[p] − i

n+ p

∣∣∣ ≤ Cpn, i = 1, . . . , n+ p, (8.136)

where Cp depends only on p. The proof of (8.136) is a matter of computations; we leave the details to the reader.

B-spline IgA collocation matrices In the IgA collocation approach based on (uniform) B-splines, the basis functions ϕ1, . . . , ϕN in(8.107) are chosen as the B-splines N2,[p], . . . , Nn+p−1,[p], i.e.,

ϕi = Ni+1,[p], i = 1, . . . , n+ p− 2. (8.137)

In this setting, N = n+p−2. Note that the boundary functions N1,[p] and Nn+p,[p] are excluded because they do not vanish on ∂([0, 1]).Moreover, following [3], the collocation points τ1 . . . , τN in (8.108) are chosen as the Greville abscissae corresponding to the B-splines(8.137), i.e.,

τi = ξi+1,[p], i = 1, . . . , n+ p− 2. (8.138)

Throughout this section, we will assume that p ≥ 2, so as to ensure that N ′′j+1,[p](ξi+1,[p]) is defined for all i, j = 1, . . . , n+ p− 2.

119

The collocation matrix (8.114) resulting from the choices of ϕi, τi as in (8.137)–(8.138) will be denoted by A[p]G,n, in order to

emphasize its dependence on the geometry map G and on the parameters n, p:

A[p]G,n =

[−aG(ξi+1,[p])N

′′j+1,[p](ξi+1,[p]) + sG(ξi+1,[p])N

′j+1,[p](ξi+1,[p]) + cG(ξi+1,[p])Nj+1,[p](ξi+1,[p])

]n+p−2

i,j=1

= D[p]n (aG)K [p]

n +D[p]n (sG)H [p]

n +D[p]n (cG)M [p]

n , (8.139)

whereD[p]n (h) = diag

i=1,...,n+p−2h(ξi+1,[p]) (8.140)

is the diagonal sampling matrix containing the samples of the function h : [0, 1]→ R at the Greville abscissae (8.138), and

K [p]n =

[−N ′′j+1,[p](ξi+1,[p])

]n+p−2

i,j=1, (8.141)

H [p]n =

[N ′j+1,[p](ξi+1,[p])

]n+p−2

i,j=1, (8.142)

M [p]n =

[Nj+1,[p](ξi+1,[p])

]n+p−2

i,j=1. (8.143)

Note that A[p]G,n can be decomposed as follows:

A[p]G,n = K

[p]G,n + Z

[p]G,n, (8.144)

whereK

[p]G,n =

[−aG(ξi+1,[p])N

′′j+1,[p](ξi+1,[p])

]n+p−2

i,j=1= D[p]

n (aG)K [p]n (8.145)

is the collocation diffusion matrix, i.e., the matrix resulting from the collocation discretization of the higher-order (diffusion) term in(8.103), and

Z[p]G,n =

[sG(ξi+1,[p])N

′j+1,[p](ξi+1,[p]) + cG(ξi+1,[p])Nj+1,[p](ξi+1,[p])

]n+p−2

i,j=1= D[p]

n (sG)H [p]n +D[p]

n (cG)M [p]n (8.146)

is the matrix resulting from the discretization of the terms in (8.103) with lower-order derivatives (i.e., the convection and reaction terms).The matrix Z [p]

G,n can be regarded as a ‘residual term’. Indeed, we shall see that the norm of Z [p]G,n is negligible with respect to the norm

of the diffusion matrix K [p]G,n when the discretization parameter n is large. In fact, after normalization by n2, it turns out that ‖ 1

n2Z[p]G,n‖

tends to 0 as n→∞ (contrary to ‖ 1n2K

[p]G,n‖, which remains bounded away from 0 and∞).

Let us now provide an approximate construction of K [p]n , M [p]

n , H [p]n . This is necessary for the GLT analysis of this section, and it is

important also in view of Section 8.5. We only construct the submatrices[(K [p]

n )ij]n−1

i,j=p,

[(H [p]

n )ij]n−1

i,j=p,

[(M [p]

n )ij]n−1

i,j=p, (8.147)

which are determined by the central basis functions (8.131) and by the central Greville abscissae (8.135). Note that the submatrix[(K

[p]n )ij

]n−1

i,j=p, when embedded in any matrix of size n + p − 2, provides an approximation of K [p]

n up to a low-rank correction. A

similar consideration applies also to the submatrices[(H

[p]n )ij

]n−1

i,j=p,[(M

[p]n )ij

]n−1

i,j=p. A direct computation based on (8.129), (8.131)–

(8.133) and (8.135) shows that

(K [p]n )ij = −n2φ′′[p]

(p+ 1

2+ i− j

)= −n2φ′′[p]

(p+ 1

2− i+ j

), i, j = p, . . . , n− 1,

(H [p]n )ij = nφ′[p]

(p+ 1

2+ i− j

)= −nφ′[p]

(p+ 1

2− i+ j

), i, j = p, . . . , n− 1,

(M [p]n )ij = φ[p]

(p+ 1

2+ i− j

)= φ[p]

(p+ 1

2− i+ j

), i, j = p, . . . , n− 1.

Since their entries depend only on the difference i− j, the submatrices (8.147) are Toeplitz matrices, and precisely

[(K [p]

n )ij]n−1

i,j=p= n2

[−φ′′[p]

(p+ 1

2− i+ j

)]n−1

i,j=p

= n2 Tn−p(fp), (8.148)

[(H [p]

n )ij]n−1

i,j=p= n

[−φ′[p]

(p+ 1

2− i+ j

)]n−1

i,j=p

= niTn−p(gp), (8.149)

[(M [p]

n )ij]n−1

i,j=p=

[φ[p]

(p+ 1

2− i+ j

)]n−1

i,j=p

= Tn−p(hp), (8.150)

120

where

fp(θ) =∑k∈Z−φ′′[p]

(p+ 1

2− k)

eikθ = −φ′′[p](p+ 1

2

)− 2

bp/2c∑k=1

φ′′[p]

(p+ 1

2− k)

cos(kθ), (8.151)

gp(θ) =∑k∈Z−φ′[p]

(p+ 1

2− k)

eikθ = −2

bp/2c∑k=1

φ′[p]

(p+ 1

2− k)

sin(kθ), (8.152)

hp(θ) =∑k∈Z

φ[p]

(p+ 1

2− k)

eikθ = φ[p]

(p+ 1

2

)+ 2

bp/2c∑k=1

φ[p]

(p+ 1

2− k)

cos(kθ); (8.153)

note that we used (8.128)–(8.129) to simplify the expressions of fp(θ), gp(θ), hp(θ). It follows from (8.148) that Tn−p(fp) is the principalsubmatrix of both 1

n2K[p]n and Tn+p−2(fp) corresponding to the set of indices p, . . . , n− 1. Similar results follow from (8.149)–(8.150),

and so we obtain1

n2K [p]n = Tn+p−2(fp) +R[p]

n , rank(R[p]n ) ≤ 4(p− 1), (8.154)

− inH [p]n = Tn+p−2(gp) + S[p]

n , rank(S[p]n ) ≤ 4(p− 1), (8.155)

M [p]n = Tn+p−2(hp) + V [p]

n , rank(V [p]n ) ≤ 4(p− 1). (8.156)

To better appreciate the above construction, let us see two examples. We only consider the case of the matrix K [p]n , since for H [p]

n andM

[p]n the matter is the same. In the first example, we fix p = 3. The matrix 1

n2K[3]n is given by

1

n2K [3]n =

1

6

33 −7 −2−9 15 −6

−6 12 −6−6 12 −6

. . . . . . . . .−6 12 −6

−6 12 −6−6 15 −9−2 −7 33

. (8.157)

The submatrix Tn−2(f3) is shown in the box and f3(θ) = 2−2 cos θ, as given by (8.151) for p = 3. In the second example, we fix p = 4.The matrix 1

n2K[4]n is given by

1

n2K [4]n =

1

96

855 −133 −71 −3−81 243 −63 −27−36 −36 132 −48 −12

−16 −44 120 −48 −12−12 −48 120 −48 −12

−12 −48 120 −48 −12. . . . . . . . . . . . . . .

−12 −48 120 −48 −12−12 −48 120 −48 −12

−12 −48 120 −44 −16−12 −48 132 −36 −36

−27 −63 243 −81−3 −71 −133 855

. (8.158)

The submatrix Tn−3(f4) is shown in the box and f4(θ) = 54 − cos θ − 1

4 cos(2θ), as given by (8.151) for p = 4.Before passing to the GLT analysis of the collocation matricesA[p]

G,n, we prove the existence of a n-independent bound for the spectral

norms of 1n2K

[p]n , 1

nH[p]n , M [p]

n . Actually, one could also prove that the components of 1n2K

[p]n , 1

nH[p]n , M [p]

n do not depend on n; see(8.157)–(8.158) for the case of the matrix 1

n2K[p]n . However, for our purposes it suffices to show that, for every p ≥ 2, there exists a

constant C [p] such that, for all n, ∥∥∥ 1

n2K [p]n

∥∥∥ ≤ C [p],∥∥∥ 1

nH [p]n

∥∥∥ ≤ C [p], ‖M [p]n ‖ ≤ C [p]. (8.159)

121

To prove (8.159), we note that K [p]n , H [p]

n , M [p]n are banded, with bandwidth bounded by 2p + 1. Indeed, in view of the local support

property (8.121), if |i − j| > p then the supports of Ni+1,[p] and Nj+1,[p] intersect in at most one point, and (K[p]n )ij = (H

[p]n )ij =

(M[p]n )ij = 0 by (8.134). Moreover, by (8.123)–(8.125), for all i, j = 1, . . . , n+ p− 2 we have

|(K [p]n )ij | = |N ′′j+1,[p](ξi+1,[p])| ≤ 4p(p− 1)n2,

|(H [p]n )ij | = |N ′j+1,[p](ξi+1,[p])| ≤ 2pn,

|(M [p]n )ij | = |Nj+1,[p](ξi+1,[p])| ≤ 1.

Hence, (8.159) follows from (2.26).

GLT analysis of B-spline IgA collocation matrices Assume that the geometry map G has the following regularity properties: G ∈C2([0, 1]) and G′(x) 6= 0 for all x ∈ [0, 1]. Under this assumption, we show that, for any p ≥ 2, 1

n2A

[p]G,n

n∼GLT fG,p (8.160)

and 1

n2A

[p]G,n

n∼σ, λ fG,p, (8.161)

where fG,p : [0, 1]× [−π, π]→ R is defined as follows:

fG,p(x, θ) = aG(x)fp(θ) =a(G(x))

(G′(x))2fp(θ), (8.162)

with fp(θ) as in (8.151). The proof of (8.160)–(8.161) consists of the following steps.


n2K

[p]G,n

∥∥∥ ≤ C (8.163)


n2Z

[p]G,n

∥∥∥ = O(n−1). (8.164)

Intuitively, (8.164) follows from (8.163) and from the fact that Z [p]G,n is the matrix resulting from the discretization of the terms in (8.103)

with lower-order derivatives: since 1n2 is the ‘correct’ normalization factor, which keeps the spectral norm of the diffusion matrix 1

n2K[p]G,n

bounded away from∞ (and from 0), we should expect that (8.164) holds.To prove (8.163), it suffices to use the regularity of G and (8.159):∥∥∥ 1

n2K

[p]G,n

∥∥∥ =∥∥∥ 1

n2D[p]n (aG)K [p]

n

∥∥∥ ≤ C [p]‖a‖∞minx∈[0,1] |G′(x)|2

.

The proof of (8.164) is similar. It suffices to use the fact that G ∈ C2([0, 1]) and (8.159):∥∥∥ 1

n2Z

[p]G,n

∥∥∥ =∥∥∥ 1

n2D[p]n (sG)H [p]

n +1

n2D[p]n (cG)M [p]

n

∥∥∥ ≤ C [p]

n

( ‖a‖∞‖G′′‖∞minx∈[0,1] |G′(x)|3

+‖s‖∞

minx∈[0,1] |G′(x)|

)+C [p]‖c‖∞

n2.

Step 2. Let us define the symmetric matrix

K[p]G,n = Dn+p−2(aG) n2 Tn+p−2(fp), (8.165)

and consider the following decomposition of 1n2A

[p]G,n:

1

n2A

[p]G,n =

1

n2K

[p]G,n +

( 1

n2K

[p]G,n −

1

n2K

[p]G,n

)+

1

n2Z

[p]G,n. (8.166)

By GLT 3, ‖ 1n2 K

[p]G,n‖ is uniformly bounded with respect to n and 1

n2 K[p]G,nn ∼GLT aG(x)fp(θ).


n2K

[p]G,n −

1

n2K

[p]G,n

∥∥∥1

= o(n). (8.167)

122

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Λ(n−2Kn[8])

graph of f8(θ)

Figure 8.8: graph of f8(θ) and spectrum of 1n2K

[8]n for n = 150.

It follows from (8.167) and (8.164) that ‖( 1n2K

[p]G,n −

1n2 K

[p]G,n) + 1

n2Z[p]G,n‖1 = o(n). Hence, ( 1

n2K[p]G,n −

1n2 K

[p]G,n) + 1

n2Z[p]G,nn ∼σ 0,

(8.160) follows from (8.166) combined with GLT 3 and GLT 5, the singular value distribution in (8.161) follows from GLT 1, and theeigenvalue distribution in (8.161) follows from GLT 2. So, once we have established (8.167), everything is proved.

To prove (8.167), we decompose the difference 1n2K

[p]G,n −

1n2 K

[p]G,n as follows:

1

n2K

[p]G,n −

1

n2K

[p]G,n =

1

n2D[p]n (aG)K [p]

n − Dn+p−2(aG) Tn+p−2(fp)

=1

n2D[p]n (aG)K [p]

n −D[p]n (aG)Tn+p−2(fp) (8.168)

+D[p]n (aG)Tn+p−2(fp)−Dn+p−2(aG)Tn+p−2(fp) (8.169)

+Dn+p−2(aG)Tn+p−2(fp)− Dn+p−2(aG) Tn+p−2(fp). (8.170)

We consider separately the three matrices in (8.168)–(8.170), and we show that their trace-norms are o(n).

• By (8.154), the rank of the matrix (8.168) is bounded by 4(p − 1). By the regularity of G and the inequalities (8.159), (5.21), thespectral norm of (8.168) is bounded by a constant C independent of n. Thus, the trace-norm of (8.168) is o(n) (actually, O(1)) by(2.27).

• By (8.136), the continuity of aG, and (5.21), the spectral norm of the matrix (8.169) is O(ωaG( 1n )), so it tends to 0. Hence, the

trace-norm of (8.169) is o(n) by (2.27).

• By GLT 3, the spectral norm of the matrix (8.170) is O(ωaG( 1n )), so it tends to 0. Hence, the trace-norm of (8.170) is o(n) by

(2.27).

In conclusion, ‖ 1n2K

[p]G,n −

1n2 K

[p]G,n‖1 = o(n).

Considering that fG,p(x, θ) is symmetric in the Fourier variable θ, its restriction to [0, 1]× [0, π] is still a singular value and spectralsymbol for the matrix-sequence 1

n2A[p]G,nn. In Figure 8.8, we considered the case where G is the identity map over Ω = [0, 1] and

a(x) = 1, b(x) = c(x) = 0 (identically); in this situation, aG(x) = 1, sG(x) = cG(x) = 0, and the matrix A[p]G,n reduces to the

collocation diffusion matrix K [p]n . We fixed the degree p = 8, and we plotted the graph of f8(θ) over [0, π], together with the pairs

( jπn+6 , λj(

1n2K

[8]n )), j = 1, . . . , n + 6, for n = 150. The eigenvalues of 1

n2K[8]n , which turn out to be real despite the nonsymmetry

of the matrix, have been arranged so as to match, as much as possible, the graph of the symbol f8(θ). We see from the figure that thespectrum behaves like a uniform sampling of f8(θ), according to Remark 2.3. However, we also observed the presence of 8 outliers,which apparently are pairwise equal: their values are approximately 1.3277, 1.3277, 2.5144, 2.5144, 6.6862, 6.6862, 31.5746, 31.5746;note that only the two outliers 1.3277, 1.3277 are visible in Figure 8.8, whereas the other ones have been cut.

Formal structure of the symbol fG,p(x, θ) We invite the reader to compare (8.162) and (8.112). It is quite remarkable that the higher-order operator, namely −aG(x)u′′(x), has a discrete spectral counterpart aG(x)fp(θ) which looks formally the same (as in the FD case;see Remark 8.1). The similarity becomes even more evident if we note that fp(θ) is the trigonometric polynomial in the Fourier variablethat arises from the discretization of the second derivative −u′′(x). Indeed, fp(θ) is the symbol of the collocation diffusion matrices

123

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p=2,3 p=4 p=5 p=6 p=7 p=8 p=9 p=10

Figure 8.9: graph of fp/Mfp for p = 2, . . . , 10.

1n2K

[p]n n, which come from the B-spline IgA collocation approximation of (8.103) in the case where a(x) = 1 identically, Ω = (0, 1),

and G is the identity map over [0, 1]; note that in this case (8.103) is the same as (8.112). Figure 8.9 shows the graph of fp(θ) normalizedby its maximum Mfp = maxθ∈[−π,π] fp(θ) for p = 2, . . . , 10. Moreover, a direct computation shows that f2(θ) = f3(θ) = 2− 2 cos θ.

Now we observe that the function hp(θ) is defined by (8.153) for all p ≥ 0 (and we have h0(θ) = h1(θ) = 1 identically), providedthat we use the standard convention that an empty sum like

∑0k=1 φ[1](1 − k) cos(kθ) equals 0.4 Using the properties of the Fourier

transforms of cardinal B-splines [30], it was proved in [35, Section 3] that, for all p ≥ 2,

fp(θ) = (2− 2 cos θ)hp−2(θ),

(2

π

)p−1

≤ hp−2(θ) ≤ hp−2(0) = 1, θ ∈ [−π, π]. (8.171)

It follows that fp is nonnegative over [−π, π] and it has a unique zero of order 2 at θ = 0, since

limθ→0

fp(θ)

θ2= 1.

These properties are not surprising, considering that fp(θ) arises from the discretization of the second derivative −u′′(x), which is adifferential operator of order 2 (see also Remark 8.1 for the FD case). Other properties of fp(θ) can be found in [35, Section 3], togetherwith interesting properties of gp(θ) and hp(θ). In particular, it was proved in [35] that fp(π)/Mfp → 0 exponentially as p→∞.

Exercise 8.5. The matrix A[p]G,n in (8.139), which we decomposed as in (8.144), can also be decomposed as follows, according to the

diffusion, convection and reaction terms:A

[p]G,n = K

[p]G,n +H

[p]G,n +M

[p]G,n, (8.172)

where the diffusion matrix K [p]G,n is defined in (8.145), and the convection and reaction matrices are given by

H[p]G,n =

[sG(ξi+1,[p])N

′j+1,[p](ξi+1,[p])

]n+p−2

i,j=1= D[p]

n (sG)H [p]n , (8.173)

M[p]G,n =

[cG(ξi+1,[p])Nj+1,[p](ξi+1,[p])

]n+p−2

i,j=1= D[p]

n (cG)M [p]n . (8.174)

Assume that G ∈ C2([0, 1]) and G′(x) 6= 0 for all x ∈ [0, 1], and let p ≥ 2. We have seen above that 1n2K

[p]G,nn ∼GLT fG,p and

1n2K

[p]G,nn ∼σ, λ fG,p. Show that:

(a) − in H

[p]G,nn ∼GLT gG,p and − i

n H[p]G,nn ∼σ, λ gG,p, where gG,p : [0, 1]× [−π, π]→ R is defined as follows:

gG,p(x, θ) = sG(x)gp(θ) =(a(G(x))G′′(x)

(G′(x))3+s(G(x))

G′(x)

)gp(θ), (8.175)

with gp(θ) as in (8.152);

(b) M [p]G,nn ∼GLT hG,p and M [p]

G,nn ∼σ, λ hG,p, where hG,p : [0, 1]× [−π, π]→ R is defined as follows:

hG,p(x, θ) = cG(x)hp(θ) = c(G(x))hp(θ), (8.176)

with hp(θ) as in (8.153).4On the contrary, the functions fp and gp are defined by (8.151) and (8.152) only for p ≥ 2, because φ′′

[1](1) and φ′

[1](1) do not exist.

124

8.4.10 Galerkin B-spline IgA approximation of convection-diffusion-reaction equationsConsider the differential problem

−(a(x)u′(x))′ + b(x)u′(x) + c(x)u(x) = f(x), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,

(8.177)

where Ω is a bounded open interval in R, f ∈ L2(Ω), and a, b, c : Ω→ R are only assumed to be in L∞(Ω). Problem (8.177) is the sameas (8.102), except for the assumptions on a, c, b, f . We consider the isogeometric Galerkin approximation of (8.177) based on uniformB-splines of degree p ≥ 1. This approximation technique is described in some detail in the next paragraph. For further details, as well asfor the motivations that led to the birth of IgA, we refer the reader to footnote 1 in Section 1.1 and, above all, to [28, 63].

Isogeometric Galerkin approximation The weak form of (8.177) reads as follows: find u ∈ H10 (Ω) such that

a(u, v) = f(v) ∀ v ∈ H10 (Ω), (8.178)

wherea(u, v) =

∫Ω

(a(x)u′(x)v′(x) + b(x)u′(x)v(x) + c(x)u(x)v(x)

)dx, f(v) =

∫Ω

f(x)v(x)dx.

In the standard Galerkin method, we look for an approximation uW of u by choosing a finite dimensional approximation space W ⊂H1

0 (Ω) and by solving the following (Galerkin) problem: find uW ∈ W such that

a(uW , v) = f(v) ∀ v ∈ W .

If ϕ1, . . . , ϕN is a basis of W , then uW =∑Nj=1 ujϕj for a unique vector u = (u1, . . . , uN )T , and, by linearity, the computation of

uW (i.e., of u) reduces to solving the linear systemAu = f ,

where

A = [a(ϕj , ϕi)]Ni,j=1 =

[∫Ω

(a(x)ϕ′j(x)ϕ′i(x) + b(x)ϕ′j(x)ϕi(x) + c(x)ϕj(x)ϕi(x)

)dx

]Ni,j=1

(8.179)

is the stiffness matrix and f =[f(ϕi)

]Ni=1

.Now, suppose that the physical domain Ω can be described by a global geometry function G : [0, 1] → Ω, which is invertible and

satisfies G(∂([0, 1])) = ∂Ω. Letϕ1, . . . , ϕN (8.180)

be a set of basis functions defined on the parametric domain [0, 1] and vanishing on the boundary ∂([0, 1]). In the isogeometric Galerkinapproach, we find an approximation uW of u by using the standard Galerkin method, in which the approximation space is chosen asW = span(ϕi, i = 1, . . . , N), with

ϕi(x) = ϕi(G−1(x)) = ϕi(x), x = G(x). (8.181)

The resulting stiffness matrixA is given by (8.179), with the basis functions ϕi defined in (8.181). Assuming thatG and ϕi, i = 1, . . . , N ,are sufficiently regular, we can apply standard differential calculus to obtain the following expression for A in terms of G and ϕi,i = 1, . . . , N :

A =

[∫[0,1]

(aG(x)ϕ′j(x)ϕ′i(x) +

b(G(x))

G′(x)ϕ′j(x)ϕi(x) + c(G(x))ϕj(x)ϕi(x)

)|G′(x)|dx

]Ni,j=1

, (8.182)

where aG(x) is the same as in (8.113), i.e.,

aG(x) =a(G(x))

(G′(x))2. (8.183)

In the IgA framework, the functions ϕi are usually B-splines or NURBS. Here, the role of the ϕi will be played by B-splines overuniform knot sequences. For the case of NURBS, we refer the reader to [49].

125

Galerkin B-spline IgA discretization matrices Similarly to the IgA collocation approach considered in Section 8.4.9, in the GalerkinB-spline IgA based on (uniform) B-splines, the functions ϕ1, . . . , ϕN are chosen as the B-splines N2,[p], . . . , Nn+p−1,[p]; see (8.137).The boundary functions N1,[p] and Nn+p,[p] are excluded because they do not vanish on ∂([0, 1]). The stiffness matrix (8.182) resultingfrom this choice of the ϕi will be denoted by A[p]

G,n:

A[p]G,n =

[∫[0,1]

(aG(x)N ′j+1,[p](x)N ′i+1,[p](x)

+b(G(x))

G′(x)N ′j+1,[p](x)Ni+1,[p](x) + c(G(x))Nj+1,[p](x)Ni+1,[p](x)

)|G′(x)|dx

]n+p−2

i,j=1

. (8.184)


AG,n = K[p]G,n + Z

[p]G,n, (8.185)

where

K[p]G,n =

[∫[0,1]

aG(x)|G′(x)|N ′j+1,[p](x)N ′i+1,[p](x)dx

]n+p−2

i,j=1

(8.186)

is the matrix resulting from the discretization of the higher-order (diffusion) term in (8.177), and

Z[p]G,n =

[∫[0,1]

(b(G(x))


)|G′(x)|dx

]n+p−2

i,j=1

(8.187)

is the matrix resulting from the discretization of the terms with lower-order derivatives (the convection and reaction terms). We will seethat, as usual, the GLT analysis of a properly scaled version of the sequence A[p]

G,nn reduces to the GLT analysis of its ‘diffusion part’

K [p]G,nn, because ‖Z [p]

G,n‖ is negligible with respect to ‖K [p]G,n‖ when n→∞.

Let

K [p]n =

[∫[0,1]

N ′j+1,[p](x)N ′i+1,[p](x)dx

]n+p−2

i,j=1

, (8.188)

H [p]n =

[∫[0,1]

N ′j+1,[p](x)Ni+1,[p](x)dx

]n+p−2

i,j=1

, (8.189)

M [p]n =

[∫[0,1]

Nj+1,[p](x)Ni+1,[p](x)dx

]n+p−2

i,j=1

. (8.190)

These matrices will play an important role in the GLT analysis of this section, and also in Section 8.5. In particular, it is necessary tounderstand their approximate structure. This is achieved by (approximately) construct them. We only construct their central submatrices[

(K [p]n )ij

]n−1

i,j=p,

[(H [p]

n )ij]n−1

i,j=p,

[(M [p]

n )ij]n−1

i,j=p, (8.191)

which are determined by the central basis functions in (8.131). For all i, j = p, . . . , n−1, by noting that [−i+p, n−i+p] ⊇ supp(φ[p]) =[0, p+ 1] and by using (8.129)–(8.130) and (8.132), we obtain

(K [p]n )ij =

∫[0,1]

N ′j+1,[p](x)N ′i+1,[p](x)dx = n2

∫[0,1]

φ′[p](nx− j + p)φ′[p](nx− i+ p)dx

= n

∫[−i+p,n−i+p]

φ′[p](t+ i− j)φ′[p](t)dt = n

∫Rφ′[p](t+ i− j)φ′[p](t)dt

= −nφ′′[2p+1](p+ 1 + i− j) = −nφ′′[2p+1](p+ 1− i+ j),

and similarly

(H [p]n )ij = φ′[2p+1](p+ 1 + i− j) = −φ′[2p+1](p+ 1− i+ j),

(M [p]n )ij = φ[2p+1](p+ 1 + i− j) =

1

nφ[2p+1](p+ 1− i+ j).

126

Since their entries depend only on the difference i− j, the submatrices (8.191) are Toeplitz matrices, and precisely[(K [p]

n )ij]n−1

i,j=p= n

[−φ′′[2p+1](p+ 1− i+ j)

]n−1

i,j=p= nTn−p(fp), (8.192)[

(H [p]n )ij

]n−1

i,j=p=[−φ′[2p+1](p+ 1− i+ j)

]n−1

i,j=p= iTn−p(gp), (8.193)[

(M [p]n )ij

]n−1

i,j=p=

1

n

[φ[2p+1](p+ 1− i+ j)

]n−1

i,j=p=

1

nTn−p(hp), (8.194)

where

fp(θ) =∑k∈Z−φ′′[2p+1](p+ 1− k)eikθ = −φ′′[2p+1](p+ 1)− 2

p∑k=1

φ′′[2p+1](p+ 1− k) cos(kθ), (8.195)

gp(θ) =∑k∈Z−φ′[2p+1](p+ 1− k)eikθ = −2

p∑k=1

φ′[2p+1](p+ 1− k) sin(kθ), (8.196)

hp(θ) =∑k∈Z

φ[2p+1](p+ 1− k)eikθ = φ[2p+1](p+ 1) + 2

p∑k=1

φ[2p+1](p+ 1− k) cos(kθ) (8.197)

(we used (8.128)–(8.129) to simplify the expressions of fp(θ), gp(θ), hp(θ)). It follows from (8.192) that Tn−p(fp) is the principalsubmatrix of both 1

nK[p]n and Tn+p−2(fp) corresponding to the set of indices p, . . . , n− 1. Similar results follow from (8.193)–(8.194),

and so1

nK [p]n = Tn+p−2(fp) +R[p]

n , rank(R[p]n ) ≤ 4(p− 1), (8.198)

−iH [p]n = Tn+p−2(gp) + S[p]

n , rank(S[p]n ) ≤ 4(p− 1), (8.199)

nM [p]n = Tn+p−2(hp) + V [p]

n , rank(V [p]n ) ≤ 4(p− 1). (8.200)

Let us see two examples. In the case p = 2, the matrix 1nK

[2]n is given by

1

nK [2]n =

1

6

8 −1 −1−1 6 −2 −1−1 −2 6 −2 −1

−1 −2 6 −2 −1. . . . . . . . . . . . . . .

−1 −2 6 −2 −1−1 −2 6 −2 −1

−1 −2 6 −1−1 −1 8

.

The submatrix Tn−2(f2) is shown in the box and f2(θ) = 1− 23 cos θ − 1

3 cos(2θ), as given by (8.195) for p = 2. In the case p = 3, thematrix 1

nK[3]n is given by

1

nK [3]n =

1

240

360 9 −60 −39 162 −8 −47 −2

−60 −8 160 −30 −48 −2−3 −47 −30 160 −30 −48 −2

−2 −48 −30 160 −30 −48 −2−2 −48 −30 160 −30 −48 −2

. . . . . . . . . . . . . . . . . . . . .−2 −48 −30 160 −30 −48 −2

−2 −48 −30 160 −30 −48 −2−2 −48 −30 160 −30 −47 −3

−2 −48 −30 160 −8 −60−2 −47 −8 162 9

−3 −60 9 360

.

The submatrix Tn−3(f3) is shown in the box and f3(θ) = 23 −

14 cos θ − 2

5 cos(2θ)− 160 cos(3θ), as given by (8.195) for p = 3.

Remark 8.7. The functions f2q+1(θ), g2q+1(θ), h2q+1(θ), defined by (8.151)–(8.153) for odd degree p = 2q + 1, coincide preciselywith the functions fq(θ), gq(θ), hq(θ) defined by (8.195)–(8.197) for p = q. In particular, the properties of the functions (8.151)–(8.153)obtained in [35, Section 3] also apply to the functions (8.195)–(8.197).

127

GLT analysis of Galerkin B-spline IgA discretization matrices Assume that the geometry map G is regular, i.e., G ∈ C1([0, 1]) andG′(x) 6= 0 for every x ∈ [0, 1]. Under this assumption, we show that, for any p ≥ 1, 1

nA

[p]G,n

n∼GLT fG,p (8.201)

and 1

nA

[p]G,n

n∼σ, λ fG,p, (8.202)

where the symbol fG,p : [0, 1]× [−π, π]→ R is defined as follows:

fG,p(x, θ) = aG(x)|G′(x)|fp(θ) =a(G(x))

|G′(x)|fp(θ), (8.203)

with fp(θ) as in (8.195). To prove (8.201)–(8.202), we follow the same procedure as in Section 8.4.7 (see Step 1 to 6).


nK

[p]G,n

∥∥∥ ≤ C (8.204)


nZ

[p]G,n

∥∥∥ = O(n−1). (8.205)

To prove (8.204), we note that K [p]G,n is a banded matrix, with bandwidth at most equal to 2p + 1. Indeed, due to the local support

property (8.121), if |i− j| > p then the supports of Ni+1,[p] and Nj+1,[p] intersect in at most one point, and (K[p]G,n)ij = 0. Moreover, by

(8.121) and (8.125), for all i, j = 1, . . . , n+ p− 2 we have

|(K [p]G,n)ij | =

∣∣∣∣∣∫

[0,1]

aG(x)|G′(x)|N ′j+1,[p](x)N ′i+1,[p](x)dx

∣∣∣∣∣ =

∣∣∣∣∣∫

[ti+1,ti+p+2]

a(G(x))

|G′(x)|N ′j+1,[p](x)N ′i+1,[p](x)dx

∣∣∣∣∣≤ 4p2n2‖a‖L∞

minx∈[0,1] |G′(x)|

∫[ti+1,ti+p+2]

dx ≤ 4p2(p+ 1)n‖a‖L∞minx∈[0,1] |G′(x)|

,

where in the last inequality we used the fact that tk+p+1 − tk ≤ (p+ 1)/n for all k = 1, . . . , n+ p; see (8.115)–(8.116). In conclusion,the components of the banded matrix 1

nK[p]G,n are bounded (in modulus) by a constant independent of n, and (8.204) follows from (2.26).

To prove (8.205), we follow the same argument as for the proof of (8.204). Due to the local support property (8.121), Z [p]G,n is banded

and, precisely, (Z[p]G,n)ij = 0 whenever |i− j| > p. Moreover, by (8.121) and (8.123)–(8.125), for all i, j = 1, . . . , n+ p− 2 we have

|(Z [p]G,n)ij | =

∣∣∣∣∣∫

[ti+1,ti+p+2]

(b(G(x))


)|G′(x)|dx

∣∣∣∣∣≤ 2p(p+ 1)‖b‖L∞ +

(p+ 1)‖c‖L∞‖G′‖∞n

,


Step 2. We define the linear operator L[p]n (·) : L1([0, 1])→ R(n+p−2)×(n+p−2),

L [p]n (h) =

[∫[0,1]

h(x)N ′j+1,[p](x)N ′i+1,[p](x)dx

]n+p−2

i,j=1

. (8.206)

By (8.186), we have K [p]G,n = L

[p]n (aG|G′|). The next three steps are devoted to show that 1

nL [p]n (h)

n∼GLT h(x)fp(θ) ∀h ∈ L1([0, 1]). (8.207)

This implies in particular that 1nK

[p]G,nn ∼GLT aG(x)|G′(x)|fp(θ) = fG,p(x, θ).

Step 3. We begin with proving that (8.207) holds in the constant-coefficient case h(x) = 1, i.e., we show that 1nL

[p]n (1)n ∼GLT

fp(θ). To prove this, we note that L[p]n (1) = K

[p]n . Hence, the relation 1

nL[p]n (1)n ∼GLT fp(θ) follows from (8.198), GLT 3 and

GLT 5.

128

Step 4. We now show that (8.207) holds in the case of a continuous function h ∈ C([0, 1]). As in Step 4 of Section 8.4.7, the proofis based on the fact that the B-spline basis functions N2,[p], . . . , Nn+p−1,[p] are ‘locally supported’. Indeed, the width of the support[ti+1, ti+p+2] of the i-th function Ni+1,[p] is bounded by (p+ 1)/n and goes to 0 when n→∞. Moreover, the support itself is localizednear the point i

n+p−2 , because

maxx∈[ti+1,ti+p+2]

∣∣∣x− i

n+ p− 2

∣∣∣ ≤ Cpn, (8.208)

for all i = 2, . . . , n+ p− 1 and for some constant Cp independent of n. By (8.125), for all i, j = 1, . . . , n+ p− 2 we have∣∣∣(L [p]n (h))ij − (Dn+p−2(h)L [p]

n (1))ij

∣∣∣ =

∣∣∣∣∣∫

[0,1]

[h(x)− h

( i

n+ p− 2

)]N ′j+1,[p](x)N ′i+1,[p](x)dx

∣∣∣∣∣≤ 4p2n2

∫[ti+1,ti+p+2]

∣∣∣h(x)− h( i

n+ p− 2

)∣∣∣dx ≤ 4p2(p+ 1)nωh

(Cpn

).

Thus, each component of the matrix 1nL

[p]n (h) − 1

nDn+p−2(h)L[p]n (1) is bounded (in modulus) by 4p2(p + 1)ωh(

Cpn ). Moreover,

1nL

[p]n (h) − 1

nDn+p−2(h)L[p]n (1) is banded with bandwith at most 2p + 1, due to the local support property of the B-spline basis

functions Ni,[p]. By (2.26), ‖ 1nL

[p]n (h) − Dn+p−2(h)L

[p]n (1)‖ ≤ 4p2(p + 1)(2p + 1)ωh(

Cpn ) → 0 as n → ∞, and 1

nL[p]n (h) −

1nDn+p−2(h)L

[p]n (1)n ∼σ 0. This implies (8.207) by 1

nL[p]n (1)n ∼GLT fp(θ), GLT 3 and GLT 5.

Step 5. Finally, we show that (8.207) holds in the general case where h ∈ L1([0, 1]). By the density of C([0, 1]) in L1([0, 1]), thereexists a sequence of continuous functions hm ∈ C([0, 1]) such that hm → h in L1([0, 1]). Clearly hm → h in measure over [0, 1]. ByStep 4, 1

nL[p]n (hm)n ∼GLT hm(x)fp(θ). In addition, 1

nL[p]n (hm)nm is an a.c.s. for 1

nL[p]n (h)n. Indeed, using (2.29) and

(8.125), we obtain

∥∥L [p]n (h)−L [p]

n (hm)∥∥

1≤n+p−2∑i,j=1

∣∣∣(L [p]n (h))ij − (L [p]

n (hm))ij

∣∣∣ =

n+p−2∑i,j=1

∣∣∣∣∣∫

[0,1]

[h(x)− hm(x)

]N ′j+1,[p](x)N ′i+1,[p](x)dx

∣∣∣∣∣≤∫

[0,1]

|h(x)− hm(x)|n+p−2∑i,j=1

|N ′j+1,[p](x)| |N ′i+1,[p](x)|dx ≤ 4p2n2

∫[0,1]

|h(x)− hm(x)|dx,

and ∥∥∥ 1

nL [p]n (h)− 1

nL [p]n (hm)

∥∥∥1≤ 4p2n‖h− hm‖L1 .

1nL

[p]n (hm)nm is then an a.c.s. for 1

nL[p]n (h)n by Corollary 4.3, and (8.207) follows from GLT 8.

Step 6. As already noted in Step 2, from (8.207) we get 1nK

[p]G,nn ∼GLT fG,p. Since we have seen in Step 1 that 1

nZ[p]G,nn ∼σ 0,

the relation 1nA

[p]G,n ∼GLT fG,p in (8.201) follows from the decomposition

1

nA

[p]G,n =

1

nK

[p]G,n +

1

nZ

[p]G,n (8.209)

(cf. (8.185)) and from GLT 3, GLT 5; and the singular value distribution in (8.202) follows from GLT 1. If b(x) = 0 identically, 1nA

[p]G,n

is symmetric and also the spectral distribution in (8.202) follows from GLT 1. If b(x) is not identically 0, the spectral distribution in(8.202) follows from GLT 2 applied to the decomposition (8.209), taking into account the symmetry of K [p]

G,n and what we have seen inStep 1.

Formal structure of the symbol fG,p(x, θ) Disregarding the regularity of a(x) and u(x), problem (8.177) can be formally rewritten asin (8.103). If, for any u : Ω→ R, we define u : [0, 1]→ R as in (8.111), then u satisfies (8.103) if and only if u satisfies the correspondingtransformed problem (8.112)–(8.113), in which the higher-order operator takes the form −aG(x)u′′(x). It is then clear that the symbolfG,p(x, θ) = aG(x)|G′(x)|fp(θ) preserves the formal structure of the higher-order operator associated with the transformed problem(8.112). However, in this Galerkin context, we notice the appearance of the determinant factor |G′(x)|, which is not present in thecollocation setting; cf. (8.203) with (8.162).

Concerning the symbol fp(θ) in (8.195), a direct computation yields f1(θ) = 2−2 cos θ. This should not come as a surprise, becausethe Galerkin B-spline IgA approximation with p = 1 (and G equal to the identity map over [0, 1]) coincides precisely with the linear FEapproximation considered in Section 8.4.7; the only (unessential) difference is that the discretization step in Section 8.4.7 was chosen ash = 1

n+1 , while in this section we have h = 1n . In particular, the B-spline basis functions of degree 1, namely N2,[1], . . . , Nn,[1], are the

hat-functions; cf. (8.119) (with p = 1) and (8.72).

129

Exercise 8.6. As the reader may have noted, this section has many connections with Section 8.4.9. In particular, this exercise is theGalerkin counterpart of the collocation Exercise 8.5. The matrix A[p]

G,n in (8.184), which we decomposed as in (8.185), can also bedecomposed as follows, according to the diffusion, convection and reaction terms:

A[p]G,n = K

[p]G,n +H

[p]G,n +M

[p]G,n, (8.210)

where the diffusion matrix K [p]G,n is defined in (8.186), while the convection and reaction matrices are given by

H[p]G,n =

[∫[0,1]

b(G(x))|G′(x)|G′(x)

N ′j+1,[p](x)Ni+1,[p](x)dx

]n+p−2

i,j=1

, (8.211)

M[p]G,n =

[∫[0,1]

c(G(x))|G′(x)|Nj+1,[p](x)Ni+1,[p](x)dx

]n+p−2

i,j=1

. (8.212)

Assume that the geometry map G is regular and let p ≥ 1. We have seen above that 1nK

[p]G,nn ∼GLT fG,p and 1

nK[p]G,nn ∼σ, λ fG,p.

Show that:

(a) −iH [p]G,nn ∼GLT gG,p and −iH [p]

G,nn ∼σ, λ gG,p, where gG,p : [0, 1]× [−π, π]→ R is defined as follows:

gG,p(x, θ) =b(G(x))|G′(x)|

G′(x)gp(θ), (8.213)

with gp(θ) as in (8.196);

(b) nM [p]G,nn ∼GLT hG,p and nM [p]

G,nn ∼σ, λ hG,p, where hG,p : [0, 1]× [−π, π]→ R is defined as follows:

hG,p(x, θ) = c(G(x))|G′(x)|hp(θ), (8.214)

with hp(θ) as in (8.197).

Compare the results of this exercise with the results of Exercise 8.5.

Exercise 8.7. Consider problem (8.177) with b(x) = 0 identically, so that the matrix A[p]G,n in (8.184) is symmetric. Show that (8.201)–

(8.202) continue to hold under the only assumption that

aG(x)|G′(x)| = a(G(x))

|G′(x)|∈ L1([0, 1]), c(G(x))|G′(x)| ∈ L1([0, 1]). (8.215)

This assumption is satisfied if, for example:

(a) a, c ∈ L1(Ω) and G is regular, i.e., G ∈ C1([0, 1]) and G′(x) 6= 0 for all x ∈ [0, 1];

(b) c ∈ L1(Ω), a ∈ L∞(Ω), G ∈ C1([0, 1]) and |G′|−1 ∈ L1([0, 1]).

Note that the condition on G in item (b) is met even in some cases in which G is not regular. For instance, it is met if Ω = (0, 1) andG(x) = xq , q > 1; in this case, G′(0) = 0 (so G is non-regular), and the mapping of the uniform mesh i

n , i = 0, . . . , n, through thefunction G is the non-uniform grid ( in )q , i = 0, . . . , n, whose points rapidly accumulate at x = 0. This induces a local refinement aroundthe site x = 0, and the choice of G is then a way to better approximate the solution in a neighboorhood of x = 0.5

8.4.11 Galerkin B-spline IgA approximation of second-order eigenvalue problemsConsider the following second-order eigenvalue problem: find eigenvalues λl and eigenfunctions ul, for l = 1, 2, . . ., such that

−(a(x)u′l(x))′ = λlc(x)ul(x), x ∈ Ω,ul(x) = 0, x ∈ ∂Ω,

(8.216)

5A natural question is the following: why should one be interested in better approximating the solution around a point? Or why should one be interested in discretizinga PDE with a grid that rapidly accumulates at a point? The answer is that this local refinement is necessary in some situations where the PDE coefficients are stronglyanisotropic. If a uniform discretization were used, the associated discretization step should be chosen very small, and this would result in a linear system with extremelylarge size: the computational cost to solve it would be unsustainable. For this reason, one adopts a local refinement, so that a coarse grid is used in the subregions of thedomain where the PDE coefficients are sufficiently smooth, and a finer grid is used only in the subregions where they are, say, ‘not well-behaved’ (e.g., unbounded becauseof a singularity or remarkably oscillatory).

130

where Ω is a bounded open interval in R, a, c ∈ L∞(Ω), and c(x) > 0 a.e. in Ω. The weak form of this problem reads as follows: findeigenvalues λl and eigenfunctions ul ∈ H1

0 (Ω), for l = 1, 2, . . ., such that∫Ω

a(x)u′l(x)w′(x)dx = λl

∫Ω

c(x)ul(x)w(x)dx ∀w ∈ H10 (Ω). (8.217)

In the Galerkin method, we choose a finite dimensional approximation space W ⊂ H10 (Ω) (of dimension, say, N ), and we look for

approximations of λl, ul by solving the following discrete problem: find pairs λl,W and ul,W ∈ W , for l = 1, . . . , N , such that∫Ω

a(x)u′l,W (x)w′(x)dx = λl,W

∫Ω

c(x)ul,W (x)w(x)dx ∀w ∈ W . (8.218)

The pairs (λl,W , ul,W ), l = 1, . . . , N , are taken as approximations to N of the (usually infinite) eigenvalue-eigenvector pairs (λl, ul),l = 1, 2, . . . If ϕ1, . . . , ϕN is a basis for W , we can identify each uW ∈ W with its coefficient vector uW = (uW ,1, . . . , uW ,N )T

relative to the basis ϕ1, . . . , ϕN. With this identification in mind, the discrete problem (8.218) is equivalent to the following generalizedeigenvalue problem: find pairs λl,W and ul,W = (ul,W ,1, . . . , ul,W ,N )T , for l = 1, . . . , N , such that

Kul,W = λl,W Mul,W , (8.219)

where

K =

[∫Ω


]Ni,j=1

, M =

[∫Ω

c(x)ϕj(x)ϕi(x)dx

]Ni,j=1

. (8.220)

The symmetric matrices K and M are referred to as the stiffness and mass matrix, respectively. Due to our assumption that c(x) > 0a.e., the mass matrix is SPD. Hence, multiplying both sides of (8.219) by M−1, we see that (8.219) can be rewritten as

M−1Kul,W = λl,W ul,W . (8.221)

This shows that the ‘discrete eigenvalues’ λl,W , l = 1, . . . , N , are nothing else than the eigenvalues of the matrix M−1K. Moreover, the‘discrete eigenfunctions’ ul,W are obtained from the eigenvectors ul,W of M−1K via the relation ul,W (x) =

∑Nj=1 ul,W ,jϕj(x).

In the isogeometric Galerkin method, we suppose that the physical domain Ω can be described by a global geometry function G :[0, 1]→ Ω, which is invertible and satisfies G(∂([0, 1])) = ∂Ω. We also assume to have a set of basis functions

ϕ1, . . . , ϕN (8.222)

defined on the parametric domain [0, 1] and vanishing on the boundary ∂([0, 1]). Then, we find approximations to the eigenvalues λl andthe eigenfunctions ul by using the Galerkin method described above, in which the approximation space is chosen as W = span(ϕi, i =1, . . . , N), with

ϕi(x) = ϕi(G−1(x)) = ϕi(x), x = G(x). (8.223)

The resulting stiffness and mass matrices K and M are given by (8.220), with the basis functions ϕi defined as in (8.223). As inSection 8.4.10, if we assume that G and ϕi, i = 1, . . . , N , are sufficiently regular, we can apply standard differential calculus to obtainfor K and M the following expressions:

K =

[∫[0,1]

a(G(x))

|G′(x)|ϕ′j(x)ϕ′i(x)dx

]Ni,j=1

, M =

[∫[0,1]

c(G(x))|G′(x)|ϕj(x)ϕi(x)dx

]Ni,j=1

. (8.224)

Let us now choose the basis functions ϕi, i = 1, . . . , N , as the B-splines Ni+1,[p], i = 1, . . . , n + p − 2, already considered inSections 8.4.9–8.4.10, and suppose that G is regular. Let

K[p]G,n =

[∫[0,1]

a(G(x))

|G′(x)|N ′j+1,[p](x)N ′i+1,[p](x)dx

]n+p−2

i,j=1

,

M[p]G,n =

[∫[0,1]

c(G(x))|G′(x)|Nj+1,[p](x)Ni+1,[p](x)dx

]n+p−2

i,j=1

,

be the resulting stiffness and mass matrix, which are the same as in (8.186) and (8.212). In this section, we perform the GLT analysisof a properly scaled version of the sequence of discretization matrices (M [p]

G,n)−1K[p]G,nn associated with the generalized eigenvalue

problem (8.221). We show that 1

n2(M

[p]G,n)−1K

[p]G,n

n∼GLT eG,p(x, θ) (8.225)

131

and 1

n2(M

[p]G,n)−1K

[p]G,n

n∼σ, λ eG,p(x, θ), (8.226)

where

eG,p(x, θ) =(hG,p(θ)

)−1fG,p(θ) =

a(G(x))

c(G(x))(G′(x))2ep(θ), ep(θ) =

(hp(θ)

)−1fp(θ), (8.227)

and fp(θ), hp(θ), fG,p(x, θ), hG,p(x, θ) are given by, respectively, (8.195), (8.197), (8.203), (8.214).To prove (8.225)–(8.226), we first recall from the analysis in Section 8.4.10 (see in particular Exercise 8.6) that 1

nK

[p]G,n

n∼GLT fG,p, nM [p]

G,nn ∼GLT hG,p. (8.228)

Hence, the GLT relation (8.225) follows directly from GLT 5, GLT 6, taking into account that hG,p(x, θ) 6= 0 a.e. by our assumptionthat c(x) > 0 a.e. and by the positivity of hp(θ) (see Remark 8.7 and (8.171)). The singuar value distribution in (8.226) follows from(8.225) and GLT 1. It only remains to prove the eigenvalue distribution in (8.226). Since the mass matrix M [p]

G,n is SPD, there exists an

SPD square root (M[p]G,n)1/2, and the matrix 1

n2

(M

[p]G,n

)−1K

[p]G,n is similar to the symmetric matrix

(nM[p]G,n)−1/2

( 1

nK

[p]G,n

)(nM

[p]G,n)−1/2.

By GLT 5, GLT 6, GLT 7, (nM

[p]G,n)−1/2

( 1

nK

[p]G,n

)(nM

[p]G,n)−1/2

n∼GLT eG,p(x, θ),

and it follows from GLT 1 that (nM

[p]G,n)−1/2

( 1

nK

[p]G,n

)(nM

[p]G,n)−1/2

n∼σ, λ eG,p(x, θ).

The eigenvalue distribution in (8.226) is then obtained as a consequence of the similarity between the matrices 1n2 (M

[p]G,n)−1K

[p]G,n and

(nM[p]G,n)−1/2

(1nK

[p]G,n

)(nM

[p]G,n)−1/2.

For p = 1, 2, 3, 4, the second equation in (8.227) gives

e1(θ) =6(1− cos θ)

2 + cos θ, (8.229)

e2(θ) =20(3− 2 cos θ − cos(2θ))

33 + 26 cos θ + cos(2θ), (8.230)

e3(θ) =42(40− 15 cos θ − 24 cos(2θ)− cos(3θ))

1208 + 1191 cos θ + 120 cos(2θ) + cos(3θ), (8.231)

e4(θ) =72(1225− 154 cos θ − 952 cos(2θ)− 118 cos(3θ)− cos(4θ))

78095 + 88234 cos θ + 14608 cos(2θ) + 502 cos(3θ) + cos(4θ). (8.232)

Eqs. (8.229)–(8.232) are the analogues of formulas (117), (130), (135), (140) obtained by engineers in [64] (see also formulas (23), (56)in [65], formulas (32), (33) in [29], and formulas (23), (24) in [80]).6 We may therefore conclude that (8.227) is a generalization of theseformulas to any degree p ≥ 1. For an extension of the results obtained in this section, we refer the reader to the engineering paper [45].

8.5 PDE discretizations: the d-dimensional caseIn this section we extend to the d-dimensional setting the study carried out in Section 8.4. More precisely, Sections 8.5.1, 8.5.2, 8.5.3 arethe d-dimensional versions of Sections 8.4.3, 8.4.9, 8.4.10, respectively. Actually, no substantial difference is encountered when passingfrom 1 to d space dimensions. In other words, all the main ideas of the GLT approach have already emerged in the 1-dimensional setting,and the GLT analysis of Sections 8.5.1–8.5.3 is conceptually identical to the GLT analysis of the corresponding 1-dimensional sectionsmentioned before. However, the d-dimensional case involves a lot of technicalities that are not visible in 1D. In order to acquire thesetechnicalities, which are necessary to transfer the GLT analysis from 1 to d dimensions, it is useful to see them in some detail. The mostimportant among them is certainly the multi-index language, which allows one to essentially maintain the 1D notation by simply turningsome letters (n, i, j, etc.) in boldface (n, i, j, etc.).

6Contrary to the B-spline IgA discretizations investigated herein and in [64], the authors of [29, 65, 80] considered NURBS IgA discretizations. However, the sameformulas are obtained in both cases. This can be easily explained in view of the results of [49], where it is shown that the symbols fp, gp, hp in (8.195)–(8.197) are exactlythe same in the B-spline and NURBS IgA frameworks.

132

In this section, we present the GLT analysis of matrix-sequences arising from the discretization of d-dimensional PDEs. The purposeis to provide the reader with the necessary technical background to face d-dimensional problems. A special attention will be devoted tothe multi-index language, which will be systematically used hereinafter. Repetitions from the 1D setting will be intentionally inserted, soas to stress better the conceptual analogy between unidimensional and multidimensional setting.

Before starting, let us outline the main general ideas of a d-dimensional GLT analysis. Consider for example a linear second-orderPDE such as

−d∑

h,k=1

ahk∂2u

∂xh∂xk+

d∑k=1

bk∂u

∂xk+ cu = f ⇐⇒ 1(A Hu)1T + b · ∇u+ cu = f, (8.233)

where A = [ahk]dh,k=1 and Hu is the Hessian of u,

(Hu)hk =∂2u

∂xh∂xk, h, k = 1, . . . , d.

The boundary conditions are not specified, because they do not affect our reasoning here. Indeed, as indicated by Section 8.4.2, differentboundary conditions only produce small-rank perturbations in the discretization matrices arising from the numerical approximation of(8.233); and small-rank perturbations do not affect the spectral distrubution of these matrices. Assume we discretize (8.233) by alocal method; to fix the ideas, here we will assume that such method is a FD scheme. The resulting discretization matrices An areparameterized by a multi-index n = (n1, . . . , nd), where ni is related to the discretization step hi in the i-th direction, and ni → ∞if and only if hi → 0 (usually, hi ∼ 1/ni). By choosing ni = n for some n ∈ N – as it normally happens in practice – we see thatn = (n, . . . , n) and, consequently, Ann is a matrix-sequence. The matrix An can be decomposed according to the terms of the PDEas follows:

An =

d∑h,k=1

Kn,hk(ahk) +

d∑k=1

Hn,k(bk) + In(c) =

d∑h,k=1

Kn,hk(ahk) + Zn,

where Zn =∑dk=1Hn,k(bk) + In(c) and Kn,hk(a), Hn,k(b), In(c) are the matrices resulting from the same FD discretization of the

separable differential operators7

−a ∂2u

∂xh∂xk, b

∂u

∂xk, cu,

respetively. It usually turns out that, after a suitable normalization that we ignore in this discussion, the matrix-sequence Znn asso-ciated with the lower-order differential operators of the PDE (8.233) is zero-distributed and the GLT analysis of Ann reduces to theGLT analysis of the matrix-sequence

∑dh,k=1Kn,hk(ahk)n associated with the higher-order differential operator of the PDE (8.233).

Moreover, the sequences Kn,hk(ahk)n often turns out to be GLT sequences (actually, sLT sequences) of the form

Kn,hk(ahk) = Dn(ahk)Tn(phk) + Zn,hk, Zn,hkn ∼σ 0,

where phk is the (separable) trigonometric polynomial that represents the FD formula used to discretize the derivative − ∂2u

∂xh∂xk. In

conclusion,Kn,hkn ∼GLT ahk(x)phk(θ) (8.234)

and, consequently,

Ann ∼GLT

d∑h,k=1

ahk(x)phk(θ) = 1(A(x) H(θ))1T , (8.235)

whereHhk = phk, h, k = 1, . . . , d.

7We say that a differential operator is separable if it is obtained by multiplying a given function with a product of partial derivatives. The general separable differentialoperator can be written as

a(x)∂r1+...+rdu

∂xr11 · · · ∂xrdd

.

An example of a non-separable differential operator is the Laplacian, which however can be written, like any other differential operator, as a sum of separable differentialoperators:

∆u =d∑k=1

∂2u

∂x2k.

The (general) message of this discussion is that a separable differential operator gives rise to a GLT (actually, sLT) sequence. As a consequence, an arbitrary differentialoperator (a sum of separable differential operators) gives rise to a sum of GLT sequences, i.e., a GLT sequence.

133

From (8.235) and GLT 1 – GLT 2 one usually arrives at the distribution relations

Ann ∼σ, λd∑

h,k=1

ahk(x)phk(θ) = 1(A(x) H(θ))1T .

Note the formal analogy between the expression of the symbol 1(A(x) H(θ))1T and the expression of the higher-order differentialoperator 1(A Hu)1T in (8.233)! Because of this analogy, and especially because of (8.234), the matrix H(θ) in the Fourier variablesθ is usually referred to as the ‘symbol of the (negative) Hessian operator’, although this terminology is clearly not rigorous from the

mathematical viewpoint. If we change the FD formulas to discretize the derivative − ∂2u

∂xh∂xk, the symbol remains the same except for

the matrix H(θ), which now collects the (separable) trigonometric polynomials associated with the new FD formulas.

8.5.1 FD discretization of convection-diffusion-reaction equationsConsider the differential problem

−∇ · A∇u+ b · ∇u+ cu = f, in (0, 1)d,u = 0, on ∂((0, 1)d),

⇐⇒

−d∑

h,k=1

∂

∂xh

(ahk

∂u

∂xk

)+

d∑k=1

bk∂u

∂xk+ cu = f, in (0, 1)d,

u = 0, on ∂((0, 1)d),(8.236)

where A : [0, 1]d → Rd×d is a symmetric matrix of functions ahk ∈ C1((0, 1)d) ∩ C([0, 1]d) whose partial derivatives ∂ahk/∂xl arebounded over (0, 1)d, b : [0, 1]d → Rd is a vector of functions bk ∈ C([0, 1]d), and c, f ∈ C([0, 1]d).

FD discretization Problem (8.236) can be reformulated as follows:

−1(A Hu)1T + s · ∇u+ cu = f, in (0, 1)d,u = 0, on ∂((0, 1)d),

⇐⇒

−d∑

h,k=1

ahk∂2u

∂xh∂xk+

d∑k=1

sk∂u

∂xk+ cu = f, in (0, 1)d,

u = 0, on ∂((0, 1)d),(8.237)

where Hu is the Hessian of u,

(Hu)hk =∂2u

∂xh∂xk, h, k = 1, . . . , d,

and s collects the coefficients of the first-order derivatives,

sk = bk −d∑

h=1

∂ahk∂xh

, k = 1, . . . , d.

We consider the classical second-order central FD discretization of (8.237). We choose n ∈ Nd and we set h = 1n+1 and xj = jh

for j = 0, . . . ,n + 1.8 Let ek be the k-th vector of the canonical basis of Rd. Then, for j = 1, . . . ,n, we can approximate the terms

8Recall from Section 2.1.1 that operations involving d-indices that have no meaning in Zd must be interpreted in the componentwise sense. In the present case,given n = (n1, . . . , nd), the vector of discretization steps h = 1

n+1and the grid points xj = jh are given by h = ( 1

n1+1, . . . , 1

nd+1) = (h1, . . . , hd) and

xj = (j1h1, . . . , jdhd).

134

appearing in (8.237) as follows:

akk∂2u

∂x2k

∣∣∣∣x=xj

≈ akk(xj)u(xj + hkek)− 2u(xj) + u(xj − hkek)

h2k

= akk(xj)u(xj+ek)− 2u(xj) + u(xj−ek)

h2k

, k = 1, . . . , d, (8.238)

ahk∂2u

∂xh∂xk

∣∣∣∣x=xj

≈ ahk(xj)

∂u

∂xh(xj + hkek)− ∂u

∂xh(xj − hkek)

2hk

≈ ahk(xj)1

2hk

[u(xj + hkek + hheh)− u(xj + hkek − hheh)

2hh

−u(xj − hkek + hheh)− u(xj − hkek − hheh)

2hh

]= ahk(xj)

u(xj+ek+eh)− u(xj+ek−eh)− u(xj−ek+eh) + u(xj−ek−eh)

4hhhk, h, k = 1, . . . , d, h 6= k, (8.239)

sk∂u

∂xk

∣∣∣∣x=xj

≈ sk(xj)u(xj + hkek)− u(xj − hkek)

2hk= sk(xj)

u(xj+ek)− u(xj−ek)

2hk, k = 1, . . . , d, (8.240)

cu|x=xj= c(xj)u(xj). (8.241)

The evaluations u(xj) of the solution of (8.237) at the grid points xj are approximated by the values uj , where uj = 0 for j ∈0, . . . ,n+ 1\1, . . . ,n, and the vector u = (u1, . . . , un)T is the solution of the linear system

−d∑k=1

akk(xj)uj+ek − 2uj + uj−ek

h2k

−d∑

h,k=1h6=k

ahk(xj)uj+ek+eh − uj+ek−eh − uj−ek+eh + uj−ek−eh

4hhhk

+

d∑k=1

sk(xj)uj+ek − uj−ek

2hk+ c(xj)uj = f(xj), j = 1, . . . ,n. (8.242)

FD discretization matrices We now want to understand the structure of the matrix An associated with the linear system (8.242). Thisis cleary important for the GLT analysis of the next paragraph. Luckily, the multi-index language allows us to provide a compact andeasy-to-manage expression of this matrix. First of all, we note that An admits the following natural decomposition:

An =

d∑h,k=1

1

hhhk

(diag

j=1,...,nahk(xj)

)Kn,hk +

d∑k=1

1

hk

(diag

j=1,...,nsk(xj)

)Hn,k +

(diag

j=1,...,nc(xj)

), (8.243)

where the matrices Kn,hk and Hn,k are defined by their action on a generic vector u ∈ RN(n), as follows:

(Kn,kku)j = −uj−ek + 2uj − uj+ek , j = 1, . . . ,n, k = 1, . . . , d, (8.244)

(Kn,khu)j = (Kn,hku)j = −1

4(uj−eh−ek − uj−eh+ek − uj+eh−ek + uj+eh+ek), j = 1, . . . ,n, 1 ≤ h < k ≤ d, (8.245)

(Hn,ku)j =1

2(−uj−ek + uj+ek), j = 1, . . . ,n, k = 1, . . . , d, (8.246)

where it is understood that ui = 0 whenever i /∈ 1, . . . ,n. Using the multi-index language, it is not difficult to see that

Kn,kk =

(k−1⊗r=1

Inr

)⊗Knk ⊗

(d⊗

r=k+1

Inr

), k = 1, . . . , d, (8.247)

Kn,kh = Kn,hk = −

(h−1⊗r=1

Inr

)⊗Hnh ⊗

(k−1⊗r=h+1

Inr

)⊗Hnk ⊗

(d⊗

r=k+1

Inr

), 1 ≤ h < k ≤ d, (8.248)

Hn,k =

(k−1⊗r=1

Inr

)⊗Hnk ⊗

(d⊗

r=k+1

Inr

), k = 1, . . . , d, (8.249)

135

where

Kn =

2 −1−1 2 −1

. . . . . . . . .−1 2 −1

−1 2

, Hn =1

2

0 −11 0 −1

. . . . . . . . .1 0 −1

1 0

.Note thatKn (resp.,Hn, In) is the diffusion (resp., convection, reaction) matrix associated with the second-order central FD discretizationof the constant-coefficient 1D diffusion equation

−u′′(x) + u′(x) + u(x) = f(x), x ∈ (0, 1),u(0) = u(1) = 0,

Let us prove, for example, eq. (8.247). Let δij = 1 if i = j and δij = 0 otherwise. By the fundamental property (2.44), for everyu ∈ RN(n) and every j = 1, . . . ,n,([(

k−1⊗r=1

Inr

)⊗Knk ⊗

(d⊗

r=k+1

Inr

)]u

)j

=

n∑`=1

[(k−1⊗r=1

Inr

)⊗Knk ⊗

(d⊗

r=k+1

Inr

)]j`

u` =

n∑`=1

(Knk)jk`k

d∏r=1r 6=k

(Inr )jr`ru`

=

n∑`=1

(Knk)jk`k

d∏r=1r 6=k

δjr`ru` = −uj−ek + 2uj − uj+ek = (Kn,kku)j ,

where the second-to-last equation is due to the fact that, when ` varies from 1 ton, the quantity (Knk)jk`k∏dr=1r 6=k

δjr`r is always equal to 0

except for ` = j−ek, j, j+ek, in which case it equals−1, 2,−1, respectively. This completes the proof of (8.247). Eqs. (8.248)–(8.249)are proved in the same way.

Since Kn = Tn(2− 2 cos θ) and Hn = iTn(sin θ), it follows from (8.247)–(8.249) and Lemma 5.3 that

Kn,kk = Tn(2− 2 cos θk) = Tn(Hkk), k = 1, . . . , d, (8.250)Kn,kh = Kn,hk = Tn(sin θh sin θk) = Tn(Hhk), 1 ≤ h < k ≤ d, (8.251)

Hn,k = iTn(sin θk), k = 1, . . . , d, (8.252)

and in particularKn,hk = Tn(Hhk), h, k = 1, . . . , d. (8.253)

GLT analysis of FD discretization matrices Let H : [0, 1]d → Rd×d the symmetric matrix of continuous functions defined by

(H(θ))kk = 2− 2 cos θk, k = 1, . . . , d, (8.254)(H(θ))kh = (H(θ))hk = sin θh sin θk, 1 ≤ h < k ≤ d. (8.255)

From now on, we assume to have a single discretization parameter nwhich varies in the infinite set of indices such thatn+1 = νn ∈ Nd,where ν = (ν1, . . . , νd) ∈ Qd is an a priori fixed vector with positive components. The relation n+ 1 = νn must be kept in mind whilereading the remainder of this section. We will prove that

n−2Ann ∼GLT f (ν) (8.256)

andn−2Ann ∼σ, λ f (ν), (8.257)

where

f (ν)(x,θ) = ν(A(x) H(θ))νT =

d∑h,k=1

νhνkahk(x)(H(θ))hk. (8.258)

Despite the technicalities intrinsic to any d-dimensional analysis, the proof of (8.256)–(8.257) is essentially the same as in the 1-dimensional case. It consists of the following steps. Throughout this proof, C denotes a generic constant independent of n.

Step 1. Decompose An as follows:An = Kn + Zn, (8.259)

136

where

Kn =

d∑h,k=1

1

hhhk

(diag

j=1,...,nahk(xj)

)Kn,hk (8.260)

is the FD diffusion matrix, resulting from the FD discretization of the higher-order term in (8.237), while

Zn =

d∑k=1

1

hk

(diag

j=1,...,nsk(xj)

)Hn,k +

(diag

j=1,...,nc(xj)

)(8.261)

is the matrix resulting from the FD discretization of the lower-order terms (the convection and reaction terms). We show that

‖n−2Kn‖ ≤ C (8.262)

and‖n−2Zn‖ = O(n−1). (8.263)

To prove (8.262), we note that ‖ diagj=1,...,n ahk(xj)‖ ≤ ‖ahk‖∞ for all h, k = 1, . . . , d. Moreover, ‖Kn‖ ≤ 4 and ‖Hn‖ ≤ 1 forall n, which, in combination with (8.247)–(8.248) and (2.45), implies that ‖Kn,hk‖ ≤ 4 for all n and h, k = 1, . . . , d. Alternatively,one could arrive at the inequality ‖Kn,hk‖ ≤ 4 by using (8.250)–(8.251) and 5.21. Since n−2/hhhk = νhνk (because h = 1

n+1 and

n+ 1 = νn), the inequality (8.262) is proved with C = 4∑dh,k=1 νhνk‖ahk‖∞.

The inequality (8.263) is proved in a similar way: we first note that ‖ diagj=1,...,n sk(xj)‖ ≤ ‖sk‖∞ and ‖ diagj=1,...,n c(xj)‖ ≤‖c‖∞; then, by (8.249), (2.45) and ‖Hn‖ ≤ 1 (or by (8.252) and (5.21)), we get ‖Hn,k‖ ≤ 1 for all n and k = 1, . . . , d; finally, usingn−2/hk = n−1νk, we see that (8.263) is met.

Step 2. Let us define the matrix

Kn =

d∑h,k=1

nhnkDn(ahk) Tn(Hhk), (8.264)

and consider the following decomposition of n−2An:

n−2An = n−2Kn + (n−2Kn + n−2Kn) + n−2Zn. (8.265)

By GLT 3 and GLT 5, ‖n−2Kn‖ ≤ C and n−2Knn ∼GLT

∑dh,k=1 νhνkahk ⊗Hhk = f (ν).

Step 3. We show that‖n−2Kn − n−2Kn‖1 = o(nd), (8.266)

so that from (8.263) we get ‖(n−2Kn − n−2Kn) + n−2Zn‖1 = o(nd) and, consequently, (n−2Kn − n−2Kn) + n−2Znn ∼σ 0.Once this is obtained, (8.256) follows from (8.265) combined with GLT 3 and GLT 5, the singular value distribution in (8.257) followsfrom GLT 1, the spectral distribution in (8.257) follows from GLT 2, and the proof is finished.

Let us then prove (8.266). We decompose the difference n−2Kn − n−2Kn as follows:

n−2Kn − n−2Kn =

d∑h,k=1

νhνk

(diag

j=1,...,nahk(xj)

)Kn,hk −

d∑h,k=1

νhνkDn(ahk) Tn(Hhk)

=

d∑h,k=1

νhνk

((diag

j=1,...,nahk(xj)

)Tn(Hhk)−Dn(ahk)Tn(Hhk)

)(8.267)

+

d∑h,k=1

νhνk

(Dn(ahk)Tn(Hhk)− Dn(ahk) Tn(Hhk)

). (8.268)

We consider separately the two summations in (8.267)–(8.268), and we show that their trace-norms are o(nd+2).

• By (5.21) and the continuity of the functions ahk, h, k = 1, . . . , d, the spectral norm of the matrix (8.267) is bounded byC∑dh,k=1 ωahk(n−1), so it tends to 0 when n→∞. Hence, the trace-norm of (8.267) is o(nd) by (2.27).

• By GLT 3, the spectral norm of the matrix (8.268) is bounded by C∑dh,k=1 ωahk(n−1), so it tends to 0 when n→∞. Hence, the

trace-norm of (8.268) is o(nd) by (2.27).

In conclusion, ‖n−2Kn − n−2Kn‖1 = o(nd).

137

Formal structure of the symbol By comparing (8.258) and (8.237), we see that the formal expression of the symbol ν(A(x)H(θ))νT

is identical to the one of the higher-order operator −1(A(x) Hu(x))1T in (8.237). The similarity becomes even more evident if:

(a) we give to every direction the same attention, by choosing ν = 1 (so that n = 1n = (n, . . . , n)). In this case, the symbol takes theform 1(A(x) H(θ))1T , which better reproduces the expression of the higher-order operator −1(A(x) Hu(x))1T ;

(b) we note that H(θ) is the matrix of (separable) trigonometric polynomials in the Fourier variables, whose components represent theFD formula used to discretize components of

−Hu =

[− ∂2u

∂xh∂xk

]dh,k=1

,

i.e., the negative Hessian operator in the physical variables x. Indeed, the trgonometric polynomial Hhk is the symbol of the

matrix-sequence Kn,hkn, which comes from the discretization of the second derivative − ∂2u

∂xh∂xk; see (8.253). For this reason,

the matrix H(θ) is sometimes referred to as the symbol of the negative Hessian operator (though, of course, this terminology is notrigorous from the mathematical viewpoint). Following the same technique as in the proof of [48, Theorem 3.4], one can show thatthe matrix H(θ) is SPSD for all θ ∈ [−π, π]d, and it is SPD for all θ ∈ [−π, π]d such that θ1 · · · θd 6= 0.

8.5.2 B-spline IgA collocation approximation of convection-diffusion-reaction equationsConsider the differential problem

−∇ · A∇u+ b · ∇u+ cu = f, in Ω,u = 0, on ∂Ω,

⇐⇒

−d∑

h,k=1

∂

∂xh

(ahk

∂u

∂xk

)+

d∑k=1

bk∂u

∂xk+ cu = f, in Ω,

u = 0, on ∂Ω,

(8.269)

where Ω is a bounded open domain in Rd, A : Ω → Rd×d is a symmetric matrix of functions ahk ∈ C1(Ω) ∩ C(Ω) whose partialderivatives ∂ahk/∂xl are bounded over Ω, b : Ω→ Rd is a vector of functions bk ∈ C(Ω), and c, f ∈ C(Ω).

Isogeometric collocation approximation Problem (8.269) can be reformulated as follows:

−1(A Hu)1T + s · ∇u+ cu = f, in Ω,u = 0, on ∂Ω,

⇐⇒

−d∑

h,k=1

ahk∂2u

∂xh∂xk+

d∑k=1

sk∂u


u = 0, on ∂Ω,

(8.270)

where Hu is the Hessian of u,

(Hu)hk =∂2u

∂xh∂xk, h, k = 1, . . . , d,

and s collects the coefficients of the first-order derivatives,

sk = bk −d∑

h=1

∂ahk∂xh

, k = 1, . . . , d.

In the standard collocation method, we choose a finite dimensional approximation space W , consisting of sufficiently smooth func-tions defined on Ω and vanishing on ∂Ω. Then, we introduce a set of N = dimW collocation points τ 1, . . . , τN ⊂ Ω, and we look fora function uW ∈ W such that

−1(A(τ i) HuW (τ i))1T + s(τ i) · ∇uW (τ i) + c(τ i)uW (τ i) = f(τ i), i = 1, . . . , N. (8.271)

The function uW is taken as an approximation to the solution of (8.270). If ϕ1, . . . , ϕN is a basis of W , then uW =∑Ni=1 uiϕi for a

unique vector u = (u1, . . . , uN )T , and, by linearity, the computation of uW (i.e., of u) reduces to solving the linear system

Au = f , (8.272)

138

where

A =[−1(A(τ i) Hϕj(τ i))1T + s(τ i) · ∇ϕj(τ i) + c(τ i)ϕj(τ i)

]Ni,j=1

=

− d∑h,k=1

ahk(τ i)∂2ϕj∂xh∂xk

(τ i) +

d∑k=1

sk(τ i)∂ϕj∂xk

(τ i) + c(τ i)ϕj(τ i)

Ni,j=1

= −d∑

h,k=1

(diag

i=1,...,Nahk(τ i)

)[ ∂2ϕj∂xh∂xk

(τ i)

]Ni,j=1

+

d∑k=1

(diag

i=1,...,Nsk(τ i)

)[∂ϕj∂xk

(τ i)

]Ni,j=1

+(

diagi=1,...,N

c(τ i))[ϕj(τ i)

]Ni,j=1

(8.273)

is the collocation matrix and f = [f(τ i)]Ni=1.

Now, suppose that the physical domain Ω can be described by a global geometry function G : [0, 1]d → Ω, which is invertible andsatisfies G(∂([0, 1]d)) = ∂Ω. Let

ϕ1, . . . , ϕN (8.274)

be a set of basis functions defined on the parametric domain [0, 1]d and vanishing on the boundary ∂([0, 1]d). Let

τ 1, . . . , τN (8.275)

be a set of N collocation points in the parametric domain [0, 1]d. In the isogeometric collocation approach, we find an approximation uW

of u by using the standard collocation method described above, in which:

• the approximation space is chosen as W = span(ϕi, i = 1, . . . , N), with

ϕi(x) = ϕi(G−1(x)) = ϕi(x), x = G(x); (8.276)

• the collocation points in the physical domain Ω are defined as follows:

τ i = G(τ i), i = 1, . . . , N. (8.277)

The resulting collocation matrix A is given by (8.273), with the basis functions ϕi and the collocation points τ i defined as in (8.276)–(8.277).

Assuming that G and ϕi, i = 1, . . . , N , are sufficiently regular, we can apply standard differential calculus to express A in terms ofG and ϕi, τ i, i = 1, . . . , N . Let us work out this expression. For any u : Ω → R consider the corresponding function defined on theparametric domain by

u : [0, 1]d → R, u(x) = u(x), x = G(x). (8.278)

In other words, u = u(G). Then, u satisfies (8.270) if and only if u satisfies the corresponding transformed problem−1(AG Hu)1T + sG · ∇u+ cGu = f(G), in (0, 1)d,u = 0, on ∂((0, 1)d),

⇐⇒

−d∑

h,k=1

aG,hk∂2u

∂xh∂xk+

d∑k=1

sG,k∂u

∂xk+ cGu = f(G), in (0, 1)d,

u = 0 on ∂((0, 1)d),

(8.279)

In (8.279), Hu is the Hessian of u, i.e.,

(Hu)ij =∂2u

∂xi∂xj, i, j = 1, . . . , d;

AG and cG are the transformed diffusion and reaction coeffiecient, which are given by

AG = (JG)−1A(G)(JG)−T =[aG,ij

]di,j=1

, cG = c(G), (8.280)

where JG is the Jacobian matrix of G,

JG =

[∂Gi∂xj

]di,j=1

=

[∂xi∂xj

]di,j=1

;

139

and sG = (sG,1, . . . , sG,d) is the transformed convection coefficient, whose expression in terms of A, s, G is complicated and hencenot reported here. For later purposes, we limit to say that, due to the assumption that the partial derivatives of the components of A arebounded over Ω, the components of sG are bounded as well, as long as G ∈ C2([0, 1]d) and det(G) 6= 0 in [0, 1]d. To realize why thisis true, we recommend that the reader have a look at the 1-dimensional case, especially at eq. (8.113), where sG was explicitly computedin terms of a, s, G. The collocation matrix A in (8.273) can be expressed in terms of G and ϕi, τ i, i = 1, . . . , N , as follows:

A =[−1(AG(τ i) Hϕj(τ i))1T + sG(τ i) · ∇ϕj(τ i) + cG(τ i)ϕj(τ i)

]Ni,j=1

= −d∑

h,k=1

(diag

i=1,...,NaG,hk(τ i)

)[ ∂2ϕj∂xh∂xk

(τ i)

]Ni,j=1

+

d∑k=1

(diag

i=1,...,NsG,k(τ i)

)[∂ϕj∂xk

(τ i)

]Ni,j=1

+(

diagi=1,...,N

cG(τ i))

[ϕj(τ i)]Ni,j=1 . (8.281)

In the context of IgA, the functions ϕi are usually tensor-product B-splines or NURBS. Here, the role of the ϕi will be played bytensor-product B-splines over uniform knot sequences (for the case of NURBS, see [49]). Moreover, following [3], the collocation pointsτ i will be chosen as the Greville abscissae corresponding to the B-splines ϕi.

Tensor-product B-splines and Greville abscissae For any pair of d-indices p,n ≥ 1 and any k = 1, . . . , d, let Nik,[pk], ik =1, . . . , nk + pk, be the B-splines of degree pk defined on the knot sequence

t1 = . . . = tpk+1 = 0 < tpk+2 < . . . < tpk+nk < 1 = tpk+nk+1 = . . . = t2pk+nk+1, (8.282)

wheretik+pk+1 =

iknk, ik = 0, . . . , nk. (8.283)

Let ξik,[pk], ik = 1, . . . , nk + pk, be the Greville abscissae associated with the B-splines Nik,[pk], ik = 1, . . . , nk + pk. Note that boththe B-splines Nik,[pk], ik = 1, . . . , nk + pk, and the Greville abscissae ξik,[pk], ik = 1, . . . , nk + pk, are defined as in Section 8.4.9 withp = pk and n = nk; see (8.117)–(8.120).

We define the tensor-product B-splines as follows:

Ni,[p] : [0, 1]d → R, Ni,[p] = Ni1,[p1] ⊗ · · · ⊗Nid,[pd], i = 1, . . . ,n+ p. (8.284)

The (tensor-product) Greville abscissa ξi,[p] associated with the tensor-product B-spline Ni,[p] is defined by

ξi,[p] = (ξi1,[p1], . . . , ξid,[pd]), i = 1, . . . ,n+ p. (8.285)

The properties of B-splines and Greville abscissae in (8.121)–(8.125) and (8.134)–(8.136) imply the following analogous propertiesof tensor-product B-splines and Greville abscissae.

• Local support property:

supp(Ni,[p]) = [ti1 , ti1+p1+1]× · · · × [tid , tid+pd+1], i = 1, . . . ,n+ p. (8.286)

• Vanishing on the boundary:Ni,[p](t) = 0, t ∈ ∂([0, 1]d), i = 2, . . . ,n+ p− 1. (8.287)

• Nonnegative partition of unity:

Ni,[p](t) ≥ 0, t ∈ [0, 1]d, i = 1, . . . ,n+ p, (8.288)n+p∑i=1

Ni,[p](t) = 1, t ∈ [0, 1]d. (8.289)

• Bounds for derivatives:

n+p∑i=1

∣∣∣∣∂Ni,[p]

∂tk(t)

∣∣∣∣ ≤ 2pknk,

n+p∑i=1

∣∣∣∣∂2Ni,[p]

∂th∂tk(t)

∣∣∣∣ ≤ 4phpknhnk, t ∈ [0, 1]d, h, k = 1, . . . , d. (8.290)

In (8.290), it is understood that the undefined values are counted as 0 in the summations.

140

• ξi,[p] lies in the support of Ni,[p],ξi,[p] ∈ supp(Ni,[p]), i = 1, . . . ,n+ p. (8.291)

• The Greville abscissae are somehow equivalent, in an asymptotic sense, to the uniform knots in [0, 1]d. More precisely,∥∥∥∥ξi,[p] −i

n+ p

∥∥∥∥∞≤ Cp

min(n), i = 1, . . . ,n+ p, (8.292)

where Cp depends only on p.

B-spline IgA collocation matrices In the IgA collocation approach based on (uniform) tensor-product B-splines, the basis functionsϕ1, . . . , ϕN in (8.274) are chosen as the tensor-product B-splines

Ni+1,[p], i = 1, . . . ,n+ p− 2, (8.293)

and the collocation points τ 1, . . . , τN in (8.275) are chosen as the Greville abscissae

ξi+1,[p], i = 1, . . . ,n+ p− 2. (8.294)

In this d-dimensional setting, we have N = N(n+ p− 2). Of course, the basis functions (8.293) and the collocation points (8.294) areordered in accordance with the standard lexicographic ordering; see (2.1). Throughout this section, we will assume that p ≥ 2, so as toensure that N ′′j+1,[p](ξi+1,[p]) is defined for all i, j = 1, . . . ,n+ p− 2.

The collocation matrix (8.281) resulting from the choices of ϕi, τi as in (8.293)–(8.294) will be denoted by A[p]G,n, in order to

emphasize its dependence on the geometry map G and on the parameters n,p:

A[p]G,n =

[−1(AG(ξi+1,[p]) HNj+1,[p](ξi+1,[p]))1

T + sG(ξi+1,[p]) · ∇Nj+1,[p](ξi+1,[p]) + cG(ξi+1,[p])Nj+1,[p](ξi+1,[p])]n+p−2

i,j=1

=

d∑h,k=1

D[p]n (aG,hk)K

[p]n,hk +

d∑k=1

D[p]n (sG,k)H

[p]n,k +D[p]

n (cG)M [p]n , (8.295)

whereD[p]n (h) = diag

i=1,...,n+p−2(h(ξi+1,[p])) (8.296)

is the d-level diagonal sampling matrix containing the samples of the function h : [0, 1]d → R at the Greville abscissae (8.294), and

K[p]n,hk =

[−∂2Nj+1,[p]

∂xh∂xk(ξi+1,[p])

]n+p−2

i,j=1

, h, k = 1, . . . , d, (8.297)

H[p]n,k =

[∂Nj+1,[p]

∂xk(ξi+1,[p])

]n+p−2

i,j=1

, k = 1, . . . , d, (8.298)

M [p]n =

[Nj+1,[p](ξi+1,[p])

]n+p−2

i,j=1. (8.299)


A[p]G,n = K

[p]G,n + Z

[p]G,n, (8.300)

where

K[p]G,n =

d∑h,k=1

D[p]n (aG,hk)K

[p]n,hk (8.301)

is the collocation diffusion matrix, resulting from the collocation discretization of the higher-order term in (8.270), and

Z[p]G,n =

d∑k=1

D[p]n (sG,k)H

[p]n,k +D[p]

n (cG)M [p]n (8.302)

is the matrix resulting from the discretization of the lower-order terms (the convection and reaction terms).

141

It is not difficult to show that the matrices K [p]n,hk, H [p]

n,k, M [p]n admit the following explicit expressions:

K[p]n,kk =

(k−1⊗r=1

M [pr]nr

)⊗K [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

), k = 1, . . . , d, (8.303)

K[p]n,kh = K

[p]n,hk = −

(h−1⊗r=1

M [pr]nr

)⊗H [ph]

nh⊗

(k−1⊗r=h+1

M [pr]nr

)⊗H [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

), 1 ≤ h < k ≤ d, (8.304)

H[p]n,k =

(k−1⊗r=1

M [pr]nr

)⊗H [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

)k = 1, . . . , d, (8.305)

M [p]n =

d⊗r=1

M [pr]nr , (8.306)

where K [p]n , H [p]

n , M [p]n are defined in (8.141)–(8.143). Let us prove, for example, eq. (8.303). By the fundamental property (2.44), for

all i, j = 1, . . . ,n+ p− 2 we have[(k−1⊗r=1

M [pr]nr

)⊗K [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

)]ij

= (K [pk]nk

)ikjk

d∏r=1r 6=k

(M [pr]nr )irjr = −N ′′jk+1,[pk](ξik+1,[pk])

d∏r=1r 6=k

Njr+1,[pr](ξir+1,[pr])

= −∂2Nj+1,[p]

∂x2k

(ξi+1,[p]) = (K[p]n,kk)ij ,

and (8.303) follows. Eqs. (8.305)–(8.306) are proved in the same way.

GLT analysis of B-spline IgA collocation matrices LetHp : [0, 1]d → Rd×d be the symmetric matrix of continuous functions definedby

(Hp)kk =

(k−1⊗r=1

hpr

)⊗ fpk ⊗

(d⊗

r=k+1

hpr

), k = 1, . . . , d, (8.307)

(Hp)kh = (Hp)hk =

(h−1⊗r=1

hpr

)⊗ gph ⊗

(k−1⊗r=h+1

hpr

)⊗ gpk ⊗

(d⊗

r=k+1

hpr

), 1 ≤ h < k ≤ d, (8.308)

where fp, gp, hp are given in (8.151)–(8.153). From now on, we assume to have a single discretization parameter n which varies in theinfinite set of indices such that n = νn ∈ Nd, where ν = (ν1, . . . , νd) ∈ Qd is an a priori fixed vector with positive components. Therelationn = νnmust be kept in mind while reading the remainder of this section. We also assume that G ∈ C2([0, 1]d) and det(JG) 6= 0over [0, 1]d. Under these assumptions, we show that, for any p ≥ 2,

n−2A[p]G,nn ∼GLT f

(ν)G,p (8.309)

andn−2A

[p]G,nn ∼σ, λ f

(ν)G,p, (8.310)

where

f(ν)G,p(x,θ) = ν

(AG(x) Hp(θ)

)νT =

d∑h,k=1

νhνkaG,hk(x)(Hp(θ))hk. (8.311)


Step 1. We show that‖n−2K

[p]G,n‖ ≤ C (8.312)

and‖n−2Z

[p]G,n‖ = O(n−1). (8.313)

142

To prove (8.312), we make use of the expressions (8.301), (8.303)–(8.304), and of the inequalities in (8.159). Combining these withthe regularity of G and eq. (2.45), and recalling that n = νn, we obtain

‖K [p]G,n‖ ≤

d∑h,k=1

‖D[p]n (aG,hk)‖ ‖K [p]

n,hk‖ ≤d∑

h,k=1

‖aG,hk‖∞C [p1] · · ·C [pd]n2k ≤ Cn2,

and (8.312) follows. Eq. (8.313) is proved in the same way: using (8.302), (8.305)–(8.306), and (8.159), we obtain

‖Z [p]G,n‖ ≤

d∑k=1

‖D[p]n (sG,k)‖ ‖H [p]

n,k‖+ ‖D[p]n (cG)‖ ‖M [p]

n ‖ ≤d∑k=1

‖sG,k‖∞C [p1] · · ·C [pd]nk + ‖cG‖∞C [p1] · · ·C [pd] ≤ Cn,

which proves (8.313).

Step 2. Let us define the symmetric matrix

K[p]G,n =

d∑h,k=1

Dn+p−2(aG,hk) nhnkTn+p−2((Hp)hk), (8.314)

and consider the following decomposition of n−2A[p]G,n:

n−2A[p]G,n = n−2K

[p]G,n +

(n−2K

[p]G,n − n

−2K[p]G,n

)+ n−2Z

[p]G,n. (8.315)

By GLT 3 and GLT 5, ‖n−2K[p]G,n‖ ≤ C and n−2K

[p]G,nn ∼GLT

∑dh,k=1 νhνkaG,hk ⊗ (Hp)hk = f

(ν)G,p.

Step 3. We show that‖n−2K

[p]G,n − n

−2K[p]G,n‖1 = o(nd), (8.316)

so that from (8.313) we get ‖(n−2K[p]G,n−n−2K

[p]G,n) +n−2Z

[p]G,n‖1 = o(nd) and (n−2K

[p]G,n−n−2K

[p]G,n) +n−2Z

[p]G,nn ∼σ 0. Once

this is obtained, (8.309) follows from (8.315) combined with GLT 3 and GLT 5, the singular value distribution in (8.310) follows fromGLT 1, the spectral distribution in (8.310) follows from GLT 2, and the proof is finished.

Let us then prove (8.316). We decompose the difference n−2K[p]G,n − n−2K

[p]G,n as follows:

n−2K[p]G,n − n

−2K[p]G,n =

d∑h,k=1

n−2D[p]n (aG,hk)K

[p]n,hk −

d∑h,k=1

Dn+p−2(aG,hk) νhνkTn+p−2((Hp)hk)

=

d∑h,k=1

(n−2D[p]

n (aG,hk)K[p]n,hk − νhνkD

[p]n (aG,hk)Tn+p−2((Hp)hk)

)(8.317)

+

d∑h,k=1

νhνk

(D[p]n (aG,hk)Tn+p−2((Hp)hk)−Dn+p−2(aG,hk)Tn+p−2((Hp)hk)

)(8.318)

+

d∑h,k=1

νhνk

(Dn+p−2(aG,hk)Tn+p−2((Hp)hk)− Dn+p−2(aG,hk) Tn+p−2((Hp)hk)

). (8.319)

We consider separately the three summations in (8.317)–(8.319), and we show that their trace-norms are o(nd+2).

• By (2.47)–(2.48), (8.154)–(8.156), (8.303)–(8.304), (8.307)–(8.308), and Lemma 5.3,

rank(n−2K

[p]n,hk − νhνkTn+p−2((Hp)hk)

)≤ N(n+ p− 2)

d∑i=1

4(pi − 1)

ni + pi − 2≤ Cnd−1, h, k = 1, . . . , d.

Hence, also the rank of the summation (8.317) is bounded by Cnd−1. A rough estimate, based on the regularity of G and on theinequalities (8.159) and (5.21), shows that the spectral norm of (8.317) is bounded by C. Thus, thanks to (2.27), the trace-norm of(8.317) is bounded by Cnd−1, and it is therefore o(nd) (actually, O(nd−1)).

• By (8.136), the continuity of the functions aG,hk, h, k = 1, . . . , d, and (5.21), the spectral norm of the matrix (8.318) is boundedby C

∑dh,k=1 ωaG,hk(n−1), so it tends to 0 when n→∞. Hence, the trace-norm of (8.318) is o(nd) by (2.27).

143

• By GLT 3, the spectral norm of the matrix (8.319) is bounded by C∑dh,k=1 ωaG,hk(n−1), so it tends to 0 when n → ∞. Hence,

the trace-norm of (8.319) is o(nd) by (2.27).

In conclusion, ‖n−2K[p]G,n − n−2K

[p]G,n‖1 = o(nd).

Exercise 8.8. Show that, for any continuous function a : [0, 1]d → R,

n−2D[p]n (a)K

[p]n,hkn ∼sLT a⊗ (Hp)hk, h, k = 1, . . . , d. (8.320)

In particular,n−2D[p]

n (aG,hk)K[p]n,hkn ∼sLT aG,hk ⊗ (Hp)hk, h, k = 1, . . . , d.

This result can be seen as a consequence of the fact that n−2D[p]n (aG,hk)K

[p]n,hk is the matrix resulting from the discretization of the

separable differential operator −aG,hk∂2u

∂xh∂xkin (8.279).9

Formal structure of the symbol f (ν)G,p By comparing (8.311) and (8.279), we see that the formal expression of the symbol ν(AG(x)

Hp(θ))νT is identical to the one of the higher-order operator −1(AG(x) Hu(x))1T in (8.279). The similarity becomes even moreevident if:

(a) we give to every direction the same attention, by choosing ν = 1 (so that n = 1n = (n, . . . , n)). In this case, the symbol takes theform 1(AG(x) Hp(θ))1T , which better reproduces the expression of the higher-order operator −1(AG(x) Hu(x))1T ;

(b) we note that Hp(θ) is the matrix of (separable) trigonometric polynomials in the Fourier variables, whose components representthe techniques used to discretize components of

−Hu =

[− ∂2u

∂xh∂xk

]dh,k=1

,

i.e., the negative Hessian operator in the parametric variables x. Indeed, as shown in Exercise 8.8, the entry (Hp)hk is the symbol

of the matrix-sequence K [p]n,hkn, which comes from the discretization of the second derivative − ∂2u

∂xh∂xk. For this reason, the

matrix Hp(θ) is sometimes referred to as the symbol of the negative Hessian operator (though, of course, this terminology is notrigorous from the mathematical viewpoint). As proved in [48, Theorem 3.4], the matrix Hp(θ) is SPSD for all θ ∈ [−π, π]d, andit is SPD for all θ ∈ [−π, π]d such that θ1 · · · θd 6= 0.

Exercise 8.9. The matrix A[p]G,n in (8.295) can be decomposed according to the diffusion, convection and reaction terms:

A[p]G,n = K

[p]G,n +H

[p]G,n +M

[p]G,n, (8.321)

where K [p]G,n is defined in (8.301), and the convection and reaction matrices are given by

H[p]G,n =

d∑k=1

D[p]n (sG,k)H

[p]n,k, (8.322)

M[p]G,n = D[p]

n (cG)M [p]n . (8.323)

Assume that G ∈ C2([0, 1]) and det(JG) 6= 0 over [0, 1]d, and let p ≥ 2, n = νn. We have seen above that n−2K[p]G,nn ∼GLT f

(ν)G,p

and n−2K[p]G,nn ∼σ, λ f

(ν)G,p. Show that:

9We say that a differential operator is separable if it is obtained by multiplying a given function with a product of partial derivatives. The general separable differentialoperator can be written as

a(x)∂r1+...+rdu

∂xr11 · · · ∂xrdd

.


∆u =d∑k=1

∂2u

∂x2k.

The (general) message of this exercise is that a separable differential operator gives rise to a sLT sequence. As a consequence, an arbitrary differential operator (a sum ofseparable differential operators) gives rise to a sum of sLT sequences, i.e., a GLT sequence.

144

(a) −in−1H[p]G,nn ∼GLT g

(ν)G,p and −in−1H

[p]G,nn ∼σ, λ g

(ν)G,p, where g(ν)

G,p : [0, 1]d × [−π, π]d → R is defined as follows:

g(ν)G,p(x,θ) =

d∑k=1

νksG,k ⊗

(k−1⊗r=1

hpr

)⊗ gpk ⊗

(d⊗

r=k+1

hpr

), (8.324)

with gp(θ), hp(θ) as in (8.152)–(8.153);

(b) M [p]G,nn ∼GLT hG,p and M [p]

G,nn ∼σ, λ hG,p, where hG,p : [0, 1]d × [−π, π]d → R is defined as follows:

hG,p(x,θ) = cG ⊗

(d⊗r=1

hpr

), (8.325)

with hp(θ) as in (8.153).

8.5.3 Galerkin B-spline IgA approximation of convection-diffusion-reaction equationsConsider the differential problem

−∇ · A∇u+ b · ∇u+ cu = f, in Ω,u = 0, on ∂Ω,

⇐⇒

−d∑

h,k=1

∂

∂xh

(ahk

∂u

∂xk

)+

d∑k=1

bk∂u


u = 0, on ∂Ω,

(8.326)

where Ω is a bounded open domain in Rd with Lipschitz boundary, A : Ω → Rd×d is a symmetric matrix of functions ahk ∈ L∞(Ω),b : Ω→ Rd is a vector of functions bk ∈ L∞(Ω), c ∈ L∞(Ω) and f ∈ L2(Ω).

Isogeometric Galerkin approximation The weak form of (8.326) consists in finding u ∈ H10 (Ω) such that

a(u, v) = f(v) ∀v ∈ H10 (Ω),

wherea(u, v) =

∫Ω

((∇u)TA∇v + (∇u)Tb v + cuv

), f(v) =

∫Ω

fv.

In the standard Galerkin method, we look for an approximation uW of u by choosing a finite dimensional approximation space W ⊂H1

0 (Ω) and by solving the following (Galerkin) problem: find uW ∈ W such that

a(uW , v) = f(v) ∀v ∈ W .

If ϕ1, . . . , ϕN is a basis of W , then uW =∑Nj=1 ujϕj for a unique vector u = (u1, . . . , uN )T , and, by linearity, the computation of

uW is equivalent to solving the linear systemAu = f ,

where

A = [a(ϕj , ϕi)]Ni,j=1 =

[∫Ω

((∇ϕj)TA∇ϕi + (∇ϕj)Tb ϕi + cϕjϕi

)]Ni,j=1

(8.327)

is the stiffness matrix and f = [f(ϕi)]Ni=1.

Now, suppose that the physical domain Ω can be described by a global geometry function G : [0, 1]d → Ω, which is invertible andsatisfies G(∂([0, 1]d)) = ∂Ω. Let

ϕ1, . . . , ϕN (8.328)

be a set of basis functions defined on the reference (parametric) domain [0, 1]d and vanishing on the boundary ∂([0, 1]d). In the isogeo-metric Galerkin approach, the approximation space is chosen as W = 〈ϕi : i = 1, . . . , N〉, with

ϕi(x) = ϕi(G−1(x)) = ϕi(x), x = G(x). (8.329)

The resulting stiffness matrixA is given by (8.327), with the basis functions ϕi defined in (8.329). Assuming that G and ϕi, i = 1, . . . , N ,are sufficiently regular, we can apply standard differential calculus to obtain the following expression for A in terms of G and ϕi,i = 1, . . . , N :

A =

[∫[0,1]d

((∇ϕj)TAG∇ϕi + (∇ϕj)T (JG)−1b(G) ϕi + c(G)ϕjϕi

)|det(JG)|

]Ni,j=1

, (8.330)

145

whereAG = (JG)−1A(G)(JG)−T =

[aG,ij

]di,j=1

, (8.331)

and JG is the Jacobian matrix of G, i.e.,

JG =

[∂Gi∂xj

]di,j=1

=

[∂xi∂xj

]di,j=1

.

In the context of IgA, the functions ϕi are usually tensor-product B-splines or NURBS. Here, the role of the ϕi will be played bytensor-product B-splines over uniform knot sequences. For the case of NURBS, we refer the reader to [49].

Galerkin B-spline IgA discretization matrices In the Galerkin B-spline IgA based on (uniform) tensor-product B-splines, the basisfunctions ϕ1, . . . , ϕN in (8.328) are chosen as the tensor-product B-splines Ni+1,[p], i = 1, . . . ,n + p − 2; see (8.284). The boundaryfunctions Ni,[p] such that ij ∈ 1, nj + pj for some j ∈ 1, . . . , d are excluded because they do not vanish on ∂([0, 1]d). The stiffnessmatrix (8.327) resulting from these choices of the ϕi will be denoted by A[p]

G,n. In multi-index notation, we have

A[p]G,n =

[∫[0,1]d

((∇Nj+1,[p])

TAG∇Ni+1,[p] + (∇Nj+1,[p])T (JG)−1b(G)Ni+1,[p] + c(G)Nj+1,[p]Ni+1,[p]

)|det(JG)|

]n+p−2

i,j=1

=

d∑h,k=1

K[p]G,n,hk +

d∑k=1

H[p]G,n,k +M

[p]G,n, (8.332)

where

K[p]G,n,hk =

[∫[0,1]d

|det(JG)|aG,hk∂Nj+1,[p]

∂xh

∂Ni+1,[p]

∂xk

]n+p−2

i,j=1

, h, k = 1, . . . , d, (8.333)

H[p]G,n,k =

[∫[0,1]d

|det(JG)|((JG)−1b(G)

)k

∂Nj+1,[p]

∂xkNi+1,[p]

]n+p−2

i,j=1

, k = 1, . . . , d, (8.334)

M[p]G,n =

[∫[0,1]d

|det(JG)|c(G)Nj+1,[p]Ni+1,[p]

]n+p−2

i,j=1

. (8.335)


A[p]G,n = K

[p]G,n + Z

[p]G,n, (8.336)

where

K[p]G,n =

d∑h,k=1

K[p]G,n,hk =

[∫[0,1]d

(∇Nj+1,[p])T |det(JG)|AG∇Ni+1,[p]

]n+p−2

i,j=1

(8.337)

is the matrix resulting from the discretization of the higher-order (diffusion) term in (8.326), and

Z[p]G,n =

d∑k=1

H[p]G,n,k +M

[p]G,n =

[∫[0,1]d

|det(JG)|((∇Nj+1,[p])

T (JG)−1b(G)Ni+1,[p] + c(G)Nj+1,[p]Ni+1,[p]

)]n+p−2

i,j=1

(8.338)

is the matrix resulting from the discretization of the terms in (8.326) with lower-order derivatives (the convection and reaction terms).Let

K[p]n,hk =

[∫[0,1]d

∂Nj+1,[p]

∂xh

∂Ni+1,[p]

∂xk

]n+p−2

i,j=1

, h, k = 1, . . . , d, (8.339)

H[p]n,k =

[∫[0,1]d

∂Nj+1,[p]

∂xkNi+1,[p]

]n+p−2

i,j=1

, k = 1, . . . , d, (8.340)

M [p]n =

[∫[0,1]d

Nj+1,[p]Ni+1,[p]

]n+p−2

i,j=1

. (8.341)

146

These matrices are obtained by formally setting |det(JG)|aG,hk = |det(JG)|((JG)−1b(G)

)k

= |det(JG)|c(G) = 1 in (8.333)–

(8.335). They will play an important role in the GLT analysis of this section. It is not difficult to see that K [p]n,hk, H [p]

n,k, M [p]n admit the

following explicit expressions:

K[p]n,kk =

(k−1⊗r=1

M [pr]nr

)⊗K [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

), k = 1, . . . , d, (8.342)

K[p]n,kh = K

[p]n,hk = −

(h−1⊗r=1

M [pr]nr

)⊗H [ph]

nh⊗

(k−1⊗r=h+1

M [pr]nr

)⊗H [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

), 1 ≤ h < k ≤ d, (8.343)

H[p]n,k =

(k−1⊗r=1

M [pr]nr

)⊗H [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

)k = 1, . . . , d, (8.344)

M [p]n =

d⊗r=1

M [pr]nr , (8.345)

where K [p]n , H [p]

n , M [p]n are defined in (8.188)–(8.190). Let us prove, for example, eq. (8.342). By the fundamental property (2.44), for

all i, j = 1, . . . ,n+ p− 2 we have[(k−1⊗r=1

M [pr]nr

)⊗K [pk]

nk⊗

(d⊗

r=k+1

M [pr]nr

)]ij

= (K [pk]nk

)ikjk

d∏r=1r 6=k

(M [pr]nr )irjr

=

∫[0,1]

N ′jk+1,[pk]N′ik+1,[pk]

d∏r=1r 6=k

∫[0,1]

Njr+1,[pr]Nir+1,[pr]

=

∫[0,1]d

∂Nj+1,[p]

∂xk

∂Ni+1,[p]

∂xk= (K

[p]n,kk)ij ,

and (8.342) follows. Eqs. (8.343)–(8.345) are proved in the same way.

GLT analysis of Galerkin B-spline IgA discretization matrices Let Hp : [0, 1]d → Rd×d be the symmetric matrix of continuousfunctions defined by

(Hp)kk =

(k−1⊗r=1

hpr

)⊗ fpk ⊗

(d⊗

r=k+1

hpr

), k = 1, . . . , d, (8.346)

(Hp)kh = (Hp)hk =

(h−1⊗r=1

hpr

)⊗ gph ⊗

(k−1⊗r=h+1

hpr

)⊗ gpk ⊗

(d⊗

r=k+1

hpr

), 1 ≤ h < k ≤ d, (8.347)

where fp, gp, hp are defined in (8.195)–(8.197). From now on, we assume to have a single discretization parameter n which varies in theinfinite set of indices such that n = νn ∈ Nd, where ν = (ν1, . . . , νd) ∈ Qd is an a priori fixed vector with positive components. Therelation n = νnmust be kept in mind while reading the remainder of this section. We also assume that G is regular, i.e., G ∈ C1([0, 1]d)and det(JG) 6= 0 over [0, 1]d. Under these assumptions, we show that

nd−2A[p]G,nn ∼GLT f

(ν)G,p (8.348)

andnd−2A

[p]G,nn ∼σ, λ f

(ν)G,p, (8.349)

where

f(ν)G,p(x,θ) =

ν(|det(JG(x))|AG(x) Hp(θ)

)νT

N(ν)=|det(JG(x))|

N(ν)

d∑h,k=1

νhνkaG,hk(x)(Hp(θ))hk. (8.350)


147

Step 1. We show that ∥∥nd−2K[p]G,n

∥∥ ≤ C (8.351)

and ∥∥nd−2Z[p]G,n

∥∥ ≤ Cn−1. (8.352)

To prove (8.351), we note that K [p]G,n is ‘banded in a d-level sense’. More precisely, if ‖i − j‖∞ > ‖p‖∞, then |ik − jk| > pk

for some k ∈ 1, . . . , d, implying that the intersection of the supports of Ni+1,[p] and Nj+1,[p] has zero measure by the local supportproperty (8.286). Thus, (K

[p]G,n)ij = 0 whenever ‖i − j‖∞ > ‖p‖∞, and this implies that the nonzero entries in each row i and

column j of K [p]G,n are at most (2‖p‖∞ + 1)d (independent of n). Moreover, using the regularity of G and eqs. (8.286), (8.290), for all

i, j = 1, . . . ,n+ p− 2 we have

∣∣(K [p]G,n)ij

∣∣ ≤ d∑h,k=1

∣∣(K [p]G,n,hk)ij

∣∣ ≤ d∑h,k=1

∫[0,1]d

|aG,hk det(JG)|∣∣∣∣∂Nj+1,[p]

∂xh

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xk

∣∣∣∣≤ C

d∑h,k=1

∫[0,1]d

∣∣∣∣∂Nj+1,[p]

∂xh

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xk

∣∣∣∣ = C

d∑h,k=1

∫supp(Ni+1,[p])

∣∣∣∣∂Nj+1,[p]

∂xh

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xk

∣∣∣∣≤ C

d∑h,k=1

4phnhpknkµd(supp(Ni+1,[p])

)≤ 4Cd2‖p‖2∞‖n‖2∞n−d ≤ Cn2−d, (8.353)

where in the last two inequalities we used the relation n = νn and the fact that tij+pj+1 − tij ≤ (pj + 1)/nj for all ij = 1, . . . , nj + pj

and all j = 1, . . . , d. Therefore, each component of the sparse matrix nd−2K[p]G,n is bounded in modulus by a constant C independent of

n, and (8.351) follows from (2.26).To prove (8.352), we follow the same argument as for the proof of (8.351). Due to the local support property, (Z

[p]G,n)ij = 0

whenever ‖i − j‖∞ > ‖p‖∞, and the nonzero entries in each row and column of Z [p]G,n are at most (2‖p‖∞ + 1)d. Moreover, for all

i, j = 1, . . . ,n+ p− 2,

∣∣(Z [p]G,n)ij

∣∣ ≤ d∑k=1

∣∣(H [p]G,n,k)ij

∣∣+∣∣(M [p]

G,n)ij∣∣

≤∫

[0,1]d

∣∣((JG)−1b(G))k

det(JG)∣∣ ∣∣∣∣∂Nj+1,[p]

∂xk

∣∣∣∣ ∣∣Ni+1,[p]

∣∣+

∫[0,1]d

|c(G) det(JG)|∣∣Nj+1,[p]

∣∣ ∣∣Ni+1,[p]

∣∣≤ C

[d∑k=1

∫[0,1]d

∣∣∣∣∂Nj+1,[p]

∂xk

∣∣∣∣ ∣∣Ni+1,[p]

∣∣+

∫[0,1]d

∣∣Nj+1,[p]

∣∣ ∣∣Ni+1,[p]

∣∣]

= C

[d∑k=1

∫supp(Ni+1,[p])

∣∣∣∣∂Nj+1,[p]

∂xk

∣∣∣∣ ∣∣Ni+1,[p]

∣∣+

∫supp(Ni+1,[p])

∣∣Nj+1,[p]

∣∣ ∣∣Ni+1,[p]

∣∣]

≤ Cµd(supp(Ni+1,[p])

) [ d∑k=1

2pknk + 1

]≤ Cn1−d,


Step 2. Let L1([0, 1]d,Rd×d) be the space of functions L : [0, 1]d → Rd×d such that Lij ∈ L1([0, 1]d) for all i, j = 1, . . . , d.Consider the linear operator

L [p]n (·) : L1([0, 1]d,Rd×d)→ RN(n+p−2)×N(n+p−2),

L [p]n (L) =

[∫[0,1]d

(∇Nj+1,[p])TL∇Ni+1,[p]

]n+p−2

i,j=1

. (8.354)

In the next four steps we show that

nd−2L [p]n (L)n ∼GLT

ν(L(x) Hp(θ))νT

N(ν)∀L ∈ L1([0, 1]d,Rd×d). (8.355)

148

Once this is proved, from K[p]G,n = L

[p]n (|det(JG)|AG) we get

nd−2K[p]G,nn ∼GLT

ν(|det(JG(x))|AG(x) Hp(θ)

)νT

N(ν)= f

(ν)G,p(x,θ).

Step 3. We first prove (8.355) in the constant-coefficient case L = Ehk, where Ehk is the d× d matrix having 1 in position (h, k) and0 elsewhere. Note that if G is the identity map on Ω = (0, 1)d and we take A = Ehk in (8.326), then we are ‘selecting’ the second-orderpartial derivative

− ∂2u

∂xh∂xk= − ∂2u

∂xh∂xk,

which is a separable differential operator.10 Substituting L = Ehk in (8.354), we see that L[p]n (Ehk) = K

[p]n,hk (cf. (8.339)). The matrix

nd−2K[p]n,hk coincides with the d-level Toeplitz matrix N(ν)−1νhνkTn+p−2((Hp)hk) up to a low-rank correction. Indeed, by (2.48),

(8.198)–(8.200), (8.342)–(8.343), (8.346)–(8.347), and Lemma 5.3, we have

rank(nd−2K

[p]n,hk −

νhνkN(ν)

Tn+p−2((Hp)hk))≤ N(n+ p− 2)

d∑i=1

4(pi − 1)

ni + pi − 2≤ Cnd−1.

It follows from GLT 3 and GLT 5 that

nd−2L [p]n (Ehk)n = nd−2K

[p]n,hkn ∼GLT

νhνkN(ν)

(Hp(θ))hk =ν(Ehk Hp(θ))νT

N(ν), (8.356)

and (8.355) is proved in the case L = Ehk, for all h, k = 1, . . . , d.

Step 4. Now we prove (8.355) in the variable-coefficient case L(x) = a(x)Ehk, where a ∈ C([0, 1]d). Note that if G is the identitymap on Ω = (0, 1)d and we take A = a(x)Ehk in (8.326), then we are ‘selecting’

− ∂

∂xh

(a(x)

∂u

∂xk

),

which coincides with the second-order separable differential operator

−a(x)∂2u

∂xh∂xk

up to a term with a lower-order derivative, namely

− ∂a

∂xh(x)

∂u

∂xk.

Given the local support of the basis functions Ni,[p], i = 2, . . . ,n + p − 1, and the fact that supp(Ni,[p]) is located near the pointi/n = (i1/n1, . . . , id/nd), we can show that

nd−2L [p]n (a(x)Ehk) = nd−2Dn+p−2(a)L [p]

n (Ehk) +Qn+p−2, (8.357)

whereQn+p−2 is a sparse matrix whose components and spectral norm are≤ Cωa(n−1); here, ωa(·) stands for the modulus of continuityof a. Since ωa(n−1)→ 0, the sequence Qn+p−2n is zero-distributed. Using (8.356), (8.357), GLT 3 and GLT 5, we get

nd−2L [p]n (a(x)Ehk)n ∼GLT

νhνkN(ν)

a(x)(Hp(θ))hk =ν(a(x)Ehk Hp(θ))νT

N(ν), (8.358)

10A differential operator is separable if it is obtained by multiplying a given function with a product of partial derivatives. The general separable differential operator canbe written as

a(x)∂r1+...+rdu

∂xr11 · · · ∂xrdd

.


∆u =

d∑k=1

∂2u

∂x2k.

What we are going to see is that a separable differential operator gives rise to a GLT sequence (or, more precisely, to a sLT sequence). As a consequence, an arbitrarydifferential operator (a sum of separable differential operators) gives rise to a sum of GLT sequences, which is again a GLT sequence by GLT 5.

149

and (8.355) is proved in the case L(x) = a(x)Ehk for all h, k = 1, . . . , d.To conclude this step, it only remains to formally prove (8.357). To this end, we will show that

‖nd−2L [p]n (a(x)Ehk)− nd−2Dn+p−2(a)L [p]

n (Ehk)‖ ≤ C ωa(n−1). (8.359)

For every i, j = 1, . . . ,n+ p− 2,∣∣(L [p]n (a(x)Ehk)−Dn+p−2(a)L [p]

n (Ehk))ij∣∣ =

∣∣(L [p]n (a(x)Ehk))ij − (Dn+p−2(a))ii(L

[p]n (Ehk))i,j

∣∣=

∣∣∣∣∫[0,1]d

[a(x)− a

(i

n+ p− 2

)]∂Nj+1,[p]

∂xs

∂Ni+1,[p]

∂xt

∣∣∣∣≤ max

x∈ supp(Ni+1,[p])

∣∣∣∣a(x)− a(

i

n+ p− 2

)∣∣∣∣ ∫[0,1]d

∣∣∣∣∂Nj+1,[p]

∂xs

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xt

∣∣∣∣ . (8.360)

A direct verification based on (8.286) shows that

maxx∈ supp(Ni+1,[p])

∥∥∥∥x− i

n+ p− 2

∥∥∥∥∞≤ C

(min

r=1,...,dnr

)−1

≤ Cn−1,

whence

maxx∈ supp(Ni,[p])

∣∣∣∣a(x)− a(

i− 1

n+ p− 2

)∣∣∣∣ ≤ Cωa(n−1).

Moreover, proceeding as in Step 1 (see (8.353)), we obtain∫[0,1]d

∣∣∣∣∂Nj+1,[p]

∂xs

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xt

∣∣∣∣ ≤ Cn−d+2.

Hence, from (8.360) we get ∣∣(L [p]n (a(x)Ehk)−Dn+p−2(a)L [p]

n (Ehk))ij∣∣ ≤ Cωa(n−1)n−d+2,

and ∣∣(nd−2L [p]n (a(x)Ehk)− nd−2Dn+p−2(a)L [p]

n (Ehk))ij∣∣ ≤ Cωa(n−1),

for every i, j = 1, . . . ,n+p−2. In addition, the matrix nd−2L[p]n (a(x)Ehk)−nd−2Dn+p−2(a)L

[p]n (Ehk), like the matrices Z [p]

G,n and

K[p]G,n already considered in Step 1, has the entry 0 in each position (i, j) whenever ‖i− j‖∞ > ‖p‖∞, due to the local support property

of tensor-product B-splines. It follows that the number of nonzero components of nd−2L[p]n (a(x)Ehk) − nd−2Dn+p−2(a)L

[p]n (Ehk)

in each row and column is at most (2‖p‖∞ + 1)d, and so∥∥nd−2L [p]n (a(x)Ehk)− nd−2Dn+p−2(a)L [p]

n (Ehk)∥∥ ≤ Cωa(n−1).

This proves (8.359).

Step 5. Now we prove (8.355) in the variable-coefficient case L(x) = a(x)Ehk, where a ∈ L1([0, 1]d). Take a sequence ammsuch that am ∈ C([0, 1]d) and am → a in L1([0, 1]d). Since am ∈ C([0, 1]d), by Step 4 we have

nd−2L [p]n (am(x)Ehk)n ∼GLT

ν(am(x)Ehk Hp(θ))νT

N(ν)(8.361)

for all m. Since am → a in L1([0, 1]d), it is clear that

ν(am(x)Ehk Hp(θ))νT

N(ν)→ ν(a(x)Ehk Hp(θ))νT

N(ν)in measure over [0, 1]d × [−π, π]d. (8.362)

We show thatnd−2L [p]

n (am(x)Ehk)nm is an a.c.s. for nd−2L [p]n (a(x)Ehk)n. (8.363)

Once this is proved, (8.355) follows from (8.361)–(8.363) and GLT 4.For every i, j = 1, . . . ,n+ p− 2,

∣∣(L [p]n (a(x)Ehk)−L [p]

n (am(x)Ehk))ij

∣∣ =

∣∣∣∣∫[0,1]d

(a− am)∂Nj+1,[p]

∂xh

∂Ni+1,[p]

∂xk

∣∣∣∣ ≤ ∫[0,1]d

|a− am|∣∣∣∣∂Nj+1,[p]

∂xh

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xk

∣∣∣∣.150

By (2.29) and (8.290), we obtain

‖L [p]n (a(x)Ehk)−L [p]

n (am(x)Ehk)‖1 ≤n+p−2∑i,j=1

∫[0,1]d

|a− am|∣∣∣∣∂Nj+1,[p]

∂xh

∣∣∣∣ ∣∣∣∣∂Ni+1,[p]

∂xk

∣∣∣∣ ≤ 4phpknhnk

∫[0,1]d

|a− am|.

In view of the equation n = νn, we arrive at

‖nd−2L [p]n (a(x)Ehk)− nd−2L [p]

n (am(x)Ehk)‖1 ≤ C N(n+ p− 2)

∫[0,1]d

|a− am|,

and (8.363) follows from Corollary 4.3.

Step 6. To prove (8.355) for an arbitrary L ∈ L1([0, 1]d,Rd×d), it suffices to observe that, by the linearity of L[p]n (·),

L [p]n (L) = L [p]

n

( d∑s,t=1

Lhk(x)Ehk

)=

d∑s,t=1

L [p]n (Lhk(x)Ehk).

Hence, (8.355) follows from Step 5 and GLT 5.

Step 7. As already noted in Step 2, from (8.355) we get nd−2K[p]G,nn ∼GLT f

(ν)G,p. Since nd−2Z

[p]G,nn ∼σ 0 by Step 1, the

relation nd−2A[p]G,n ∼GLT f

(ν)G,p in (8.348) follows from the decomposition

nd−2A[p]G,n = nd−2K

[p]G,n + nd−2Z

[p]G,n (8.364)

(cf. (8.336)) and from GLT 3, GLT 5; and the singular value distribution in (8.349) follows from GLT 1. If b = 0 identically, nd−2A[p]G,n

is symmetric and also the spectral distribution in (8.349) follows from GLT 1. If b is not identically 0, the spectral distribution in (8.349)follows from GLT 2 applied to the decomposition (8.364), taking into account the symmetry of K [p]

G,n and what we have seen in Step 1.

Formal structure of the symbol f (ν)G,p(x,θ) Problem (8.326) can be formally rewritten as in (8.270). If, for any u : Ω → R,

we define u : [0, 1]d → R as in (8.278), then u satisfies (8.270) if and only if u satisfies the corresponding transformed problem(8.279), in which the higher-order operator takes the form −1(AG(x) Hu(x))1T . It is then clear that the symbol f (ν)

G,p(x,θ) =

N(ν)−1ν(|det(JG(x))|AG(x) Hp(θ))νT preserves the formal structure of the higher-order operator associated with the transformedproblem (8.279), especially when ν = 1, i.e., when we choose the same discretization step in each direction xj , j = 1, . . . , d. However,in this Galerkin context, we notice the appearance of the determinant factor |det(JG(x))|, which is not present in the collocation setting;cf. (8.350) with (8.311).

151

Conclusions and perspectives

In this work, we developed the theory of GLT sequences. We made a significant review of the original theory, by ‘correcting’ all therelevant definitions and by generalizing and/or simplifying a lot of key results. We also extended the theory itself: the main novelties ofthis work are the content of Sections 7.3–7.4, Theorem 7.9 and the new proof of Theorem 7.10. Finally, we provided a precise summaryof the theory of GLT sequences in Section 7.7, with the purpose of giving to the reader an easy-to-use GLT manual. Some hints on howto use this manual in practical applications were given in Chapter 8.

We conclude this work with a list of possible future lines of research.

1. Extend Theorem 8.1 following the suggestions in Section 8.2.2. The goal is to show that the variable-coefficient Toeplitz sequenceAn(a)n is a GLT sequence with symbol a(x, x, θ) for a set of functions a(x, y, θ) larger than W (refer to Section 8.2 for thenotation we are using here).

2. Prove (or disprove) the conjecture formulated at the end of Section 8.3, which we report here for the reader’s convenience.

Conjecture. Suppose that A(i)n n ∼GLT κi for i = 1, . . . , k, where A(1)

n , . . . , A(k)n are Hermitian positive definite, and let

G(A(1)n , . . . , A

(k)n ) be the geometric (Karcher) mean of A(1)

n , . . . , A(k)n . Then G(A

(1)n , . . . , A

(k)n )n ∼GLT (κ1, . . . , κk)1/k. In

particular, G(A(1)n , . . . , A

(k)n )n ∼σ, λ (κ1, . . . , κk)1/k.

3. Develop the theory of block GLT sequences. Multilevel block Toeplitz matrices, defined, e.g., in [100] or [43, Section 1.4.1],naturally arise in the numerical approximation of constant-coefficient systems of PDEs and, surprisingly enough, also in two othercontexts:

(a) in the FE approximation of constant-coefficient PDEs (see [55] or [43, Chapter 3]);(b) in the approximation of constant-coefficient eigenvalue problems by IgA collocation/Galerkin methods based on B-splines

with reduced smoothness [45].

Among other things, it was shown in [45, 55] that block Toeplitz structures allow one to provide a symbol-based explanation of theso-called spectral (acoustical/optical) branches, numerically observed by engineers in [64, 65].

In the case of nonconstant-coefficients, the counterpart of (multilevel) block Toeplitz sequences would be block GLT sequences, inthe same way as GLT sequences are the counterpart of (multilevel) Toeplitz sequences; to understand better the latter point, examinecarefully the procedure followed in Section 8.4.1 (Method 1). Hence, a theory of block GLT sequences, already introduced in [90,Section 3.3], would be the ideal framework to deal with variable-coefficient differential problems approximated by either FEs ofany regularity or IgA based on B-splines with reduced smoothness, as well as to face the linear systems of nonconstant-coefficientPDEs. We note that the first step of such a theory has already been made. Indeed, thanks to the work of Böttcher, Silbermann,Miranda, and Tilli (see [24, 74, 100]), we know that any sequence of multilevel block Toeplitz matrices Tn(f)n, generated bya multivariate matrix-valued function f : [−π, π]d → Cs×s with components fij ∈ L1([−π, π]d), has an asymptotic spectral andsingular value distribution described by f ; we refer the reader to [100] for the meaning of this statement. In particular, in [24,p. 202] and [74] the monolevel block case was disposed of, and Tilli [100] finally proved the result in the multilevel block caseunder the sole assumption that the entries of f are in L1([−π, π]d).

4. Revisit (and extend) the work of [90, Section 3.1.4] concerning reduced GLT sequences. A suitable theory of reduced GLTsequences would allow one to deal with sequences of discretization matrices associated with the FE approximation of PDEs definedon non-rectangular domains Ω. Actually, also the theory of GLT sequences allows one to deal with such sequences, but under theadditional assumption that: (a) the non-rectangular domain Ω is exactly decribed by a regular geometry map G : Ω → Ω definedon a rectangular domain Ω; (b) the basis functions used in the FE approximation are defined as the ‘G-deformations’ of basisfunctions defined over Ω. The assumptions (a) and (b) are satisfied in the IgA approach [28], but not in the general FE setting.

5. Try to design an automatic procedure for computing the symbol of a sequence of PDE discretization matrices, assuming to knowthat it is a GLT sequence. The idea would be to express the symbol as a function of the higher-order differential operator associatedwith the PDE, of the related coefficient, and of the used approximation technique. Some hints in this direction are given in [89,Section 2] and [90, Question 3.1].

152

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Solutions to some of the exercises

Exercise 4.1 The answer is ‘no’, as shown by the following example. Let Ann be a matrix-sequence, with An of size n, and letBn,m = An + cn,mIn, where cn,m is a coefficient depending on both n and m. It is clear that

‖An −Bn,m‖p = |cn,m|n1/p.

If we choose cn,m so that0 ≤ cn,m ≤ ε(m)nα−1/p, lim

m→∞ε(m) = 0, (E.1)

then‖An −Bn,m‖p ≤ ε(m)nα. (E.2)

However, this is not enough to ensure that Bn,mnm is not an a.c.s. for Ann. Indeed, since α > 1/p by assumption, there are manychoices for the coefficient cn,m such that (E.1) (and hence (E.2)) is satisfied but Bn,mnm is not an a.c.s. for Ann. For example,take any cn,m satisfying (E.1) and such that, for every m,

limn→∞

cn,m = cm ≥ c > 0, (E.3)

with c independent of m. A possible choice is cn,m = ε(m)nα−1/p with ε(m) = 1/m. Then Bn,mnm is not an a.c.s. for Ann.Let us prove it formally. An elegant and short proof can be given as follows, on the basis of the topological results presented in Section 4.1.If Rn,m and Nn,m are any two matrices such that

cn,mIn = An −Bn,m = Rn,m +Nn,m,

then, by the minimax principle for singular values (Theorem 2.5),

cn,m = σi(Rn,m +Nn,m)

= maxV subspace of Cn

dimV=i

minx∈V‖x‖=1

‖Rn,mx +Nn,mx‖

≤ maxV subspace of Cn

dimV=i

minx∈V‖x‖=1

‖Rn,mx‖+ ‖Nn,m‖

= σi(Rn,m) + ‖Nn,m‖, (E.4)

for all i = 1, . . . , n. If rank(Rn,m) < n, then at least one singular value σi(Rn,m) is zero, and by (E.3)–(E.4) we have ‖Nn,m‖ ≥cn,m → cm ≥ c > 0. In view of (4.3) and (4.6), this implies that da.c.s.(Ann, Bn,mn) ≥ min(c, 1) > 0 for all m and, consequently,Bn,mnm is not an a.c.s. for Ann.

Exercise 8.1 In view of GLT 5, it suffices to prove that An(a)n ∼GLT a(x, x, θ) for any a of the form a(x, y, θ) = α(x, y)β(θ),with α ∈ C([0, 1]2) and β ∈ L2([−π, π]). Fix then a(x, y, θ) = α(x, y)β(θ) with α ∈ C([0, 1]2) and β ∈ L2([−π, π]). Let

am(x, y, θ) = α(x, y)pm(θ),

where pm is any trigonometric polynomial such that pm → β in L2([−π, π]). We show that:

i. An(am)n ∼GLT am(x, x, θ) for every m ∈ N;


iii. An(am)na.c.s.−→ An(a)n.

158

Once this is done, the relation An(a)n ∼GLT a(x, x, θ) follows from GLT 8. Since am belongs to the space W defined in (8.4)(note that only a finite number of Fourier coefficients of am(x, y, ·) is nonzero), item i follows from Theorem 8.1. Since pm → β inL2([−π, π]), am(x, x, θ) = α(x, x)pm(θ) → α(x, x)β(θ) = a(x, x, θ) in L2([0, 1] × [−π, π]), and so am(x, x, θ) → a(x, x, θ) inmeasure. This proves item ii. To prove item iii, we note that, by definition,

An(a) =

[α( i− 1

n− 1,j − 1

n− 1

)]ni,j=1

Tn(β), An(am) =

[α( i− 1

n− 1,j − 1

n− 1

)]ni,j=1

Tn(pm),

and

An(a)−An(am) = An(a− am) =

[α( i− 1

n− 1,j − 1

n− 1

)]ni,j=1

Tn(β − pm).

The Frobenius norm of An(a)−An(am) can then be estimated by means of Theorem 5.2:

‖An(a)−An(am)‖22 ≤ ‖α‖2∞,[0,1]2‖Tn(β − pm)‖22 ≤ ‖α‖2∞,[0,1]2‖β − pm‖2L2n.

Item iii now follows from Corollary 4.3, taking into account that ‖β − pm‖L2 → 0.

Exercise 8.2 In view of GLT 5, it suffices to prove that An(a)n ∼GLT a(x, x, θ) for any a of the form a(x, y, θ) = α(x)β(y)γ(θ),with bounded measurable α, β : [0, 1] → C and γ ∈ L1([−π, π]). Fix then a(x, y, θ) = α(x)β(y)γ(θ) with bounded measurableα, β : [0, 1]→ C and γ ∈ L1([−π, π]). Let

am(x, y, θ) = α(x)β(y)pm(θ),

where pm is any trigonometric polynomial such that pm → γ in L1([−π, π]). We show that:

i. An(am)n ∼GLT am(x, x, θ) for every m ∈ N;


iii. An(am)na.c.s.−→ An(a)n.

Once this is done, the relation An(a)n ∼GLT a(x, x, θ) follows from GLT 8. Since am belongs to the space W defined in (8.4)(note that only a finite number of Fourier coefficients of am(x, y, ·) is nonzero), item i follows from Theorem 8.1. Since pm → γ inL1([−π, π]), am(x, x, θ) = α(x)β(x)pm(θ) → α(x)β(x)γ(θ) = a(x, x, θ) in L1([0, 1] × [−π, π]), and so am(x, x, θ) → a(x, x, θ) inmeasure. This proves item ii. To prove item iii, let

∆n(g) = diagi=1,...,n

g( i− 1

n− 1

), g : [0, 1]→ C, n ≥ 2.

By definition,An(a) = ∆n(α)Tn(γ)∆n(β), An(am) = ∆n(α)Tn(pm)∆n(β)

andAn(a)−An(am) = An(a− am) = ∆n(α)Tn(γ − pm)∆n(β).

The trace norm of An(a)−An(am) can then be estimated by means of the Hölder-type inequality (2.30) and Theorem 5.2:

‖An(a)−An(am)‖1 ≤ ‖∆n(α)‖ ‖∆n(β)‖ ‖Tn(γ − pm)‖1 ≤ ‖α‖∞‖β‖∞‖γ − pm‖L1n.

Item iii now follows from Corollary 4.3, taking into account that ‖γ − pm‖L1 → 0.

159

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