operator methods in wavelets, tilings, and frames

626

Operator Methods in Wavelets,Tilings, and Frames

AMS Special SessionHarmonic Analysis of Frames, Wavelets, and Tilings

April 13–14, 2013Boulder, Colorado

Veronika FurstKeri A. Kornelson

Eric S. WeberEditors

American Mathematical Society

626






American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash C. Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 41Axx, 42Axx, 42Cxx, 43Axx, 46Cxx,47Axx, 94Axx.

Library of Congress Cataloging-in-Publication Data

Operator methods in wavelets, tilings, and frames / Veronika Furst, Keri A. Kornelson, Eric S.Weber, editors.

pages cm. – (Contemporary mathematics ; volume 626)“AMS Special Session on Harmonic Analysis of Frames, Wavelets, and Tilings, April 13-14,2013, Boulder, Colorado.”Includes bibliographical references.ISBN 978-1-4704-1040-7 (alk. paper)1. Frames (Combinatorial analysis) 2. Wavelets (Mathematics) I. Furst, Veronika, 1979- editor

of compilation. II. Kornelson, Keri A., 1967- editor of compilation. III. Weber, Eric S., 1972-editor of compilation.

QA403.3.O64 2014511′.6–dc23 2014009729

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/626

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10 9 8 7 6 5 4 3 2 1 19 18 17 16 15 14

Contents

Preface vii

Participants ix

Phase retrieval by vectors and projectionsPeter G. Casazza and Lindsey M. Woodland 1

Scalable frames and convex geometryGitta Kutyniok, Kasso A. Okoudjou, and Friedrich Philipp 19

Dilations of frames, operator-valued measures and bounded linear mapsDeguang Han, David R. Larson, Bei Liu, and Rui Liu 33

Images of the continuous wavelet transformMahya Ghandehari and Keith F. Taylor 55

Decompositions of generalized wavelet representationsBradley Currey, Azita Mayeli, and Vignon Oussa 67

Exponential splines of complex orderPeter Massopust 87

Local translations associated to spectral setsDorin Ervin Dutkay and John Haussermann 107

Additive spectra of the 14 Cantor measure

Palle E. T. Jorgensen, Keri A. Kornelson,

and Karen L. Shuman 121

Necessary density conditions for sampling and interpolation in de Brangesspaces

Sa’ud al-Sa’Di and Eric Weber 129

Dynamical sampling in hybrid shift invariant spacesRoza Aceska and Sui Tang 149

Dynamical sampling in infinite dimensions with and without a forcing termJacqueline Davis 167

v

Preface

Frames were first introduced by Duffin and Schaeffer in 1952 in the context ofnonharmonic Fourier series but have enjoyed widespread interest in recent years,particularly as a unifying concept. Indeed, mathematicians with backgrounds asdiverse as classical and modern harmonic analysis, Banach space theory, operatoralgebras, and complex analysis have recently worked in frame theory. The presentvolume contains papers expositing the theme of operator theoretic methods in frametheory in four specific contexts: frame constructions, wavelet theory, tilings, andsampling theory.

There are numerous constructions of frames, as there are numerous situationsin which frame theory plays a central role, and each of these situations requires aframe with different characteristics. The paper by Casazza and Woodland discussesframe constructions, and associated projections, which allow for the reconstructionof an unknown vector using the magnitude of frame coefficients without the phase.The paper by Kutyniok, Okoudjou, and Phillips concerns frames which can bepreconditioned via scalar multiplication to obtain a tight frame. The paper byHan, Larson, Liu and Liu approaches the idea of a frame in a generalized sense, inwhich the frame is given by a set of operators, not a set of vectors.

Although the first wavelet was introduced by Haar in 1909, wavelet analysisofficially took off with the pioneering work of Daubechies, Grossman, and Meyer inthe 1980s. The main attractiveness of a wavelet is its simultaneous localization ofa square-integrable function in both time and frequency. Its “zooming” capabilityis formalized in the definition of a multiresolution analysis. Ghandehari and Taylorgeneralize the classical dilation and translation operators by considering a unitaryrepresentation of a locally compact group G and defining a wavelet to be a vectorin the associated Hilbert space for which a reconstruction formula holds in a weaksense. Their focus is how the images of the corresponding continuous wavelettransform, as subspaces of L2(G), change and are related to one another, as aconsequence of varying the wavelet. The paper by Currey, Mayeli, and Oussa alsogeneralizes the wavelet representation of the subgroup of the ax + b group that isisomorphic to the subgroup of unitary operators generated by the classical dilationand translation. The authors replace the Hilbert space L2(Rn) by L2(N) for asimply connected, connected nilpotent Lie group N . They define a correspondingwavelet representation and analyze its direct integral decomposition, particularlyfor non-commutative N . In the paper by Massopust, exponential splines of complexorder extend the class of exponential B-splines of order n for n ∈ N and polynomialB-splines of complex order. The new class of splines defines multiresolution analysesof L2(R) and corresponding wavelet bases.

vii

viii PREFACE

The Fuglede conjecture from 1974 presents the connection that is often, butnot always, present between sets that tile Rd by translation and the existence ofspectral sets associated with the tiling. The conjecture is resolved for at leastdimension 3 but not dimensions 2 and 1. The paper by Dutkay and Haussermanconsiders tiling sets in dimension 1. The authors present properties of unitarygroups of local translations acting on subsets of the real line and draw connectionsto tilings. The Fuglede conjecture created increased interest in the presence or lackof Fourier bases or Fourier frames with respect to a variety of measures. The paperby Jorgensen, Kornelson, and Shuman presents spectra on a fractal measure spaceand gives structural information about the connections between different spectraon the same space.

Sampling theory concerns the reconstruction of an unknown function from itsknown samples at certain points in its domain. This idea can be traced back toCauchy, where the unknown function was a trigonometric polynomial, but in themodern context, sampling theory can be described in terms of frames. In this form,the main problem is when a certain operator possesses a generalized inverse. Thepaper by al-Sa’di and Weber gives necessary conditions which guarantee that thisoperator does possess a generalized inverse, where the unknown function belongs toa Hilbert space of entire functions. The papers by Aceska and Tang, and by Davisconcern the variation on sampling theory in which some of the known samples ofthe unknown function are obtained after an operator acts upon the function. Inthe paper by Aceska and Tang, the space of functions is a hybrid shift invariantspace, and the operator which acts in between successive sampling operations is aconvolution operator. In the paper by Davis, the function space of square-summablesequences, and the operator acting between sampling operations may involve anonlinear forcing term. In both papers, the essential question is: When does thematrix representation for an operator possess an appropriate submatrix with ageneralized inverse?

This collection of papers covers a wide variety of topics, including: convex ge-ometry, direct integral decompositions, Beurling density, operator-valued measures,splines, and more. These topics arise naturally in the study of frames, which againis the unifying theme in this volume. In nearly all of the papers, ideas and resultsfrom operator theory are the crucial tools in solving the problems in the study offrames. This volume will be of interest to researchers in frame theory, and alsoto those in approximation theory, representation theory, functional analysis, andharmonic analysis.

Veronika FurstKeri Kornelson

Eric Weber

Participants

Speakers and titles from the AMS Special Session “Harmonic Analysis of Frames,Wavelets, and Tilings” from the AMS Western Sectional Meeting, Boulder, CO,April 13–14, 2013.

Marcin BownikExistence of Frames with Prescribed Norms and Frame Operator

Peter G. CasazzaFusion Frames for Wireless Sensor Networks

Jacqueline DavisDynamical Sampling

Dorin DutkayThe Fuglede Conjecture in Dimension One

Matthew FickusCharacterizing Completions of Finite Frames

Deguang HanSpectrally Optimal Frames for Erasures

John HaussermannTiling Properties of Spectra of Measures

John JasperSpectra of Frame Operators with Prescribed Frame Norms

Palle JorgensenTilings in Wavelet Theory: IFS Measures and Wavelet Packets

Chun-Kit LaiSpectral Property of Cantor Measures with Consecutive Digits

David R. LarsonFrames, Dilations and Operator-Valued Measures

Peter MassopustExponential Splines with Complex Order

Azita MayeliBracket Map for the Heisenberg Group and the Characterization of CyclicSubspaces

ix

x PARTICIPANTS

Kathy D. MerrillSimple n-Dimensional Wavelet Sets

Vignon S OussaParseval Frame Wavelets on Some Non-Abelian Nilpotent Matrix Groups

Gabriel PicioroagaOrthonormal Bases Generated by Cuntz Algebras

Benjamin PurkisConstructing Projective Multiresolution Analyses over Irrational RotationAlgebras

Darrin SpeegleLinear Independence of Time-Frequency Translates of Functions with Faster thanExponential Decay

Tim WertzLocalization of Matrix Factorizations

Contemporary MathematicsVolume 626, 2014http://dx.doi.org/10.1090/conm/626/12501

Phase retrieval by vectors and projections

Peter G. Casazza and Lindsey M. Woodland

Abstract. The mathematical study of phase retrieval was started in 2006in a landmark paper of Balan, Casazza and Edidin. This quickly became aheavily studied topic with implications for many areas of research in bothapplied mathematics and engineering. We highlight the major advances inphase retrieval given by vectors developed since 2006 which have had significantmathematical impact. We then discuss recent developments in a new area of

study pertaining to phase retrieval given by projections. We will also give anextensive overview of the papers in both vector and projection phase retrieval.

1. INTRODUCTION

Signal reconstruction has been a longstanding problem in engineering and hasapplications to a wide array of problems. However, when a signal is received,usually there is a loss of information making the reconstruction of the desired signala challenging task.

Traditionally, and as described in [8], signal reconstruction consisted of threesteps: first, the input signal is linearly transformed from its input domain (e.g.,time, or space) into a transformed domain (e.g., time-frequency, time-scale, space-scale etc.); second, a (nonlinear) estimation operator is applied in this representa-tion domain; third, a (left) inverse of the linear transformation at step one is appliedto the signal obtained at step two in order to synthesize the estimated signal in theinput domain.

Some linear transformations which have been used for signal reconstruction arethe windowed Fourier transform, wavelet filterbanks, and local cosine basis as seenin [8,50,52,59]. Likewise, many signal estimators have been proposed and studiedin the literature, some of them statistically motivated; e.g., Wiener (MMSE) filter,Maximum A Posteriori (MAP), Maximum Likelihood (ML) etc., others having arather ad-hoc motivation; e.g., spectral subtraction, psychoacoustically motivatedaudio and video estimators etc., as described in [8].

For years, knowledge of the phase of a signal, or an estimation thereof, wasseen to be a necessary component when reconstructing a signal. However, oftentimes, as a signal is passed through a linear system, the phase of the signal is lostand only the absolute values of linear measurement coefficients, called intensity

2010 Mathematics Subject Classification. Primary 46G10, 46L10, 46L51, 47A20; Secondary42C15, 46B15, 46B25, 47B48.

The authors are supported by NSF DMS 1307685; NSF ATD 1042701; AFOSR DGE51;FA9550-11-1-0245; and NSF ATD 1321779.

c©2014 American Mathematical Society

1

2 PETER G. CASAZZA AND LINDSEY M. WOODLAND

measurements, are known. Many researchers believed that signal reconstructionshould be possible without the use of phase. And many believed that with enoughinformation and redundancy then the use of phase should no longer be necessary.The process of reconstructing a signal from intensity measurements without the useof phase is known as phase retrieval and is currently a popular topic of research.

Phase retrieval is a challenging problem and has a wide range of applicationsin numerous fields. As such it has been studied by Engineers, Mathematiciansand Physicists alike. Important applications of phase retrieval occur in optics withapplications to X-ray crystallography, electron microscopy and coherence theory[7, 32, 33, 38, 39, 43, 47–49, 57, 61]. There are also applications in the areas ofdiffractive imaging [12,13,21], astronomical imagining [24,33], X-ray tomography[26], and speech recognition technology [9,31,52–54,59], just to name a few. Forother references regarding advances in phase retrieval consider [28,30,35,37,41,42,45,46].

Moreover, this problem of phase retrieval is very similar to a problem in quan-tum theory known as state tomography. As described in [10], a pure quantum stateis given by a rank-one projection on a finite-dimensional Hilbert space, or equiv-alently, by the vectors in the range of this projection. A state is experimentallyaccessible only through the magnitudes of its Hilbert–Schmidt inner products withother states. These inner products of projections can be interpreted as the squaredmagnitudes of the inner products of corresponding normalized vectors in the re-spective range of the projections. Thus, reconstructing a pure quantum state isthe same as finding a vector, up to a unimodular constant, from the magnitudes oflinear transform coefficients. For more information on quantum state tomographysee [10,23,34,36,51,55,56,58].

It is clear that phase retrieval is an important topic of research in numerousfields and as such there have been and will continue to be many influential papers inthis area. In what follows, we highlight some major mathematical contributions inthe area of phase retrieval. Section 2 discusses the advances in phase retrieval whenthe intensity measurements are given by the magnitudes of the inner products ofthe signal with a collection of vectors. The main papers we highlight in this sectionare [8] by Balan, Casazza and Edidin; [6] by Bandeira, Cahill, Mixon and Nelson;[4,5,10] by Balan, Bodmann, Casazza and Edidin; [17] by Candes, Strohmer andVoroninski; and [15] by Candes, Eldar, Strohmer and Voroninski. We also discusscontributions from a few other notable papers. Recently there have been newdevelopments in phase retrieval and instead of reconstructing a signal using vectors,researchers are looking at reconstructing a signal via the magnitudes of projections.We follow up in section 3 by highlighting the contributions in phase retrieval whenreconstructing a signal via projections. Here we highlight [2] by Bachoc and Ehler;and [14] by Cahill, Casazza, Peterson and Woodland. We conclude this section bylisting some of the open problems discussed in [14].

2. PHASE RETRIEVAL BY VECTORS

Signal reconstruction has been a long standing area of research in many dif-ferent scientific fields and for years people have developed numerous methods forreconstructing a signal; however, many of them included phase. Researchers be-lieved that with enough redundancy, one should be able to reconstruct a signalwithout using phase. In 2006, Balan, Casazza and Edidin in [8] gave a unique

PHASE RETRIEVAL BY VECTORS AND PROJECTIONS 3

mathematical approach for phase retrieval, which quickly turned into an industry.Through the use of redundant systems, known as frames, they showed for the firsttime that a signal can be reconstructed (up to a unimodular constant) withoutusing noisy phase.

Before we discuss their phase retrieval results, a few definitions from frametheory are necessary. For a more complete study of Frame Theory, the reader isencouraged to look at [18,19,22].

Definition 2.1. Given a Hilbert space H with scalar product <,> and a count-able or finite set I. A set of vectors Φ = {ϕi}i∈I is called a frame if there exist twopositive constants A,B such that for all x ∈ H,

A||x||2 ≤∑i∈I

|〈x, ϕi〉|2 ≤ B||x||2.

Where the values (〈x, ϕi〉)Mi=1 are called the frame coefficients of the vector x withrespect to the frame Φ.

Remark 2.2. When I is a finite set, then Φ = {ϕi}i∈I is just a spanning setfor H.

Definition 2.3. A frame Φ is tight or A-tight if A = B.

Phase retrieval is the problem of recovering a signal from the absolute values oflinear measurement coefficients (frame coefficients) called intensity measurements.Note that multiplying a signal by a unimodular constant does not affect thesecoefficients, so we seek signal recovery modulo a unimodular constant. In [8], theydetermine what kind of reconstruction is possible if we only have knowledge of theabsolute values of the frame coefficients.

Definition 2.4. A set of vectors Φ = {ϕi}Mi=1 in RN (or CN ) gives phaseretrieval if for all x, y ∈ RN (or CN) satisfying |〈x, ϕi〉| = |〈y, ϕi〉| for all i =1, . . . ,M , then x = cy where c = ±1 in RN (and c ∈ T1 in CN where T1 is thecomplex unit circle).

In [8], they prove that a remarkably small number of vectors are needed forphase retrieval and actually show that this small set of vectors is a generic frame.

Definition 2.5. A generic frame Φ = {ϕi}Mi=1 is an open, dense subset of theset of all M -element frames in HN .

In [6] they define a generic frame as such: Recall that a real algebraic variety isthe set of common zeros of a finite set of polynomials with real coefficients. Takingall real algebraic varieties in RN to be closed sets defines the Zariski topology onRN . Viewing the frame Φ as a member of R2MN , then we say a generic Φ isany member of some undisclosed, nonempty, Zariski-open subset of R2MN . SinceZariski-open sets are either empty or dense with full measure, then genericity is astrong property.

Theorem 2.6. [8] Let Φ = {ϕi}Mi=1 be a frame for HN .

• For HN = RN , a generic frame of at least 2N − 1 vectors does phaseretrieval.

• For HN = CN , a generic frame of at least 4N − 2 vectors does phaseretrieval.


Theorem 2.7. [8] There does not exist an M -element frame in RN with M ≤2N − 2 which gives phase retrieval.

As seen in Theorem 2.6, [8] gives upper bounds on the number of vectors neededto give phase retrieval in RN and in CN . However, they also completely classifythe real case by showing that 2N − 1 vectors is the necessary and sufficient boundfor phase retrieval in RN .

Another great contribution of [8] is that they give a precise and easily under-standable classification for when a real frame gives phase retrieval. But before wegive this classification, we need some definitions.

Definition 2.8. Given a frame Φ = {ϕi}Mi=1 in HN , the spark of Φ is definedas the cardinality of the smallest linearly dependent subset of Φ. In particular, whenspark(Φ) = N + 1, every subset of size N is linearly independent and Φ is said tobe full spark.

Definition 2.9. A frame Φ = {ϕi}Mi=1 in HN is said to have the comple-ment property if for all subsets S ⊆ {1, . . . ,M}, either span({ϕi}i∈S) = HN orspan({ϕi}i∈Sc) = HN .

Remark 2.10. A full spark frame Φ = {ϕi}Mi=1 in HN with M = 2N − 1 hasthe complement property.

A nice classification for when a real frame gives phase retrieval is as follows:

Theorem 2.11. [8] A frame Φ = {ϕi}Mi=1 in RN gives phase retrieval if andonly if it has the complement property. In particular, a full spark frame with 2N−1vectors does phase retrieval. Moreover, no set of 2N−2 or less vectors can do phaseretrieval.

This was the first classification for when signals in RN can be reconstructedwithout using phase; however, the proof of Theorem 2.11 uses results from algebraicgeometry and therefore was not implementable.

In light of this nice classification of real frames which give phase retrieval inTheorem 2.11, researchers were hoping for a similar result for complex frames.For the complex case there has yet to be a complete classification for when a frameadmits phase retrieval. However, there have been improvements to the upper boundof 4N − 2 used in Theorem 2.6. Earlier in 2013, B. Bodmann and N. Hammenconstructed a 4N −4 element complex frame which gives phase retrieval, as seen in[11]. Recently, Conca, Edidin, Hering and Vinzant improved the bound of 4N − 2and gave the following sufficient condition for the new bound of 4N − 4.

Theorem 2.12. [29] If M ≥ 4N − 4, then the generic complex M -elementframe in CN admits phaseless reconstruction.

For necessity, Heinosaari, Mazzarella and Wolf in [40] gave the following lowerbound for the number of vectors necessary for phase retrieval in the complex case.

Theorem 2.13. If a frame {ϕi}Mi=1 in CN gives phase retrieval, then M ≥4N − c logN , for some c ∈ C.

An interesting problem that is now a popular topic of research is to find theexact necessary and sufficient bound for the number of vectors giving phase retrievalin the complex case. Bandeira, Cahill, Mixon, and Nelson in [6] made stridestowards this problem.


In [6], they proved the first characterization for when a complex frame givesphase retrieval by showing when a non-linear map is injective.

Theorem 2.14. [6] Consider Φ = {ϕi}Mi=1 in CN and the mapping A : CN/T1 →RM defined by (A(x))(i) := |〈x, ϕi〉|2 for i = {1, · · · ,M}. Viewing {ϕiϕ

∗i u}Mi=1 as

vectors in R2N , denote S(u) :=spanR{ϕiϕ∗i u}Mi=1. Then the following are equiva-

lent:

(1) A is injective.(2) dimS(u) ≥ 2N − 1 for every u ∈ CN − {0}.(3) S(u) =spanR{iu}⊥ for every u ∈ CN − {0}.

Note that when A is injective, then the frame Φ gives phase retrieval. In thereal case, the complement property is equivalent to having span{ϕiϕ

∗i u}Mi=1 = RN

for all u ∈ RN−{0}, thus there is a connection between Theorem 2.11 and Theorem2.14. However, as nice as this theorem is, it still does not give us a concrete way ofchecking if a complex frame gives phase retrieval.

Recall in Theorem 2.11 that the complement property is necessary and sufficientfor injectivity in the real case. In the complex case, [6] proves the necessity of thecomplement property for injectivity; however, the complement property is not astrong enough property to be a sufficient condition. The problem for finding animplementable necessary and sufficient condition for injectivity in the complex caseis still open.

Theorem 2.15. [6] Consider Φ = {ϕi}Mi=1 in CN and the mapping A : CN/T1 →RM defined by (A(x))(i) := |〈x, ϕi〉|2. If A is injective then Φ satisfies the comple-ment property.

A useful lemma to determine the injectivity of A developed in [6] and relatedto a result in [40] is as follows:

Lemma 2.16. [6] A is not injective if and only if there exists a matrix of rank 1or 2 in the null space of A, where A is the super analysis operator A : HN×N → RM

given by (AH)(i) = 〈H,ϕiϕ∗i 〉HS and 〈 , 〉HS is the Hilbert–Schmidt inner product.

Another contribution of [6] is their attempt to show that 4N − 4 vectors inCN is the necessary and sufficient bound to give phase retrieval. In this paperthey discuss previous works and explanations as to why they believe this to be thecorrect bound. The following conjecture is presented in [6].

Conjecture 2.17. [6] Consider Φ = {ϕi}Mi=1 in CN and the mapping A :CN/T1 → RM defined by (A(x))(i) := |〈x, ϕi〉|2 for i = {1, · · · ,M}. If N ≥ 2,then the following statements hold:

(1) If M < 4N − 4 then A is not injective.(2) If M ≥ 4N − 4 then A is injective for a generic Φ.

In [6] they prove their conjecture for the cases when N = 2 and when N = 3.Note, as mentioned previously, in [29] they prove part (2) of Conjecture 2.17.

In addition to the authors of [6] and [8], there have been many people whohave found or attempted to give a complete classification for when a signal can bereconstructed in a real or complex Hilbert space, without using phase. However, theresults are usually theoretic and are not able to be easily used in applications. Morespecifically, the proofs in [8] use techniques of Algebraic Geometry that cannot


be implemented. In lieu of this, Bodmann, Casazza, Edidin and Balan, in [10],developed an algorithm for phase retrieval using discrete chirps. However, this

method requires N(N+1)2 vectors in RN and N2 vectors in CN , which is larger than

the bounds of 2N − 1 in RN and 4N − 4 in CN found in [8] and [29].In [10] they develop implementable ways of reconstructing a signal using the

absolute value of the frame coefficients of the signal under a linear map. To dothis, they show an equivalence between the problem of reconstructing a vector xin HN to the problem of reconstructing the rank one Hermitian operator xx∗ inHN×N . This transforms the highly non-linear problem of reconstructing x up to aunimodular constant from |〈x, ϕi〉|2, into a linear one where they reconstruct xx∗

from its expansion with respect to an operator-valued frame S = {Si}Mi=1.

Definition 2.18. [10] Let Φ = {ϕi}Mi=1 be a frame for a real or complex Hilbertspace HN . Let Si = ϕiϕ

∗i denote the rank-one Hermitian operator associated with

each ϕi. Let X be the span of the family S = {Si}Mi=1, equipped with the Hilbert-Schmidt inner product. We say that {Si}Mi=1 is the operator-valued frame for Xassociated with {ϕi}Mi=1. Also, the Grammian H of S has entries Hj,k = tr[SjSk] =|〈ϕj , ϕk〉|2.

As noted in [10], if the operator-valued frame {Si}Mi=1 associated with Φ hasmaximal span, then we can reconstruct any operator xx∗ from its Hilbert–Schmidtinner products with the family {Si}Mi=1. Since the values of these inner productsare tr[xx∗Si] = |〈x, ϕi〉|2 and xx∗ determines x up to a unimodular constant, thisamounts to reconstructing x from the magnitudes of its frame coefficients withrespect to the frame Φ. Hence, the following reconstruction algorithm uses theoperator-valued frame S to reconstruct xx∗.

Theorem 2.19. [10] Let HN be an N-dimensional real or complex Hilbert spaceand Φ = {ϕi}Mi=1 an M

N -tight frame such that the associated operator-valued frame Shas maximal span. Let Q be the pseudo-inverse of the Grammian H, so HQH = H,

and denote the canonical dual of S as R, containing operators Rj =∑M

k=1 Qj,kSk.Given a vector x ∈ HN , then

xx∗ =

M∑j=1

|〈x, ϕj〉|2Rj .

In [10] they also specialize the general result of Theorem 2.19 to equiangularframes and mutually unbiased bases, which is presented in the next corollary.

Definition 2.20. A frame Φ = {ϕi}Mi=1 in HN is equiangular if for all 1 ≤i = j ≤ n, |〈ϕi, ϕj〉| = c for some nonzero constant c.

Mutually unbiased bases form a frame which is composed of a collection oforthonormal bases for a Hilbert space in such a way that for any two vectors fromdifferent orthonormal bases, the magnitude of their inner products are constant.This type of frame has an associated operator-valued frame with maximal span.

Definition 2.21. [10] Let HN be a real or complex Hilbert space of dimension

N and let {ejk}Nk=1 be an orthonormal basis for HN for each j = {1, . . . , P}. A

family of vectors {ejk} in HN indexed by k ∈ K = {1, 2, . . . , N} and j ∈ J ={1, 2, . . . , P} is said to form P mutually unbiased bases if for all j, j′ ∈ J and for

all k, k′ ∈ K the magnitude of the inner product between ejk and ej′

k′ is given by


|〈ejk, ej′

k′〉| = δk,k′δj,j′ +1√N(1 − δj,j′), where Kroecker’s δ symbol is one when its

indices are equal and zero otherwise.

Definition 2.22. [10] In CN , if N is prime then there exists a maximal set ofmutually unbiased bases called discrete chirps.

Corollary 2.23. [10] Let HN be a complex Hilbert space. If Φ = {ϕi}Mi=1

is an equal-norm, equiangular MN -tight frame or a tight frame formed by mutually

unbiased bases, and the associated operator-valued frame S has maximal span, thenthe reconstruction identity becomes

xx∗ =N(N + 1)

M

M∑i=1

|〈x, ϕj〉|2(ϕjϕ

∗j − I

N + 1

).

Corollary 2.23 gives an algorithm for reconstructing x, up to a unimodularconstant, by considering one non-vanishing row of the N ×N matrix xx∗ [10].

Much like the results found in [10], the same authors published other resultswhich give algorithms for reconstruction and they are found in [4]. In [4] theygive a nice reconstruction formula for a signal in a finite dimensional Hilbert space,equipped with a frame which has a maximal number of mutually unbiased bases.

Definition 2.24. [4] The self-adjoint rank-one operators Qx associated to x ∈H are given by Qx(y) = 〈y, x〉x, for y ∈ H.

Definition 2.25. [4] Let HN be a complex Hilbert space of dimension N . Let

J = {1, 2, · · · , N + 1} and K = {1, 2, · · · , N}. If the family of vectors {ejk : j ∈J, k ∈ K} forms N + 1 mutually unbiased bases in HN , and ω is a primitive N-throot of unity, then we denote

Bjk =

1√N

N∑l=1

ωklP jl ,

where for each k ∈ K and j ∈ J, P jk is the rank-one orthogonal projection onto the

span of ejk.

Theorem 2.26. [4] Given a family of vectors {ejk : j ∈ J, k ∈ K} that formN + 1 mutually unbiased bases in CN , a primitive N-th root of unity ω and theassociated operators {Bj

k}, then for all x ∈ CN ,

Qx =||x||2N

I +1√N

N+1∑j=1

N−1∑k=1

N∑l=1

ω−kl|〈x, ejl 〉|2Bjk.

Hence, Theorem 2.26 gives a nice reconstruction formula for the rank one op-erator Qx, from which we can factorize and easily obtain x up to a unimodularconstant.

Another contribution from the same authors is found in [5] where they showthat a generic frame gives phase retrieval in a polynomial number of steps.

Theorem 2.27. [5]

(1) If HN is a real N-dimensional Hilbert space, M ≥ N(N+1)2 and Φ =

{ϕi}Mi=1 is a generic frame then a vector x ∈ HN can be reconstructed(up to sign) from the set {|〈x, ϕi〉|}Mi=1 in a polynomial number O(N6) ofsteps.


(2) If HN is a complex N-dimensional Hilbert space and M ≥ N2 and Φ ={ϕi}Mi=1 is a generic frame then a vector x ∈ HN can be reconstructed(up to multiplication by a root of unity) from the set {|〈x, ϕi〉|}Mi=1 in apolynomial number O(N6) of steps.

Following [4,5,10], Candes, Strohmer, and Voroninski in [17] also viewed in-tensity measurements as Hilbert-Schmidt inner products of self-adjoint Hermitianmatrices given by xx∗ in order to reconstruct a vector x when only the magnitudes ofthe frame coefficients were known. In [17], the authors used semidefinite program-ming and developed a new process for phase retrieval which requires O(N logN)frame vectors to be independently and uniformly chosen on the unit sphere. Theyprove that it suffices to solve a convex program of trace minimization in order tosolve the more complicated nonconvex problem of rank minimization, which is seento be equivalent to phase retrieval.

As seen in [15, 17, 20, 44], phase retrieval is equivalent to the nonconvexproblem of finding X, subject to A(X) = b,X � 0, and rank(X) = 1, whereA : HN×N → RM is given by X → {ϕ∗

iXϕi}Mi=1 and X � 0 means that X ispositive semidefinite. Once we solve for X = xx∗ we can then reconstruct x up toa unimodular constant, as seen above and in [10]. One way to do this would be tosolve the combinatorial problem of rank minimization by minimizing the rank(X)subject to A(X) = b and X � 0. Since this is NP-hard, [17] suggests solvingthe convex program of trace-minimization which is to minimize Tr(X) subject toA(X) = b and X � 0. If the solution to this trace minimization has rank one, thenwe can factorize it and recover x up to a unimodular constant.

In [17], they coin the term PhaseLift and define it as such: lift the problem ofrecovering a vector from quadratic constraints into that of recovering a rank-onematrix from affine constraints, and relax the combinatorial problem into a conve-nient convex program. Since solving the nonconvex rank minimization problem isequivalent to phase retrieval, then in order to instead use the easier convex problemof trace minimization, we need to determine when the rank minimization problemand the trace minimization problem have the same unique solution.

Theorem 2.28. [17] Consider an arbitrary signal x ∈ RN or CN and supposewe choose unit norm measurement vectors {ϕi}Mi=1 independently and uniformly inRN or CN . If M ≥ c0N logN , where c0 is a sufficiently large constant, then in boththe real and complex cases, PhaseLift recovers x (up to a unimodular constant) from

{|〈x, ϕi〉|2}Mi=1 with probability at least 1 − 3e−γ MN , where γ is a positive absolute

constant. Or stated another way, the trace minimization program: minimize Tr(X)subject to A(X) = b and X � 0, has a unique solution obeying X = xx∗ with

probability at least 1− 3e−γ MN .

In 2012, Candes and Li were able to improve this bound in [16].

Corollary 2.29. The condition that M ≥ c0N logN in Theorem 2.28 can bereplaced by M ≥ c0N as shown in [16].

In [25] and [60] they also develop similar results and refinements of Theorem2.28.

While [8, 10] made great contributions in the area of phase retrieval, theirframe theoretic approaches require very specific types of measurements which limitthe applicability of their solutions. Also, the approach developed in [17] requires


that the measurement vectors be sampled independently and uniformly at randomon the unit sphere in order for PhaseLift to recover a signal modulo a unimodularconstant. For more general results, at about the same time as [17] came out,Candes, Strohmer and Voroninski along with Eldar published another paper, [15],that utilized Phaselift. In [15], they developed a new approach to phase retrievalwhich required a smaller number of measurements for phase retrieval than that of[10] and placed no constraints on the signal to be reconstructed.

Recall that phase retrieval is the process of recovering a signal from the mag-nitude squared of its Fourier transform. But, when only using these magnitudes,we lose information about the phase of the signal. In [15], they suggest using acollection of masks to alter the diffraction pattern of the signal, where a diffrac-tion pattern is the magnitude squared of the signal’s Fourier transform. They thencombine the information gained from each diffraction pattern and are able to re-construct the signal. In order to do this, after the diffraction patterns are collected,they are then analyzed via a convex program of trace-norm minimization to recoverthe signal, much like that of [17]. Throughout this process, there are no constraintson the desired signal, and so their method works for more general scenarios of phaseretrieval.

One main result of [15], which demonstrates the above idea is as follows. In[15], they construct three masks and show that the diffraction patterns from thesemasks will reconstruct almost any signal up to a unimodular constant. Suppose x isa one dimensional signal, let FN be the N ×N unitary Discrete Fourier Transform(DFT), and let D be the modulation D =diag({ei2πt/N}0≤t≤N−1). Consider thefollowing collection of the 3N real valued measurements A(x) = {|FNx|2, |FN (x+Dsx)|2, |FN (x − iDsx)|2}, where s is a nonnegative integer. Note that these mea-surements can be obtained via the three masks 1, 1+ei2πst/N and 1+ei2π(st/N−1/4).

Theorem 2.30. [15] Suppose the DFT of x ∈ CN does not vanish. Then xcan be recovered up to a unimodular constant from the 3N real numbers A(x) if andonly if s and N are coprime. Conversely, if the DFT vanishes at two frequencypoints k and k′ such that k − k′ = s mod N , then recovery is not possible from the3N real numbers A(x).

They also prove a similar result to Theorem 2.30 for when the signal x ishigher dimensional. Notice that Theorem 2.30 requires only a small number ofintensity measurements in order to reconstruct the signal, which is advantageousin application.

In [15] they also give results which deal with noise interference, and give nu-merical simulations using one and two dimensional signals with and without noiseto illustrate the effctiveness of PhaseLift. Within these numerical simulations itis also seen that PhaseLift may be too slow to be affective in certain large scalesituations. One solution to this is an idea of “polarization” as developed in [1] byAlexeev, Bandeira, Fickus and Mixon. The phase retrieval method created in [1]does reconstruct a signal much more quickly than that of PhaseLift; however, thereis a trade-off, the polarization technique puts constraints on the measurementswhere as PhaseLift, as described in [15,17], is much more flexible.

Phase retrieval is a widely studied field and one that has greatly fluorishedin the past few years. Thus far we have discussed some important mathematicalpapers that have had an impact in the area of phase retrieval by vectors, where a


signal is reconstructed from the magnitudes of its frame coefficients, with respectto some frame.

As of now, we would like to point out one technicality of phase retrieval, thatbeing the difference between phase retrieval and phaseless reconstruction. Sincethe beginning of the mathematical study of this subject, phase retrieval and phase-less reconstruction have been used interchangably; but as it turns out they aretechnically different. To see this we give the following example.

Example 2.31. In R3, let {ei}3i=1 be the standard unit basis and define {ϕi}2i=1 ⊂R3 so that Φ = {e1, e2, e3, ϕ1, ϕ2} is a full spark collection of vectors.

By Theorem 2.11, since Φ is a collection of 5 full spark vectors in R3 then itgives phaseless reconstruction.

Next, set Ψ = {e1, e2, ϕ1, ϕ2}. Since 4 < 5 = 2N − 2 for N = 3 in R3, then Ψcannot do phaseless reconstruction. However, given that ||x|| = 1 we can retrievethe phase using Ψ, which we now show.

Note |〈x, e3〉|2 = 1 − |〈x, e2〉|2 − |〈x, e1〉|2. Hence we know |〈x, e1〉|2, |〈x, e2〉|2and |〈x, e3〉|2 and so Ψ gives phase retrieval.

Therefore, Ψ cannot do phaseless reconstruction because it cannot identify thenorm of the vector from its intensity measurements; but, Ψ gives phase retrieval.

Determining collections of vectors which allow phase retrieval and/or phase-less recontruction is a very popular topic of research, as seen above. However,in some instances a signal must be reconstructed from higher dimensional spaces.One question which a few people are now studying is, what if we are not given aframe and are instead given a collection of subspaces and are asked what kind ofreconstruction is possible when using the subspace components?

3. PHASE RETRIEVAL BY PROJECTIONS

Given a signal x in a Hilbert space, intensity measurements may also be thoughtof as norms of x under rank one projections. Here the spans of measurementvectors serve as the one dimensional range of the projections. In some applicationshowever, a signal must be reconstructed from the norms of higher dimensionalcomponents. For example, in X-ray crystallography such a problem arises withcrystal twinning [27]. In this scenario, there exists a similar phase retrieval problem:Given subspaces {Wi}Mi=1 of an N -dimensional Hilbert space HN and orthogonalprojections Pi : HN → Wi, can we recover any x ∈ HN (up to a unimodularconstant) from the measurements {||Pix||}Mi=1?

This problem was recently studied by Bachoc and Ehler in [2]. In [2], theyaddressed the following questions: Under what conditions are we able to reconstructa signal from the norms squared of its k-dimensional subspace components? Ifsome of the norms of these subspace components are erased, then what type ofreconstruction is possible? Bachoc and Ehler make the following discovery in answerto these questions. They proved that given a collection of equidimensional subspaces{Wi}Mi=1 in RN satisfying certain conditions, then for any signal x, any subset of{||Pi(x)||}Mi=1 which has cardinality at least M − p will reconstruct a finite list ofpossible candidate signals, one of which is the correct signal.

Before stating their results explicitly, we will first give some necessary back-ground.


Definition 3.1. Let {Wi}Mi=1 be a collection of subspaces in RN and definePi to be the orthogonal projection on Wi for each i. Let {ωi}Mi=1 be a collection ofpositive weights. Then {(Wi, ωi)}Mi=1 is called a fusion frame if there are positiveconstants A and B such that

A||x||2 ≤M∑i=1

ωi||Pi(x)||2 ≤ B||x||2,

for all x ∈ RN .

Remark 3.2. If A = B, any signal x ∈ RN can be reconstructed from itssubspace components via the formula

x =1

A

M∑i=1

ωiPi(x).

A natural question to then ask, given Remark 3.2, is when are we able toreconstruct a signal x from only having knowledge of {||Pi(x)||}Mi=1, given that somuch information is now lost? This is the problem of phase retrieval by projections.

Definition 3.3. A collection of orthogonal projections {Pi}Mi=1 onto subspaces{Wi}Mi=1 gives phase retrieval in RN (or CN) if and only if for all x, y ∈ RN (orCN) satisfying ||Pix||2 = ||Piy||2 for all i ∈ {1, · · · ,M} then x = ±y in RN (orx = cy in CN where c ∈ T1 the complex unit circle).

The authors of [2] found conditions for the subspaces which allow phase retrievalin RN and their methods require the use of a tight p-fusion frame which is also acubature of strength 4 for Gk,N , which we will now define.

Definition 3.4. Define Gk,N to be the real Grassmann space which consistsof all k-dimensional subspaces of RN .

Definition 3.5. Let {Wi}Mi=1 ⊂ Gk,N and let {ωi}Mi=1 be a collection of positiveweights and p a positive integer. Then {(Wi, ωi)}Mi=1 is called a p-fusion frame ifthere exist positive constants A and B such that

A||x||2p ≤M∑i=1

ωi||PWi(x)||2p ≤ B||x||2p,

for all x ∈ RN . If A = B, then {(Wi, ωi)}Mi=1 is called a tight p-fusion frame.

Remark 3.6. If {(Wi, ωi)}Mi=1 is a tight p-fusion frame, then it is also a tightl-fusion frame for all integers 1 ≤ l ≤ p, with tight l-fusion frame bound Al =(k/2)l(N/2)l

∑Mi=1 ωi.

Definition 3.7. Define σk to be the probability measure on Gk,N induced bythe Haar measure on the real orthogonal group O(RN ).

Definition 3.8. [2] Let {Wi}Mi=1 ⊂ Gk,N and {ωi}Mi=1 be a collection of positive

weights such that∑M

i=1 ωi = 1. Then {(Wi, ωi)}Mi=1 is called a cubature of strength2p for Gk,N if∫

Gk,N

f(V )dσk(V ) =

M∑i=1

ωif(Wi), for all f ∈ Pol≤2p(Gk,N ).


In [2], they developed a two step algorithm which reconstructs ±x from a collec-tion of M − p elements of {||Pi(x)||2}Mi=1. Without loss of generality, they assumedthat the first p norms have been erased, so they recover ±x from the knowledge of{||Pi(x)||2}Mi=p+1. In the first step, they compute the erased values ti := ||Pi(x)||2,1 ≤ i ≤ p. In the second step, they reconstruct ±x from {||Pi(x)||2}Mi=1 by com-puting Px from each of the possible candidates for {||Pi(x)||2}Mi=1.

Algorithm 1:[2]

(1) Define ti := {||Pi(x)||2}Mi=1.(2) Compute the set S of solutions to the algebraic system of equations in the

unknowns T1, . . . , Tp given by

p∑i=1

ωiTli =

(k/2)l(N/2)l

−n∑

i=p+1

ωitli, for 1 ≤ l ≤ p.

(3) Define α = 2k(N−k)N(N+2)(N−1) and β = k(kN+k−2)

N(N+2)(N−1) . Then for every (t1, . . . , tp) ∈S, compute

P =1

α

M∑i=1

ωitiPi −β

αId.

(4) Then if P is a projection of rank 1, compute a unit vector ψ spanning itsimage and add ±ψ to the list of possible candidates L.

The following theorem illustrates that through the use of Algorithm 1, a col-lection of k-dimensional subspaces in RN under certain contraints, can reconstructany signal x ∈ SN−1.

Theorem 3.9. [2] Let {Wi}Mi=1 ⊂ Gk,N and let {ωi}Mi=1 be a collection ofpositive weights and p a positive integer. Let {(Wi, ωi)}Mi=1 be a tight p-fusion framethat is also a cubature of strength 4 for Gk,N . For x ∈ SN−1, Algorithm 1 outputsa list, L, of at most 2p! elements in SN−1 containing x.

In [2], they also develop a result similar to Theorem 2.28 from [17]. Thismethod of reconstruction replaces the algebraic reconstruction formula of step 2 inAlgorithm 1 with a semidefinite program similar to that in [17].

Definition 3.10. [2] Let HN×N denote the collection of symmetric matricesin RN×N . For {Wi}Mi=1 ⊂ Gk,N , define the operator

FM : HN×N → RM , X → N

k(〈X,Pi〉)Mi=1.

The goal is to reconstruct any x ∈ SN−1 from f := Nk (||Pi(x)||2)Mi=1 = FM (Px) ∈

RM .Define the following rank minimization problem, which is similar to [17]: min-

imize the rank(X) for X ∈ HN×N subject to FM (X) = f and X � 0. Where wedefine X � 0 to mean that X is positive semidefinite. Note that Px is a solution tothis rank minimization problem if the k-dimensional subspaces {Wi}Mi=1 span RN .

Similar to our discussion for [17], since rank minimization is NP-hard, then itis sometimes replaced with the convex program of trace minimization, which is as


follows: minimize the Tr(X) for X ∈ HN×N subject to FM (X) = f and X � 0.Note that for k = 1, this is equivalent to the work in [17].

For k > 1, they prove the following:

Theorem 3.11. [2] There are constants c1, c2 > 0 such that, if x ∈ RN ,M ≥ c1N and {Wi}Mi=1 ⊂ Gk,N are chosen independently, identically distributed

according to σk, then with probability 1 − e−c2M/N the matrix xx∗ is the uniquesolution to the trace minimization problem: minimize the Tr(X) for X ∈ HN×N

subject to FM (X) = f and X � 0.

Recall that in [16] and as Corollary 2.29 states, in the one dimensional caseCandes and Li required the cardinality of the random intensity measurements toscale linearly with the dimension of the signal space. Similarly, we see from Theorem3.11 that signal reconstruction from the norms of equidimensional random subspacecomponents is possible with the cardinality of measurements scaling linearly withthe dimension.

The authors of [2] made great contributions in the area of phase retrieval byprojections; however, one draw back from their work is that the subspaces used forreconstruction were required to be equidimensional real subspaces.

Recently, another paper concerning phase retrieval by projections came aboutwhich addressed real and complex subspaces and did not require the subspaces tobe equidimensional. Much like [8], in [14] Cahill, Casazza, Peterson and Woodlandsought to better characterize the subspaces {Wi}Mi=1 of HN for which the measure-ments {||Pix||}Mi=1 were injective for all x ∈ HN ; hence yielding reconstruction of xup to a unimodular constant.

In [14], they developed a surprisingly small upper bound for the number ofsubspaces of arbitrary rank which are needed to give phase retrieval in both thereal and complex cases.

Theorem 3.12. [14] Phase retrieval in RN is possible using 2N − 1 subspaceseach of any dimension less than N .

Theorem 3.13. [14] Phase retrieval in CN is possible using 4N − 3 subspaceseach of any dimension less than N .

Remark 3.14. The upper bound given in Theorem 3.12 is identical to thenecessary and sufficient bound of 2N − 1 for the one dimensional real vector case.Also, note that the bound given in Theorem 3.13 is very close to the conjecturedbound of 4N − 4 for the one dimensional complex vector case.

Theorem 3.12 and Theorem 3.13 define upper bounds on the number of sub-spaces needed for phase retrieval; but they do not give any information on howto construct subspaces which allow phase retrieval. This next theorem sheds somelight on how to construct each of these subspaces by relating them to the onedimensional phase retrieval problem.

Theorem 3.15. [14] Let {Wi}Mi=1 be subspaces of RN . The following are equiv-alent:

(a) {Wi}Mi=1 allows phase retrieval in RN .

(b) For every orthonormal basis {ϕi,d}Di

d=1 of Wi, the set {ϕi,d}M, Di

i=1,d=1 allows phase

retrieval in RN and thus has the complement property.


After constructing a way to establish if subspaces give phase retrieval, theauthors of [14] then proceeded to establish a useful result which defines whensubspaces do not give phase retrieval.

Lemma 3.16. [14] Subspaces {Wi}Mi=1 in RN do not allow phase retrieval if andonly if there exists nonzero u, v ∈ RN with u ⊥ v such that ||Piu|| = ||Piv|| for alli = 1, . . . ,M .

Notice that Lemma 3.16 is more useful than it may appear. According to thedefinition for subspaces allowing phase retrieval, it is clear that subspaces {Wi}Mi=1

do not give phase retrieval if and only if there exists x, y ∈ RN such that x = ±y and||Pix|| = ||Piy|| for all i = 1, · · · ,M . However, Lemma 3.16 proves that a strongercondition actually occurs. It shows that if phase retrieval fails then there existsnonzero perpendicular vectors u ⊥ v satisfying ||Piu|| = ||Piv|| for all i = 1, . . . ,M ,which is stronger than implying the existence of nonzero vectors u = ±v satisfying||Piu|| = ||Piv|| for all i = 1, . . . ,M .

After giving some characterizations for when subspaces allow phase retrieval,the authors of [14] addressed the following question: Given subspaces which yieldphase retrieval, can those subspaces be used to find other subspaces which yieldphase retrieval? The answer is yes, and is given in the following theorem.

Theorem 3.17. [14] Suppose {Wi}Mi=1 are subspaces allowing phase retrievalfor RN with associated orthogonal projections Pi. Let {W ′

i}Mi=1 be subspaces withassociated orthogonal projections Qi. Then there exists an ε > 0 such that when||Pi −Qi|| < ε for all i = 1, · · · ,M , then {W ′

i}Mi=1 allow phase retrieval.

Recently, [3] developed similar results to Theorem 3.17 for the one dimensionalphase retrieval case. In [3], Balan showed that if a spanning set Φ of M vectors inCN allows vector reconstruction from the magnitudes of its coefficients, then thereis a perturbation bound ρ so that any spanning set within ρ from Φ has the sameproperty.

Lastly, [14] highlights several open questions concerning phase retrieval fromthe norms of subspace components and they add their progress towards answeringeach problem.

Problem 3.18. What is the minimal number M such that {Wi}Mi=1 allow phaseretrieval in RN? Does this number depend upon the dimensions of the subspaces?

Problem 3.19. Does a generic choice of 2N − 1 projections in the real case or4N − 3 projections in the complex case allow phase retrieval?

Problem 3.20. Can phase retrieval be done with random subspaces of RN?

Problem 3.21. Classify subspaces {Wi}Mi=1 such that {W⊥i }Mi=1 allow phase

retrieval.

Problem 3.22. Show random subspaces or find examples of non-structuredsubspaces of arbitrary dimension which allow phase retrieval.

Problem 3.23. Find examples of (or classify) the subspaces {Wi}Mi=1 whichallow phase retrieval but the span of their associated projections {Pi}Mi=1 is notequal to the span of any M rank one projections.

Problem 3.24. For the complex case, is the minimal number of projections ofarbitrary rank needed for phase retrieval less than or equal to the minimal numberof rank one projections needed for phase retrieval.


Phase retrieval by vectors and by projections are both important topics of re-search and we have discussed some meaningful applications and influential papersin these areas. We have highlighted many mathematical results in vector phaseretrieval that have came about in the past decade and have referenced numerouspapers not only in mathematics but in many scientific fields which studied vectorphase retrieval. While numerous researchers have been and will continue to studyphase retrieval by vectors and the applications therein, phase retrieval by projec-tions is currently a growing area of study as well. We have highlighted a few of themajor mathematical results developed in the area of phase retrieval by projectionsand have listed a few problems in this area, which we hope will be solved.

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Department of Mathematics, University of Missouri, Columbia, Missouri 65211-4100

E-mail address: [email protected]

Department of Mathematics, University of Missouri, Columbia, Missouri 65211-4100



Scalable frames and convex geometry

Gitta Kutyniok, Kasso A. Okoudjou, and Friedrich Philipp

Abstract. The recently introduced and characterized scalable frames can beconsidered as those frames which allow for perfect preconditioning in the sensethat the frame vectors can be rescaled to yield a tight frame. In this paper wedefine m-scalability, a refinement of scalability based on the number of non-zero weights used in the rescaling process, and study the connection betweenthis notion and elements from convex geometry. Finally, we provide results onthe topology of scalable frames. In particular, we prove that the set of scalableframes with “small” redundancy is nowhere dense in the set of frames.

1. Introduction

Frame theory is nowadays a standard methodology in applied mathematics andengineering. The key advantage of frames over orthonormal bases is the fact thatframes are allowed to be redundant, yet provide stable decompositions. This is acrucial fact, for instance, for applications which require robustness against noise orerasures, or which require a sparse decomposition (cf. [3]).

Tight frames provide optimal stability, since these systems satisfy the Parsevalequality up to a constant. Formulated in the language of numerical linear algebra,a tight frame is perfectly conditioned, since the condition number of its analysisoperator is one. Thus, one key question is the following: Given a frame Φ ={ϕk}Mk=1 ⊂ RN , M ≥ N , say, can the frame vectors ϕk be modified so that theresulting system forms a tight frame? Again in numerical linear algebra terms, thisquestion can be regarded as a request for perfect preconditioning [1, 4]. Since aframe is typically designed to accommodate certain requirements of an application,this modification process should be as careful as possible in order not to change theproperties of the system too drastically.

One recently considered approach consists in multiplying each frame vector bya scalar/a weight. Notice that this process does not even disturb sparse decom-position properties at all, hence it might be considered ‘minimally invasive’. Theformal definition was given in [8] by the authors and E.K. Tuley (see also [9]). Inthat paper, a frame, for which scalars exist so that the scaled frame forms a tightframe, was coined scalable frame. Moreover, in the infinite dimensional situation,various equivalent conditions for scalability were provided, and in the finite dimen-sional situation, a very intuitive geometric characterization was proven. In fact,

2010 Mathematics Subject Classification. Primary 42C15, 52B11; Secondary 15A03, 65F08.Key words and phrases. Scalable frames, tight frames, preconditioning, Farkas’s lemma.


19

20 GITTA KUTYNIOK, KASSO A. OKOUDJOU, AND FRIEDRICH PHILIPP

this characterization showed that a frame is non-scalable, if the frame vectors donot spread ‘too much’ in the space. This seems to indicate that there exist relationsto convex geometry.

Scalable frames were then also investigated in the papers [6] and [2]. In [6], theauthors analyzed the problem by making use of the properties of so-called diagramvectors [7], whereas [2] gives a detailed insight into the set of weights which can beused for scaling.

The contribution of the present paper is three-fold. First, we refine the defini-tion of scalability by calling a (scalable) frame m-scalable, if at most m non-zeroweights can be used for the scaling. Second, we establish a link to convex geome-try. More precisely, we prove that this refinement leads to a reformulation of thescalability question in terms of the properties of certain polytopes associated toa nonlinear transformation of the frame vectors. This nonlinear transformation isrelated but not equivalent to the diagram vectors used in the results obtained in[6]. Using this reformulation, we establish new characterizations of scalable framesusing convex geometry, namely convex polytopes. Third, we investigate the topo-logical properties of the set of scalable frames. In particular, we prove that in theset of frames in RN with M frame vectors the set of scalable frames is nowheredense if M < N(N +1)/2. We wish to mention, that the results stated and provedin this paper were before announced in [10].

The paper is organized as follows. In Section 2, we introduce the requirednotions with respect to frames and their (m-)scalability as well as state some basicresults. Section 3 is devoted to establishing the link to convex geometry and derivenovel characterizations of scalable frames using this theory. Finally, in Section 4,we study the topology of the set of scalable frames.

2. Preliminaries

First of all, let us fix some notations. If X is any set whose elements areindexed by xj , j ∈ J , and I ⊂ J , we define XI := {xi : i ∈ I}. Moreover, for theset {1, . . . , n}, n ∈ N, we write [n].

A set Φ = {ϕk}Mk=1 ⊂ RN , M ≥ N is called a frame, if there exist positiveconstants A and B such that

(2.1) A‖x‖2 ≤M∑k=1

|〈x, ϕk〉|2 ≤ B‖x‖2

holds for all x ∈ RN . Constants A and B as in (2.1) are called frame bounds ofΦ. The frame Φ is called tight if A = B is possible in (2.1). In this case we have

A = 1N

∑M=1 ‖ϕk‖2. A tight frame with A = B = 1 in (2.1) is called Parseval frame.

We will sometimes identify a frame Φ = {ϕk}Mk=1 ⊂ RN with the N×M matrixwhose kth column is the vector ϕk. This matrix is called the synthesis operator ofthe frame. The adjoint ΦT of Φ is called the analysis operator. Using the analysisoperator, the relation (2.1) reads

A‖x‖2 ≤ ‖ΦTx‖2 ≤ B‖x‖2.

Hence, a frame Φ is tight if and only if some multiple of ΦT is an isometry. Theset of frames for RN with M elements will be denoted by F(M,N). We say that aframe Φ ∈ F(M,N) is degenerate if one of its frame vectors is the zero-vector. If

SCALABLE FRAMES AND CONVEX GEOMETRY 21

X (M,N) is a set of frames in F(M,N), we denote by X ∗(M,N) the set of the non-degenerate frames in X (M,N). For example, F∗(M,N) is the set of non-degenerateframes in F(M,N). For more details on frames, we refer the reader to [3,5]

Let us recall the following definition from [8, Definition 2.1].

Definition 2.1. A frame Φ = {ϕk}Mk=1 for RN is called scalable, respec-tively, strictly scalable, if there exist nonnegative, respectively, positive, scalarsc1, . . . , cM ∈ R such that {ckϕk}Mk=1 is a tight frame for RN . The set of scal-able, respectively, strictly scalable, frames in F(M,N) is denoted by SC(M,N),respectively, SC+(M,N).

In order to gain a better understanding of the structure of scalable frames werefine the definition of scalability.

Definition 2.2. Let M,N,m ∈ N be given such that N ≤ m ≤ M . A frameΦ = {ϕk}Mk=1 ∈ F(M,N) is said to be m-scalable, respectively, strictly m-scalable,if there exists a subset I ⊆ [M ], #I = m, such that ΦI is a scalable frame, respec-tively, a strictly scalable frame for RN . We denote the set of m-scalable frames,respectively, strictly m-scalable frames in F(M,N) by SC(M,N,m), respectively,SC+(M,N,m).

It is easily seen that for m ≤ m′ we have that SC(M,N,m) ⊂ SC(M,N,m′).Therefore,

SC(M,N) = SC(M,N,M) =

M⋃m=N

SC(M,N,m).

In the sequel, if no confusion can arise, we often only write F , SC, SC+, SC(m),and SC+(m) instead of SC(M,N), SC+(M,N), SC(M,N,m), and SC+(M,N,m),respectively. The notations F∗, SC∗, SC∗

+, SC(m)∗, and SC+(m)∗ are to be readanalogously.

Note that for a frame Φ ∈ F to be m-scalable it is necessary that m ≥ N . Inaddition, Φ ∈ SC(M,N) holds if and only if T (Φ) ∈ SC(M,N) holds for one (andhence for all) orthogonal transformation(s) T on RN ; cf. [8, Corollary 2.6].

If M ≥ N , we have Φ ∈ SC(M,N,N) if and only if Φ contains an orthogonalbasis of RN . This completely characterizes the set SC(M,N,N) of N -scalableframes for RN consisting of M vectors. For frames with M = N + 1 vectors in RN

we have the following result:

Proposition 2.3. Let N ≥ 2 and Φ = {ϕk}N+1k=1 ∈ F∗ with ϕk = ±ϕ� for

k = �. If Φ ∈ SC+(N + 1, N,N) then Φ /∈ SC+(N + 1, N).

Proof. If Φ ∈ SC+(N + 1, N,N), then Φ must contain an orthogonal basis.By applying some orthogonal transformation and rescaling the frame vectors, wecan assume without loss of generality that {ϕk}Nk=1 = {ek}Nk=1 is the standardorthonormal basis of RN , and that ϕN+1 = ±ek for each k = 1, 2, . . . , N , with‖ϕN+1‖ = 1. Thus, Φ can be written as Φ =

[IdN ϕN+1

], where IdN is the

N ×N identity matrix.

Assume that there exists {λk}N+1k=1 ⊂ (0,∞) such that Φ = {λkϕk}N+1

k=1 is a

tight frame, i.e. ΦΦT = A IdN . Using a block multiplication this equation can berewritten as

Λ + λ2N+1ϕN+1ϕ

TN+1 = A IdN


where Λ = diag(λ2k) is the N × N diagonal matrix with λ2

k, k = 1, . . . , N , on itsdiagonal. Consequently,

λ2k + λ2

N+1ϕ2N+1,k = A for k = 1, . . . , N and

λ2N+1ϕN+1,�ϕN+1,k = 0 for k = �.

But λN+1 > 0 and so all but one entry in ϕN+1 vanish. Since ϕN+1 is a unitnorm vector, we see that ϕN+1 = ±ek for some k ∈ [N ] which is contrary to theassumption, so Φ cannot be strictly (N + 1)-scalable. �

3. Scalable Frames and Convex Polytopes

Our characterizations of m-scalable frames will be stated in terms of certainconvex polytopes and, more generally, using tools from convex geometry. Therefore,we collect below some key facts and properties needed to state and prove our results.For a detailed treatment of convex geometry we refer to [11,13,14].

3.1. Background on Convex Geometry. In this subsection, let E be a reallinear space, and let X = {xi}Mi=1 be a finite set in E. The convex hull generatedby X is the compact convex subset of E defined by

co(X) :=

{M∑i=1

αixi : αi ≥ 0,

M∑i=1

αi = 1

}.

The affine hull generated by X is defined by

aff(X) :=

{M∑i=1

αixi :

M∑i=1

αi = 1

}.

Hence, we have co(X) ⊂ aff(X). Recall that for fixed a ∈ aff(X), the set

V (X) := aff(X)− a = {y − a : y ∈ aff(X)}is a subspace of E (which is independent of a ∈ aff(X)) and that one defines

dimX := dim co(X) := dimaff(X) := dimV (X).

We shall use Caratheodory’s Theorem for convex polytopes (see, e.g., [13, Theorem2.2.12]) in deciding whether a frame is scalable:

Theorem 3.1 (Caratheodory). Let X = {x1, . . . , xk} be a finite subset of Ewith d := dimX. Then for each x ∈ co(X) there exists I ⊂ [k] with #I = d + 1such that x ∈ co(XI).

The relative interior of the polytope co(X) denoted by ri co(X), is the interiorof co(X) in the topology induced by aff(X). We have that ri co(X) = ∅ as long as#X ≥ 2; cf. [13, Lemma 3.2.8]. Furthermore,

ri co(X) =

{M∑i=1

λixi : λi > 0,M∑i=1

λi = 1

},

see [14, Theorem 2.3.7]. Moreover, the interior of co(X) in E is non-empty if andonly if aff(X) = E.

The following lemma characterizes dimX in terms of dim spanX.

Lemma 3.2. Let X be a finite set of points in E. Put m := dim spanX. ThendimX ∈ {m− 1,m}. Moreover, the following statements are equivalent:


(i) dimX = m− 1.(ii) For all linearly independent X ′ ⊂ X with dim spanX ′ = m we have X \

X ′ ⊂ aff(X ′).(iii) For some linearly independent X ′ ⊂ X with dim spanX ′ = m we have

X \X ′ ⊂ aff(X ′).

Proof. Let X = {x1, . . . , xk}. First of all, we observe that for a linearlyindependent set X ′ = {xi1 , . . . , xim} as in (ii) or (iii) we have

dimV (X ′) = dim span{xil − xi1 : l = 2, . . . ,m} = m− 1.

Therefore, V (X ′) ⊂ V (X) ⊂ spanX implies m − 1 ≤ dimX ≤ m. Let us nowprove the moreover-part of the lemma.

(i)⇒(ii). Assume that dimX = m−1 and let X ′ = {xi1 , . . . , xim} be a linearlyindependent set as in (ii). From dimV (X) = dimX = m − 1 we obtain V (X) =V (X ′). Therefore, for each xj ∈ X \X ′ there exist μ2, . . . , μm ∈ R such that

xj − xi1 =m∑i=2

μi(xi − xi1) =m∑i=2

μixi −(

m∑i=2

μi

)xi1 .

And this implies

xj =

(1−

m∑i=2

μi

)xi1 +

m∑i=2

μixi ∈ aff(X ′).

(ii)⇒(iii). This is obvious.(iii)⇒(i). Let X ′ = {xi1 , . . . , xim} be a linearly independent set as in (iii). If

x ∈ X\X ′, then we have x ∈ aff(X ′) by (iii). Consequently, there exist λ1, . . . , λm ∈R with

∑ml=1 λl = 1 such that x =

∑ml=1 λlxil . Hence, we obtain

x− xi1 =

m∑l=1

λlxil −(

m∑l=1

λl

)xi1 =

m∑l=1

λl(xil − xi1) ∈ V (X ′).

This implies V (X) = V (X ′) and hence (i). �

In the sequel we will have to deal with a special case of the situation inLemma 3.2, where X is a set of rank-one orthogonal projections acting on a realor complex Hilbert space H. In this case, E is the set consisting of the selfadjointoperators in H which is a real linear space.

Corollary 3.3. Let X be a finite set consisting of rank-one orthogonal pro-jections acting on a Hilbert space H. Then we have

dimX = dim spanX − 1.

Proof. Let X = {P1, . . . , Pk}, m := dim spanX, and let X ′ ⊂ X be a lin-early independent subset of X such that dim spanX ′ = m. Without loss of gen-erality assume that X ′ = {P1, . . . , Pm}. Let j ∈ {m + 1, . . . , k}. Then there existλ1, . . . , λm ∈ R such that Pj =

∑mi=1 λiPi. This implies

1 = TrPj = Tr

(m∑i=1

λiPi

)=

m∑i=1

λi Tr(Pi) =m∑i=1

λi,

which shows that Pj ∈ aff(X ′). The statement now follows from Lemma 3.2. �


3.2. Scalability in Terms of Convex Combinations of Rank-One Ma-trices. Here, and for the rest of this paper, for a frame Φ = {ϕi}Mi=1 in F(M,N)we set

XΦ := {ϕiϕTi : i ∈ [M ]}.

This is a subset of the space of all real symmetric N ×N -matrices which we shalldenote by SN . We shall also denote the set of positive multiples of the identity byI+ := {α IdN : α > 0}.

Proposition 3.4. For a frame Φ ∈ F(M,N) the following statements areequivalent:

(i) Φ is scalable, respectively, strictly scalable.(ii) I+ ∩ co(XΦ) = ∅, respectively, I+ ∩ ri co(XΦ) = ∅.

Proof. Assume that the frame Φ = {ϕi}Mi=1 is scalable. Then there existnon-negative scalars c1, . . . , cM such that

M∑i=1

ciϕiϕTi = Id .

Put α :=∑M

i=1 ci. Then α > 0 and with λi := α−1ci we have

M∑i=1

λiϕiϕTi = α−1 Id and

M∑i=1

λi = 1.

Hence α−1 Id ∈ co(XΦ). The converse direction is obvious. �

As pointed out earlier, for m ≤ m′ we have SC(m) ⊂ SC(m′). Given Φ ∈SC(M,N) = SC(M), there exists m ≤ M such that such that Φ ∈ SC(m); e.g.,we can always take m = M . However, the next result gives a “canonical” integerm = mΦ that is in a way “optimal”.

Proposition 3.5. For a frame Φ = {ϕk}Mk=1 ∈ F , put m = mΦ := dim spanXΦ.Then the following statements are equivalent:

(i) Φ is scalable.(ii) Φ is m-scalable.

Proof. Clearly, (ii) implies (i). Conversely, let Φ = {ϕi}Mi=1 be scalable. Afterpossibly removing zero vectors from the frame and thereby reducing M (which doesnot affect the value of m), we may assume that Φ is unit-norm. By Proposition3.4, there exists α > 0 such that α IdN ∈ co(XΦ). Therefore, from Theorem 3.1 itfollows that there exists I ⊂ [M ] with #I = dimXΦ+1 such that α IdN ∈ co(XΦI

).Hence, ΦI is scalable by Proposition 3.4. And since dimXΦ = dim spanXΦ − 1 byCorollary 3.3, the claim follows. �

As XΦ ⊂ SN and dimSN = N(N + 1)/2, we immediately obtain the followingcorollary.

Corollary 3.6. For M ≥ N(N + 1)/2 we have

SC(M,N) = SC(M,N,

N(N + 1)

2

).


3.3. Convex Polytopes Associated with m-Scalable Frames. Let Φ ={ϕk}Mk=1 be a frame for RN . Then the analysis operator of the scaled frame{ckϕk}Mk=1 is given by CΦT , where C is the diagonal matrix with the values ckon its diagonal. Hence, the frame Φ is scalable if and only if

(3.1) ΦC2ΦT = A IdN ,

where A > 0. Similarly, Φ is m-scalable if and only if (3.1) holds with C = diag(c),where c ∈ [0,∞)M such that ‖c‖0 ≤ m. Here, we used the so-called “zero-norm”(which is in fact not a norm), defined by

‖x‖0 := #{k ∈ [n] : xk = 0}, x ∈ Rn.

Comparing corresponding entries from left- and right-hand sides of (3.1), it is seenthat for a frame to be m-scalable it is necessary and sufficient that there exists avector u = (c21, c

22, . . . , c

2M )T with ‖u‖0 ≤ m which is a solution of the following

linear system of N(N+1)2 equations in M unknowns:

(3.2)

⎧⎪⎪⎨⎪⎪⎩M∑j=1

ϕj(k)2yj = A for k = 1, . . . , N,

M∑j=1

ϕj(�)ϕj(k)yj = 0 for �, k = 1, . . . , N, k > �.

Subtraction of equations in the first system in (3.2) leads to the new homoge-neous linear system

(3.3)

⎧⎪⎪⎨⎪⎪⎩M∑j=1

(ϕj(1)

2 − ϕj(k)2)yj = 0 for k = 2, . . . , N,

M∑j=1

ϕj(�)ϕj(k)yj = 0 for �, k = 1, . . . , N, k > �.

It is not hard to see that we have not lost information in the last step, hence Φ ism-scalable if and only if there exists a nonnegative vector u ∈ RM with ‖u‖0 ≤ mwhich is a solution to (3.3). In matrix form, (3.3) reads

F (Φ)u = 0,

where the (N − 1)(N + 2)/2×M matrix F (Φ) is given by

F (Φ) =(F (ϕ1) F (ϕ2) . . . F (ϕM )

),

where F : RN → Rd, d := (N − 1)(N + 2)/2, is defined by

F (x) =

⎛⎜⎜⎜⎝F0(x)F1(x)

...FN−1(x)

⎞⎟⎟⎟⎠ , F0(x) =

⎛⎜⎜⎜⎝x21 − x2

2

x21 − x2

3...

x21 − x2

N

⎞⎟⎟⎟⎠ , Fk(x) =

⎛⎜⎜⎜⎝xkxk+1

xkxk+2

...xkxN

⎞⎟⎟⎟⎠ ,

and F0(x) ∈ RN−1, Fk(x) ∈ RN−k, k = 1, 2, . . . , N − 1.Summarizing, we have just proved the following proposition.

Proposition 3.7. A frame Φ for RN is m-scalable, respectively, strictly m-scalable if and only if there exists a nonnegative u ∈ kerF (Φ) \ {0} with ‖u‖0 ≤ m,respectively, ‖u‖0 = m.


We will now utilize the above reformulation to characterize m-scalable framesin terms of the properties of convex polytopes of the type co(F (ΦI)), I ⊂ [M ]. Oneof the key tools will be Farkas’ lemma (see, e.g., [11, Lemma 1.2.5]).

Lemma 3.8 (Farkas’ Lemma). For every real N ×M -matrix A exactly one ofthe following cases occurs:

(i) The system of linear equations Ax = 0 has a nontrivial nonnegative solu-tion x ∈ RM (i.e., all components of x are nonnegative and at least oneof them is strictly positive.)

(ii) There exists y ∈ RN such that AT y is a vector with all entries strictlypositive.

In our first main result we use the notation co(A) for a matrix A which wesimply define as the convex hull of the set of column vectors of A.

Theorem 3.9. Let M ≥ m ≥ N ≥ 2, and let Φ = {ϕk}Mk=1 be a frame for RN .Then the following statements are equivalent:

(i) Φ is m-scalable, respectively, strictly m-scalable,(ii) There exists a subset I ⊂ [M ] with #I = m such that 0 ∈ co(F (ΦI)),

respectively, 0 ∈ ri co(F (ΦI)).(iii) There exists a subset I ⊂ [M ] with #I = m for which there is no h ∈ Rd

with 〈F (ϕk), h〉 > 0 for all k ∈ I, respectively, with 〈F (ϕk), h〉 ≥ 0 for allk ∈ I, with at least one of the inequalities being strict.

Proof. (i)⇔(ii). This equivalence follows directly if we can show the followingequivalences for Ψ ⊂ Φ:

0 ∈ co(F (Ψ)) ⇐⇒ kerF (Ψ) \ {0} contains a nonnegative vector and

0 ∈ ri co(F (Ψ)) ⇐⇒ kerF (Ψ) contains a positive vector.(3.4)

The implication “⇒” is trivial in both cases. For the implication “⇐” in the firstcase let I ⊂ [M ] be such that Ψ = ΦI , I = {i1, . . . , im}, and let u = (c1, . . . , cm)T ∈kerF (Ψ) be a non-zero nonnegative vector. Then A :=

∑mk=1 ck > 0 and with

λk := ck/A, k ∈ [m], we have∑m

k=1 λk = 1 and∑m

k=1 λkF (ϕik) = A−1F (Ψ)u = 0.Hence 0 ∈ co(F (Ψ)). The proof for the second case is similar.

(ii)⇔(iii). In the non-strict case this follows from (3.4) and Lemma 3.8. In thestrict case this is a known fact; e.g., see [14, Lemma 3.6.5]. �

We now derive a few consequences of the above theorem. A given vector v ∈ Rd

defines a hyperplane by

H(v) = {y ∈ Rd : 〈v, y〉 = 0},

which itself determines two open convex cones H−(v) and H+(v), defined by

H−(v) = {y ∈ Rd : 〈v, y〉 < 0} and H+(v) = {y ∈ Rd : 〈v, y〉 > 0}.

Using these notations we can restate the equivalence (i)⇔(iii) in Theorem 3.9 asfollows:

Proposition 3.10. Let M ≥ N ≥ 2, and let m be such that N ≤ m ≤ M .Then a frame Φ = {ϕk}Mk=1 for RN is m-scalable if and only if there exists a subsetI ⊂ [M ] with #I = m such that

⋂i∈I H

+(F (ϕi)) = ∅.


Remark 3.11. In the case of strict m-scalability we have the following neces-sary condition: If Φ is strictly m-scalable, then there exists a subset I ⊂ [M ] with#I = m such that

⋂i∈I H

−(F (ϕi)) = ∅.

Remark 3.12. When M ≥ d+ 1 = N(N + 1)/2, we can use properties of theconvex sets H±(F (ϕk)) to give an alternative proof of Corollary 3.6. For this, letthe frame Φ = {ϕk}Mk=1 for RN be scalable. Then, by Proposition 3.10 we have

that⋂M

k=1 H+(F (ϕk)) = ∅. Now, Helly’s theorem (see, e.g., [11, Theorem 1.3.2])

implies that there exists I ⊂ [M ] with #I = d+1 such that⋂

i∈I H+(F (ϕi)) = ∅.

Exploiting Proposition 3.10 again, we conclude that Φ is (d+ 1)-scalable.

The following result is an application of Proposition 3.10 which provides asimple condition for Φ /∈ SC(M,N).

Proposition 3.13. Let Φ = {ϕk}Mk=1 be a frame for RN , N ≥ 2. If there existsan isometry T such that T (Φ) ⊂ RN−2 ×R2

+, then Φ is not scalable. In particular,Φ is not scalable if there exist i, j ∈ [N ], i = j, such that ϕk(i)ϕk(j) > 0 for allk ∈ [M ].

Proof. Without loss of generality, we may assume that Φ ⊂ RN−2 × R2+, cf.

[8, Corollary 2.6]. Let {ek}dk=1 be the standard ONB for Rd. Then for each k ∈ [M ]we have that

〈ed, F (ϕk)〉 = ϕk(N − 1)ϕk(N) > 0.

Hence, ed ∈⋂

i∈[M ] H+(F (ϕi)). By Proposition 3.10, Φ is not scalable. �

The characterizations stated above can be recast in terms of the convex coneC(F (Φ)) generated by F (Φ). We state this result for the sake of completeness.But first, recall that for a finite subset X = {x1, . . . , xM} of Rd the polyhedral conegenerated by X is the closed convex cone C(X) defined by

C(X) =

{M∑i=1

αixi : αi ≥ 0

}.

Let C be a cone in Rd. The polar cone of C is the closed convex cone C◦ definedby

C◦ := {x ∈ RN : 〈x, y〉 ≤ 0 for all y ∈ C}.The cone C is said to be pointed if C ∩ (−C) = {0}, and blunt if the linear spacegenerated by C is RN , i.e. spanC = RN .

Corollary 3.14. Let Φ = {ϕk}Mk=1 ∈ F∗, and let N ≤ m ≤ M be fixed. Thenthe following conditions are equivalent:

(i) Φ is strictly m-scalable .(ii) There exists I ⊂ [M ] with #I = m such that C(F (ΦI)) is not pointed.(iii) There exists I ⊂ [M ] with #I = m such that C(F (ΦI))

◦ is not blunt.(iv) There exists I ⊂ [M ] with #I = m such that the interior of C(F (ΦI))

◦ isempty.

Proof. (i)⇔(ii). By Proposition 3.7, Φ is strictly m-scalable if and only ifthere exist I ⊂ [M ] with #I = m and a nonnegative u ∈ kerF (ΦI) \ {0} with‖u‖0 = m. By [13, Lemma 2.10.9], this is equivalent to the cone C(F (ΦI)) beingnot pointed. This proves that (i) is equivalent to (ii).


(ii)⇔(iii). This follows from the fact that the polar of a pointed cone C isblunt and vice versa, as long as C◦◦ = C, see [13, Theorem 2.10.7]. But in our caseC(F (ΦI))

◦◦ = C(F (ΦI)), see [13, Lemma 2.7.9].(iii)⇒(iv). If C(F (ΦI))

◦ is not blunt, then it is contained in a proper hyperplaneof Rd whose interior is empty. Hence, also the interior of C(F (ΦI))

◦ must be empty.(iv)⇒(iii). We use a contra positive argument. Assume that C(F (Φ))◦ is

blunt. This is equivalent to spanC(F (Φ))◦ = Rd. But for the nonempty coneC(F (Φ))◦ we have aff(C(F (Φ))◦) = spanC(F (Φ))◦. Hence, aff(C(F (Φ))◦) = Rd,and so the relative interior of C(F (Φ))◦ is equal to its interior, which therefore isnonempty. �

The main idea of the previous results is the characterization of (m-)scalabilityof Φ in terms of properties of the convex polytopes co(F (ΦI)). However, it seemsmore “natural” to seek assumptions on the convex polytopes co(ΦI) that will ensurethat co(F (Φ)) satisfy the conditions in Theorem 3.9 hold. Proposition 3.13, whichgives a condition on Φ that precludes it to be scalable, is a step in this direction.

Nonetheless, we address the related question of whether F (Φ) is a frame for Rd

whenever Φ is a scalable frame for RN . This depends clearly on the redundancyof Φ as well as on the map F . In particular, we finish this section by giving acondition which ensures that F (Φ) is always a frame for Rd when M ≥ d + 1. Inorder to prove this result, we need a few preliminary facts.

For x = (xk)Nk=1 ∈ RN and h = (hk)

dk=1 ∈ Rd, we have that

(3.5) 〈F (x), h〉 =N∑�=2

h�−1(x21 − x2

�) +

N−1∑k=1

N∑�=k+1

hk(N−1−(k−1)/2)+�−1xkx�.

The right-hand side of (3.5) is obviously a homogeneous polynomial of degree 2

in x1, x2, . . . , xN . We shall denote the set of all polynomials of this form by PN2 .

It is easily seen that PN2 is isomorphic to the subspace of real symmetric N × N

matrices whose trace is 0. Indeed, for each N ≥ 2, and each p ∈ PN2 ,

p(x) =

N∑�=2

a�−1(x21 − x2

�) +

N−1∑k=1

N∑�=k+1

ak(N−(k+1)/2)+�−1xkx�,

we have p(x) = 〈Qpx, x〉, where Qp is the symmetric N ×N -matrix with entries

Qp(1, 1) =

N−1∑k=1

ak, Qp(�, �) = −a�−1 for � = 2, 3, . . . , N

and

Qp(k, �) =1

2ak(N−(k+1)/2)+�−1 for k = 1, . . . , N − 1, � = k + 1, . . . , N.

In particular, the dimension of PN2 is d = (N + 2)(N − 1)/2.

Proposition 3.15. Let M ≥ d + 1 where d = (N − 1)(N + 2)/2, and Φ ={ϕk}Mk=1 ∈ SC+(d+ 1) \ SC(d). Then F (Φ) is a frame for Rd.

Proof. Let I ⊂ [M ], #I = d + 1, be an index set such that ΦI is strictlyscalable. Assume that there exists h ∈ Rd such that 〈F (ϕk), h〉 = 0 for each k ∈ I.By (3.5) we conclude that ph(ϕk) = 0 for all k ∈ I, where ph is the polynomial in


PN2 on the right hand side of (3.5). Hence 〈Qph

ϕk, ϕk〉 = 0 for all k ∈ I. Now, wehave

(3.6) 〈ϕkϕTk , Qph

〉HS = Tr(ϕkϕTkQph

) = 〈Qphϕk, ϕk〉 = 0 for all k ∈ I.

But as ΦI is not d-scalable (otherwise, Φ ∈ SC(d)) it is not m-scalable for everym ≤ d. Thus, Proposition 3.5 yields that

dim span{ϕkϕTk : k ∈ I} = d+ 1.

Equivalently, {ϕkϕTk : k ∈ I} is a basis of the (d + 1)-dimensional space SN .

Therefore, from (3.6) we conclude that Qph= 0 which implies ph = 0 (since p → Qp

is an isomorphism) and thus h = 0.Now, it follows that F (ΦI) spans Rd which is equivalent to F (ΦI) being a frame

for Rd. Hence, so is F (Φ). �

4. Topology of the Set of Scalable Frames

In this section, we present some topological features of the set SC(M,N).Hereby, we identify frames in F(M,N) with real N × M -matrices as we alreadydid before, see, e.g., (3.1) in subsection 3.3. Hence, we consider F(M,N) as theset of all matrices in RN×M of rank N . Note that under this identification theorder of the vectors in a frame becomes important. However, it allows us to endowF(M,N) with the usual Euclidean topology of RN×M .

In [8] it was proved that SC(M,N) is a closed set in F(M,N) (in the relativetopology of F(M,N)). The next proposition refines this fact.

Proposition 4.1. Let M ≥ m ≥ N ≥ 2. Then SC(M,N,m) is closed inF(M,N).

Proof. We prove the assertion by establishing that the complement F\SC(m)is open, that is, if Φ = {ϕk}Mk=1 ∈ F is a frame which is not m-scalable, we provethat there exists ε > 0 such that for any collection Ψ = {ψk}Mk=1 of vectors in RN

for which‖ϕk − ψk‖ < ε for all k ∈ [M ],

we have that Ψ is a frame which is not m-scalable. Thus assume that Φ = {ϕk}Mk=1

is a frame which is not m-scalable and define the finite set I of subsets by

I := {I ⊂ [M ] : #I = m}.By Proposition 3.10, for each I ∈ I there exists yI ∈

⋂k∈I H

+(F (ϕk)), that is,mink∈I〈yI , F (ϕk)〉 > 0. By the continuity of the map F , there exists ε > 0 suchthat for each {ψk}Mk=1 ⊂ RN with ‖ψk − ϕk‖ < ε for all k ∈ [M ] we still havemink∈I〈yI , F (ψk)〉 > 0. We can choose ε > 0 sufficiently small to guarantee thatΨ = {ψk}Mk=1 ∈ F . Again from Proposition 3.10 we conclude that Ψ is not m-scalable for any N ≤ m ≤ M . Hence, SC(m) is closed in F . �

The next theorem is the main result of this section. It shows that the set ofscalable frames is nowhere dense in the set of frames unless the redundancy of theconsidered frames is disproportionately large.

Theorem 4.2. Assume that 2 ≤ N ≤ M < d + 1 = N(N + 1)/2. ThenSC(M,N) does not contain interior points. In other words, for the boundary ofSC(M,N) we have

∂SC(M,N) = SC(M,N).


For the proof of Theorem 4.2 we shall need two lemmas. Recall that for a frameΦ = {ϕk}Mk=1 ∈ F we use the notation

XΦ = {ϕiϕTi : i ∈ [M ]}.

Lemma 4.3. Let {ϕk}Mk=1 ⊂ RN be such that dim spanXΦ < N(N+1)2 . Then

there exists ϕ0 ∈ RN with ‖ϕ0‖ = 1 such that ϕ0ϕT0 /∈ spanXΦ.

Proof. Assume the contrary. Then each rank-one orthogonal projection is anelement of spanXΦ. But by the spectral decomposition theorem every symmet-ric matrix in RN×N is a linear combination of such projections. Hence, spanXΦ

coincides with the linear space SN of all symmetric matrices in RN×N . Therefore,

dim spanXΦ =N(N + 1)

2,

which is a contradiction. �

The following lemma shows that for a genericM -element set Φ = {ϕi}Mi=1 ⊂ RN

(or matrix in RN×M , if the ϕi are considered as columns) the subspace spanXΦ

has the largest possible dimension.

Lemma 4.4. Let D := min{M,N(N + 1)/2}. Then the set{Φ ∈ RN×M : dim spanXΦ = D

}is dense in RN×M .

Proof. Let Φ = {ϕi}Mi=1 ∈ RN×M and ε > 0. We will show that there existsΨ = {ψi}Mi=1 ∈ RN×M with ‖Φ − Ψ‖ < ε and dim spanXΨ = D. For this, setW := spanXΦ and let k be the dimension of W . If k = D, nothing is to prove.Hence, let k < D. Without loss of generality, assume that ϕ1ϕ

T1 , . . . , ϕkϕ

Tk are

linearly independent. By Lemma 4.3 there exists ϕ0 ∈ RN with ‖ϕ0‖ = 1 such thatϕ0ϕ

T0 /∈ W . For δ > 0 define the symmetric matrix

Sδ := δ(ϕk+1ϕ

T0 + ϕ0ϕ

Tk+1

)+ δ2ϕ0ϕ

T0 .

Then there exists at most one δ > 0 such that Sδ ∈ W (regardless of whetherϕk+1ϕ

T0 + ϕ0ϕ

Tk+1 and ϕ0ϕ

T0 are linearly independent or not). Therefore, we find

δ > 0 such that δ < ε/M and Sδ /∈ W . Now, for i ∈ [M ] put

ψi :=

{ϕi if i = k + 1

ϕk+1 + δϕ0 if i = k + 1

and Ψ := {ψi}Mi=1. Let λ1, . . . , λk+1 ∈ R such that

k+1∑i=1

λiψiψTi = 0.

Then, since ψk+1ψTk+1 = ϕk+1ϕ

Tk+1 + Sδ, we have that

λk+1Sδ = −k+1∑i=1

λiϕiϕTi ∈ W ,

which implies λk+1 = 0 and therefore also λ1 = . . . = λk = 0. Hence, we havedim spanXΨ = k+1 and ‖Φ−Ψ‖ < ε/M . If k = D−1, we are finished. Otherwise,repeat the above construction at most D − k − 1 times. �


Remark 4.5. For the case M ≥ N(N + 1)/2, Lemma 4.4 has been proved in[2, Theorem 2.1]. In the proof, the authors note that XΦ spans SN if and onlyif the frame operator of XΦ (considered as a system in SN ) is invertible. But thedeterminant of this operator is a polynomial in the entries of ϕi, and the complementof the set of roots of such polynomials is known to be dense.

Proof of Theorem 4.2. Assume the contrary. Then, by Lemma 4.4, thereeven exists an interior point Φ = {ϕi}Mi=1 ∈ SC(M,N) of SC(M,N) for which thelinear space W := spanXΦ has dimension M . Since Φ is scalable, there existsc1, . . . , cM ≥ 0 such that

M∑i=1

ciϕiϕTi = Id .

Without loss of generality we may assume that c1 > 0.By Lemma 4.3 there exists ϕ0 ∈ RN with ‖ϕ0‖ = 1 such that ϕ0ϕ

T0 /∈ W . As

in the proof of Lemma 4.4, we set

Sδ := δ(ϕ1ϕ

T0 + ϕ0ϕ

T1

)+ δ2ϕ0ϕ

T0 .

Then, for δ > 0 sufficiently small, Sδ /∈ W and Ψ := {ϕ1 + δϕ0, ϕ2, . . . , ϕM} ∈SC(M,N). Hence, there exist c′1, . . . , c

′M ≥ 0 such that

M∑i=1

ciϕiϕTi = Id = c′1(ϕ1 + δϕ0)(ϕ1 + δϕ0)

T +

M∑i=2

c′iϕiϕTi =

M∑i=1

c′iϕiϕTi + c′1Sδ.

This implies c′1Sδ ∈ W , and thus c′1 = 0. But then we have

c1ϕ1ϕT1 +

M∑i=2

(ci − c′i)ϕiϕTi = 0,

which yields c1 = 0 as the matrices ϕ1ϕT1 , . . . , ϕMϕT

M are linearly independent. Acontradiction. �

ACKNOWLEDGMENTS

G. Kutyniok acknowledges support by the Einstein Foundation Berlin, byDeutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, and by the DFG Re-search Center Matheon “Mathematics for key technologies” in Berlin. F. Philippis supported by the DFG Research Center Matheon. K. A. Okoudjou was sup-ported by ONR grants N000140910324 and N000140910144, by a RASA from theGraduate School of UMCP and by the Alexander von Humboldt foundation. Hewould also like to express his gratitude to the Institute for Mathematics at the Uni-versitat Osnabruck and the Institute of Mathematics at the Technische UniversitatBerlin for their hospitality while part of this work was completed.

References

[1] V. Balakrishnan and S. Boyd, Existence and uniqueness of optimal matrix scalings, SIAM J.Matrix Anal. Appl. 16 (1995), no. 1, 29–39, DOI 10.1137/S0895479892235393. MR1311416(95j:65043)

[2] J. Cahill and X. Chen, A note on scalable frames, Proceedings of the 10th InternationalConference on Sampling Theory and Applications, (2013), 93–96

[3] Peter G. Casazza and Gitta Kutyniok (eds.), Finite frames: Theory and Applications, Appliedand Numerical Harmonic Analysis, Birkhauser/Springer, New York, 2013. MR2964005


[4] Ke Chen, Matrix preconditioning techniques and applications, Cambridge Monographs onApplied and Computational Mathematics, vol. 19, Cambridge University Press, Cambridge,2005. MR2169217 (2006e:65001)

[5] Ole Christensen,An introduction to frames and Riesz bases, Applied and Numerical HarmonicAnalysis, Birkhauser Boston, Inc., Boston, MA, 2003. MR1946982 (2003k:42001)

[6] M. S. Copenhaver, Y. H. Kim, C. Logan, K. Mayfield, S. K. Narayan, M.J. Petro, and J.Sheperd, Diagram vectors and tight frame scaling in finite dimensions, Oper. Matrices, 8

(2014), no. 1, 73–88, DOI 10.7153/oam-08-02.[7] Deguang Han, Keri Kornelson, David Larson, and Eric Weber, Frames for undergraduates,

Student Mathematical Library, vol. 40, American Mathematical Society, Providence, RI,2007. MR2367342 (2010e:42044)

[8] Gitta Kutyniok, Kasso A. Okoudjou, Friedrich Philipp, and Elizabeth K. Tuley, Scalableframes, Linear Algebra Appl. 438 (2013), no. 5, 2225–2238, DOI 10.1016/j.laa.2012.10.046.MR3005286

[9] G. Kutyniok, K. A. Okoudjou, and F. Philipp, Perfect preconditioning of frames by a di-agonal operator, Proceedings of the 10th International Conference on Sampling Theory andApplications, (2013), 85–88.

[10] G. Kutyniok, K. A. Okoudjou, and F. Philipp, Preconditioning of frames, Proc. SPIE, 8858,(2013), Wavelets and Sparsity XV, DOI 10.1117/12.2022667

[11] Jirı Matousek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212,Springer-Verlag, New York, 2002. MR1899299 (2003f:52011)

[12] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, PrincetonUniversity Press, Princeton, N.J., 1970. MR0274683 (43 #445)

[13] Josef Stoer and Christoph Witzgall, Convexity and optimization in finite dimensions. I, DieGrundlehren der mathematischen Wissenschaften, Band 163, Springer-Verlag, New York-Berlin, 1970. MR0286498 (44 #3707)

[14] Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford Uni-versity Press, New York, 1994. MR1443208 (98h:52001)

Technische Universitat Berlin, Institut fur Mathematik, Strasse des 17. Juni 136,

10623 Berlin, Germany


University of Maryland, Department of Mathematics, College Park, Maryland

20742


Technische Universitat Berlin, Institut fur Mathematik, Strasse des 17. Juni 136,

10623 Berlin, Germany



Dilations of frames, operator-valued measuresand bounded linear maps

Deguang Han, David R. Larson, Bei Liu, and Rui Liu

Abstract. We will give an outline of the main results in our recent AMSMemoir, and include some new results, exposition and open problems. In thatmemoir we developed a general dilation theory for operator- valued measuresacting on Banach spaces where operator-valued measures (or maps) are notnecessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation toa projection-valued measure acting on a Banach space, and every boundedlinear map, again not necessarily completely bounded, on a Banach algebrahas a bounded homomorphism dilation acting on a Banach space. Here thedilation space often needs to be a Banach space even if the underlying space isa Hilbert space, and the projections are idempotents that are not necessarilyself-adjoint. These results lead to some new connections between frame theoryand operator algebras, and some of them can be considered as part of theinvestigation about “non-commutative” frame theory.

1. Introduction

Frame theory belongs to the area of applied harmonic analysis, but its under-pinnings involve large areas of functional analysis including operator theory, anddeep connections with the theory of operator algebras on Hilbert space. For in-stance, recently the Kadison–Singer “Extension of pure states on von Neumannalgebras” problem has been solved and its solution is known to have wide ramifica-tions in frame theory due mainly to the research and excellent exposition of Casazzaand of Weaver. The purpose of the present article is to give a good exposition ofsome recent work of the authors that establishes some deep connections betweenframe theory on the one hand and operator-valued measures and maps between vonNeumann algebras on the other hand. While our work has little or nothing to dodirectly with the above-mentioned extension of pure states problem, it represents aseparate instance of a rather deep connection between frame theory and operatoralgebras, and this is the point of this article. We show that it may have something

2010 Mathematics Subject Classification. Primary 46G10, 46L07, 46L10, 46L51, 47A20; Sec-ondary 42C15, 46B15, 46B25, 47B48.

Key words and phrases. Operator-valued measures, von Neumann algebras, dilations, normalmaps, completely bounded maps, frames.

Acknowledgements: The authors were all participants in the NSF funded Workshop in Anal-ysis and Probability at Texas A&M University. The first author acknowledges partial support bya grant from the NSF. The third and fourth authors received partial support from the NSFC.


33

34 DEGUANG HAN, DAVID R. LARSON, BEI LIU, AND RUI LIU

to do with another problem of Kadison: the “similarity problem”. At the leastit indicates the possibility of another possible approach to that problem. And itdoes show that the ideas implicit in frame theory belong to the underpinnings of asignificant part of modern mathematics.

Before discussing our results from [18] we need an exposition of some of thepreliminaries leading up to them.

2. Frames, Framings, and Operator-Valued Measures

A frame F for a Hilbert space H is a sequence of vectors {xn} ⊂ H indexed bya countable index set J for which there exist constants 0 < A ≤ B < ∞ such that,for every x ∈ H,

(2.1) A‖x‖2 ≤∑n∈J

| 〈x, xn 〉 |2 ≤ B‖x‖2

The optimal constants are known as the upper and lower frame bounds. A frameis called tight if A = B, and is called a Parseval frame if A = B = 1. If we onlyrequire that a sequence {xn} satisfies the upper bound condition in (2.1), then{xn} is called a Bessel sequence. A frame which is a basis is called a Riesz basis.Orthonormal bases are special cases of Parseval frames. A Parseval frame {xn} fora Hilbert space H is an orthonormal basis if and only if each xn is a unit vector.

For a Bessel sequence {xn}, its analysis operator Θ is a bounded linear operatorfrom H to �2(N) defined by

(2.2) Θx =∑n∈N

〈x, xn 〉 en,

where {en} is the standard orthonormal basis for �2(N). It is easily verified that

Θ∗en = xn, ∀n ∈ N

The Hilbert space adjoint Θ∗ is called the synthesis operator for {xn}. The positiveoperator S := Θ∗Θ : H → H is called the frame operator, or sometimes the Besseloperator if the Bessel sequence is not a frame, and we have

(2.3) Sx =∑n∈N

〈x, xn 〉xn, ∀x ∈ H.

A sequence {xn} is a frame for H if and only if its analysis operator Θ isbounded, injective and has closed range, which is, in turn, equivalent to the condi-tion that the frame operator S is bounded and invertible. In particular, {xn} is aParseval frame for H if and only if Θ is an isometry or equivalently if S = I.

Let S be the frame operator for a frame {xn}. Then the lower frame bound is1/||S−1|| and the upper frame bound is ||S||. From (2.3) we obtain the reconstruc-tion formula (or frame decomposition):

x =∑n∈N

⟨x, S−1xn

⟩xn, ∀x ∈ H

or equivalently

x =∑n∈N

〈 x, xn 〉S−1xn, ∀x ∈ H.

DILATIONS AND BOUNDED LINEAR MAPS 35

The frame {S−1xn} is called the canonical or standard dual of {xn}. In thecase that {xn} is a Parseval frame for H, we have that S = I and hence x =∑

n∈N 〈 x, xn 〉xn, ∀x ∈ H. More generally, if a Bessel sequence {yn} satisfiesx =

∑n∈N 〈x, yn 〉xn, ∀x ∈ H, where the convergence is in norm of H, then {yn}

is called an alternate dual of {xn}. (Then {yn} is also necessarily a frame.) Thecanonical and alternate duals are often simply referred to as duals, and {xn, yn} iscalled a dual frame pair. It is a well-known fact that that a frame {xn} is a Rieszbasis if and only if {xn} has a unique dual frame.

There is a geometric interpretation of Parseval frames and general frames. LetP be an orthogonal projection from a Hilbert space K onto a closed subspace H, andlet {un} be a sequence in K. Then {Pun} is called the orthogonal compression of{un} under P , and correspondingly {un} is called an orthogonal dilation of {Pun}.We first observe that if {un} is a frame for K, then {Pun} is a frame for H withframe bounds at least as good as those of {un} (in the sense that the lower framecannot decrease and the upper bound cannot increase). In particular, {Pun} is aParseval frame forH when {un} is an orthonormal basis for K; i.e., every orthogonalcompression of an orthonormal basis (resp. Riesz basis) is a Parseval frame (resp.frame) for the projection subspace. The converse is also true: Every frame can beorthogonally dilated to a Riesz basis, and every Parseval frame can be dilated to anorthonormal basis. This was apparently first shown explicitly by Han and Larsonin Chapter 1 of [17]. There, with appropriate definitions it had an elementarytwo-line proof. And as noted by several authors, it can be alternately derived byapplying the Naimark (Neumark) Dilation theorem for operator-valued measuresby first passing from a frame sequence to a natural discrete positive operator-valuedmeasure on the power set of the index set. So it is sometimes referred to as theNaimark dilation theorem for frames. In fact, this is the observation that inspiredmuch of the work in [18].

For completeness we formally state this result:

Proposition 2.1. [17] Let {xn} be a sequence in a Hilbert space H. Then

(1) {xn} is a Parseval frame for H if and only if there exists a Hilbert spaceK ⊇ H and an orthonormal basis {un} for K such that xn = Pun, whereP is the orthogonal projection from K onto H.

(2) {xn} is a frame for H if and only if there exists a Hilbert space K ⊇ Hand a Riesz basis {vn} for K such that xn = Pvn, where P again is theorthogonal projection from K onto H.

The above dilation result was later generalized in [4] to dual frame pairs.

Theorem 2.2. Suppose that {xn} and {yn} are two frames for a Hilbert spaceH. Then the following are equivalent:

(1) {yn} is a dual for {xn};(2) There exists a Hilbert space K ⊇ H and a Riesz basis {un} for K such

that xn = Pun, and yn = Pu∗n, where {u∗

n} is the (unique) dual of theRiesz basis {un} and P is the orthogonal projection from K onto H.

As in [4], a framing for a Banach space X is a pair of sequences {xi, yi} with{xi} in X, {yi} in the dual space X∗ of X, satisfying the condition that

x =∑i

〈x, yi〉xi,


where this series converges unconditionally for all x ∈ X.The definition of a framing is a natural generalization of the definition of a

dual frame pair. Assume that {xi} is a frame for H and {yi} is a dual frame for{xi}. Then {xi, yi} is clearly a framing for H. Moreover, if αi is a sequence of non-zero constants, then {αixi, α

−1i yi} (called a rescaling of the pair) is also a framing,

although it is easy to show that it need not be a pair of frames, even if {αixi},{α−1

i yi} are bounded sequence.We recall that a sequence {zi} in a Banach space Z is called a Schauder basis

(or just a basis) for Z if for each z ∈ Z there is a unique sequence of scalars {αi}so that z =

∑i αizi. The unique elements z∗i ∈ Z∗ satisfying

(2.4) z =∑i

z∗i (z)zi,

for all z ∈ Z, are called the dual (or biorthogonal) functionals for {zi}. If the seriesin (2.4) converges unconditionally for every z ∈ Z, we call {zi, z∗i } an unconditionalbasis for Z.

We also have an unconditional basis constant for an unconditional basis givenby:

UBC(zi) = sup{‖∑i

bizi‖ : ‖∑i

aizi‖ = 1, |bi| ≤ |ai|, ∀i}.

If {zi, z∗i } is an unconditional basis for Z, we can define an equivalent norm on Zby:

‖∑i

aizi‖1 = sup{‖∑i

biaizi‖ : |bi| ≤ 1, ∀i}.

Then {zi, z∗i } is an unconditional basis for Z with UBC(zi) = 1. In this case wejust call {zi} a 1-unconditional basis for Z.

Definition 2.3. [4] A sequence {xi}i∈N in a Banach space X is a projectiveframe for X if there is a Banach space Z with an unconditional basis {zi, z∗i } withX ⊂ Z and a (onto) projection P : Z → X so that Pzi = xi for all i ∈ N. If {zi} isa 1-unconditional basis for Z and ‖P‖ = 1, we will call {xi} a projective Parsevalframe for X.

In this case, we have for all x ∈ X that

x =∑i

〈x, z∗i 〉zi = Px =∑i

〈x, z∗i 〉Pzi =∑i

〈x, z∗i 〉xi,

and this series converges unconditionally in X. So this definition recaptures theunconditional convergence from the Hilbert space definition.

We note that there exist projective frames in the sense of Definition 2.3 foran infinite dimensional Hilbert space that fail to be frames. We think they occurin abundance, but specific examples are hard to prove. A concrete example iscontained in [18, Chapter 5].

Definition 2.4. [4] A framing model is a Banach space Z with a fixed uncon-ditional basis {ei} for Z. A framing modeled on (Z, {ei}i∈N) for a Banach space Xis a pair of sequences {yi} in X∗ and {xi} in X so that the operator θ : X → Zdefined by

θu =∑i∈N

〈u, yi〉ei,


is an into isomorphism and Γ : Z → X given by

Γ(∑i∈N

aiei) =∑i∈N

aixi

is bounded and Γθ = IX .

In this setting, Γ becomes the reconstruction operator for the frame. Thefollowing result due to Casazza, Han and Larson [4] shows that these three methodsfor defining a frame on a Banach space are really the same.

Proposition 2.5. Let X be a Banach space and {xi} be a sequence of elementsof X. The following are equivalent:

(1) {xi} is a projective frame for X.(2) There exists a sequence yi ∈ X∗ so that {xi, yi} is a framing for X.(3) There exists a sequence yi ∈ X∗ and a framing model (Z, {ei}) so that

{xi, yi} is a framing modeled on (Z, {ei}).The proof of the implication from (1) to (2) is trivial: If {zi} is an unconditional

basis for a Banach space Z and P is a bounded projection from Z to a closedsubspace X with xi = Pei, then (xi, yi) is a framing for X, where yi = P ∗z∗i and{z∗i } is the (unique) dual basis of {zi}. One of the main contributions of paper [4]was to show that every framing can be obtained in this way.

Theorem 2.6 (Corollary 4.7 of [4]). Suppose that {xi, yi} is a framing for X.Then there exist a Banach space Z containing X and an unconditional basis {zi, z∗i }for Z such that xi = Pzi and yi = P ∗z∗i , where P is a bounded projection from Zonto X.

The definition of (discrete) frames has a natural generalization.

Definition 2.7. Let H be a separable Hilbert space and Ω be a σ-locallycompact (σ-compact and locally compact) Hausdorff space endowed with a positiveRadon measure μ with supp(μ) = Ω. A weakly continuous function F : Ω → H iscalled a continuous frame if there exist constants 0 < C1 ≤ C2 < ∞ such that

C1‖x‖2 ≤∫Ω

|〈x,F(ω)〉|2dμ(ω) ≤ C2‖x‖2, ∀x ∈ H.

If C1 = C2 then the frame is called tight. Associated to F is the frame operatorSF : H → H defined in the weak sense by

〈SF(x), y〉 :=∫Ω

〈x,F(ω)〉 · 〈F(ω), y〉dμ(ω).

It follows from the definition that SF is a bounded, positive, and invertibleoperator. We define the following transform associated to F ,

VF : H → L2(Ω, μ), VF (x)(ω) := 〈x,F(ω)〉.This operator is called the analysis operator in the literature and its adjoint operatoris given by

V ∗F : L2(Ω, μ) → H, 〈V ∗

F (f), x〉 :=∫Ω

f(ω)〈F(ω), x〉dμ(ω).

Then we have SF = V ∗FVF , and

〈x, y〉 =∫Ω

〈x,F(ω)〉 · 〈G(ω), y〉dμ(ω),(2.5)


where G(ω) := S−1F F(ω) is the standard dual of F . A weakly continuous function

F : Ω → H is called Bessel if there exists a positive constant C such that∫Ω

|〈x,F(ω)〉|2dμ(ω) ≤ C‖x‖2, ∀x ∈ H.

It can be easily shown that if F : Ω → H is Bessel, then it is a frame for H ifand only if there exists a Bessel mapping G such that the reconstruction formula(2.5) holds. This G may not be the standard dual of F . We will call (F ,G) a dualpair.

A discrete frame is a Riesz basis if and only if its analysis operator is surjective.But for a continuous frame F , in general we don’t have the dilation space to beL2(Ω, μ). In fact, this could happen only when μ is purely atomic. Thereforethere is no Riesz basis type dilation theory for continuous frames (however, wewill see later that in contrast the induced operator-valued measure does have aprojection valued measure dilation). The following modified dilation theorem isdue to Gabardo and Han [12]:

Theorem 2.8. Let F be a (Ω, μ)-frame for H and G be one of its duals. Supposethat both VF (H) and VG(H) are contained in the range space M of the analysisoperator for some (Ω, μ)-frame. Then there is a Hilbert space K ⊃ H and a (Ω, μ)-

frame F for K with P F = F , P G = G and VF (H) = M, where G is the standard

dual of F and P is the orthogonal projection from K onto H.

Let Ω be a compact Hausdorff space, and let B be the σ-algebra of all the Borelsubsets of Ω. A B(H)-valued measure on Ω is a mapping E : B → B(H) that isweakly countably additive; i.e., if {Bi} is a countable collection of disjoint Borelsets with union B, then

〈E(B)x, y〉 =∑i

〈E(Bi)x, y〉

holds for all x, y in H. The measure is called bounded provided that

sup{‖E(B)‖ : B ∈ B} < ∞,

and we let ‖E‖ denote this supremum. The measure is called regular if for all x, yin H, the complex measure given by

μx,y(B) = 〈E(B)x, y〉(2.6)

is regular.Given a regular bounded B(H)-valued measure E, one obtains a bounded,

linear map

φE : C(Ω) → B(H)

by

〈φE(f)x, y〉 =∫Ω

f d μx,y.(2.7)

Conversely, given a bounded, linear map φ : C(Ω) → B(H), if one definesregular Borel measures {μx,y} for each x, y in H by the above formula (2.7), thenfor each Borel set B, there exists a unique, bounded operator E(B), defined byformula (2.6), and the map B → E(B) defines a bounded, regular B(H)-valuedmeasure. There is a one-to-one correspondence between the bounded, linear maps


of C(Ω) into B(H) and the regular bounded B(H)-valued measures. Such measuresare called

(i) spectral if E(B1 ∩B2) = E(B1) · E(B2),(ii) positive if E(B) ≥ 0,(iii) self-adjoint if E(B)∗ = E(B),

for all Borel sets B,B1 and B2. Note that if E is spectral and self-adjoint, thenE(B) must be an orthogonal projection for all B ∈ B, and hence E is positive.

In the commutative C∗ theory, compactness is usually used as above becausewhen viewing a unital C∗-algebra as C(Ω) there is no loss in generality in takingΩ to be compact, because if needed it can be taken to be βΩ – the Stone-Cechcompactification of Ω. This is because the C∗-algebras C(Ω) and C(βΩ) are ∗-isomorphic. Having Ω compact makes the integration theory representation oflinear maps and the connection between linear maps on C(Ω) and operator-valuedmeasures very elegant.

But in our theory, the basic connection to frame theory is essentially lost ifwe replace the index set of the frame with its Stone-cech compactification. Inthe continous frame case it is more natural to assume Ω is σ-locally compact (asin Definition 2.7), and in the general dilation theory we need to use the generalmeasurable space setting (as in Definition 3.2) to preserve our basic connectionswith the frame theory.

Both discrete and continuous framings induce operator-valued measures in anatural way.

Example 2.9. Let {xi}i∈J be a frame for a separable Hilbert space H. Let Σbe the σ-algebra of all subsets of J. Define the mapping

E : Σ → B(H), E(B) =∑i∈B

xi ⊗ xi

where x ⊗ y is the mapping on H defined by (x ⊗ y)(u) = 〈u, y〉x. Then E is aregular, positive B(H)-valued measure.

Similarly, suppose that {xi, yi}i∈J is a non-zero framing for a separable Hilbertspace H. Define the mapping

E : Σ → B(H), E(B) =∑i∈B

xi ⊗ yi,

for all B ∈ Σ. Then E is a B(H)-valued measure.

Example 2.10. Let X be a Banach space and Ω be a σ-locally compact Haus-dorff space. Let μ be a Borel measure on Ω. A continuous framing on X is a pairof maps (F ,G),

F : Ω → X, G : Ω → X∗,

such that the equation⟨E(F ,G)(B)x, y

⟩=

∫B

〈x,G(ω)〉〈F(ω), y〉 dμ(ω)

for x ∈ X, y ∈ X∗, and B a Borel subset of Ω, defines an operator-valued probabilitymeasure on Ω taking value in B(X). In particular, we require the integral on theright to converge for each B ⊂ Ω. We have

E(F ,G)(B) =

∫B

F(ω)⊗ G(ω) dE(ω)(2.8)


where the integral converges in the sense of Bochner. In particular, since E(F ,G)(Ω) =IX , we have for any x ∈ X that

〈x, y〉 =∫Ω

〈x,G(ω)〉〈F(ω), y〉 dE(ω).

Remark 2.11. We point out that there exists an operator space with a (finitedimensional) projection valued (purely atomic) probability measure that does notadmit a framing. Let X be the space of all compact operators T on �2 which havea triangular representing matrix with respect to the unit vector basis; i.e.,

Ten =

n∑m=1

an,mem

for all n ∈ N. Let Xn be the subspace of X consisting of those T ∈ X such thatTej = 0 for j = n (i.e., for which aj,m = 0 unless j = n). It is clear that Xn isisometric to �n2 , n = 1, 2, . . .. Moreover, it is trivial to check that {Xn}∞n=1 formsan unconditional finite dimensional decomposition of X which naturally induces aprojection valued probability measure. Let Pn be the canonical projection from Xonto Xn satisfy:

(i) dim(Pn(X)) = dim(Xn) = n for all n ∈ N;(ii) PnPm = PmPn = 0 for any n = m ∈ N;(iii) x =

∑∞n=1 Pn(x) for every x ∈ X.

Let Σ be the σ-algebra of all subsets of N. Define E : Σ → B(X) by E(B) =∑n∈B Pn. Then E is a projection valued probability measure with dim(E({n})) =

n. Nevertheless, it follows from the results of [13] that X does not have an uncon-ditional basis and it is not even complemented in a space with an unconditionalbasis. Thus, by Proposition 2.5, X does not have a framing.

Let A be a unital C∗- algebra. An operator-valued linear map φ : A → B(H)is said to be positive if φ(a∗a) ≥ 0 for every a ∈ A, and it is called completelypositive (cp for abbreviation) if for every n-tuple a1, ..., an of elements in A, thematrix (φ(a∗i aj)) is positive in the usual sense that for every n-tuple of vectorsξ1, ..., ξn ∈ H, we have

n∑i,j=1

〈φ(aia∗j )ξj , ξi〉 ≥ 0(2.9)

or equivalently, (φ(a∗i aj)) is a positive operator on the Hilbert space H⊗Cn ([23]).Let A be a C∗- algebra. We use Mn to denote the set of all n × n complex

matrices, and Mn(A) to denote the set of all n × n matrices with entries from A.Given two C∗-algebras A and B and a map φ : A → B, obtain maps φn : Mn(A) →Mn(B) via the formula

φn((ai,j)) = (φ(ai,j)).

The map φ is called completely bounded (cb for abbreviation) if φ is bounded and‖φ‖cb = supn ‖φn‖ is finite.

3. Dilations of Operator-Valued Measures

Possibly the first well-known dilation result for operator-valued measures is dueto Naimark.


Theorem 3.1 (Naimark’s Dilation Theorem). Let E be a regular, positive,B(H)-valued measure on Ω. Then there exist a Hilbert space K, a bounded linearoperator V : H → K, and a regular, self-adjoint, spectral, B(K)-valued measure Fon Ω, such that

E(B) = V ∗F (B)V.

From Naimark’s dilation Theorem, we know that every regular positive operator-valued measure (OVM for abbreviation) can be dilated to a self-adjoint, spectraloperator-valued measure on a larger Hilbert space. But not all of the operator-valued measures can have a Hilbert dilation space. Such an example was con-structed in [18] in which we constructed an operator-valued measure induced bythe framing that does not have a Hilbert dilation space. The construction is basedon an example of Osaka [22] of a normal non-completely bounded map from �∞(N)into B(H). In fact operator-valued measures that admit Hilbert space dilations arethe ones that are closely related to completely bounded measures and maps.

Now let Ω be a compact Hausdorff space, let E be a bounded, regular, operator-valued measure on Ω, and let φ : C(Ω) → B(H) be the bounded, linear mapassociated with E by integration. So for any f ∈ C(Ω),

〈φ(f)x, y〉 =∫Ω

f d μx,y,

whereμx,y(B) = 〈E(B)x, y〉

The OVM E is called completely bounded when φ is completely bounded. UsingWittstock’s decomposition theorem, E is completely bounded if and only if it canbe expressed as a linear combination of positive operator-valued measures.

Let {xi}i∈J be a non-zero frame for a separable Hilbert space H. Let Σ be theσ-algebra of all subsets of J, and

E : Σ → B(H), E(B) =∑i∈B

xi ⊗ xi

Since E is a regular, positive B(H)-valued measure, by Naimark’s dilation Theorem3.1, there exists a Hilbert space K, a bounded linear operator V : H → K, and aregular, self-adjoint, spectral, B(K)-valued measure F on J, such that

E(B) = V ∗F (B)V.

This Hilbert space K can be �2, and the atoms xi ⊗ xi of the measure dilate torank-1 projections ei ⊗ ei, where {ei} is the standard orthonormal basis for �2.That is K can be the same as the dilation space in Proposition 2.1 (ii).

In the case that {xi, yi}i∈J is a non-zero framing for a separable Hilbert spaceH, and E(B) =

∑i∈B xi ⊗ yi for all B ∈ Σ, E is a B(H)-valued measure. In [18]

we showed that this E also has a dilation space Z. But this dilation space is notnecessarily a Hilbert space, in general, it is a Banach space and consistent withProposition 2.5. The dilation is essentially constructed using Proposition 2.5 (ii),where the dilation of the atoms xi ⊗ yi corresponds to the projection ui ⊗ u∗

i and{ui} is an unconditional basis for the dilation space Z.

Framings are the natural generalization of discrete frame theory (more specifi-cally, dual-frame pairs) to non-Hilbertian settings. Even if the underlying space is aHilbert space, the dilation space for framing induced-operator-valued measures canfail to be Hilbertian. This theory was originally developed by Casazza, Han and


Larson in [4] as an attempt to introduce frame theory with dilations into a Banachspace context. The initial motivation of this investigation was to completely under-stand the dilation theory of framings. In the context of Hilbert spaces, we realizedthat the dilation theory for discrete framings from [4] induces a dilation theory fordiscrete operator-valued measures that may fail to be completely bounded.

These examples inspired us to consider Banach space dilation theory for arbi-trary operator-valued measures.

Definition 3.2. LetX and Y be Banach spaces, and let (Ω,Σ) be a measurablespace. A B(X,Y )-valued measure on Ω is a map E : Σ → B(X,Y ) that is countablyadditive in the weak operator topology; that is, if {Bi} is a disjoint countablecollection of members of Σ with union B, then

y∗(E(B)x) =∑i

y∗(E(Bi)x)

for all x ∈ X and y∗ ∈ Y ∗.

We will use the symbol (Ω,Σ, E) if the range space is clear from context, or(Ω,Σ, E,B(X,Y )), to denote this operator-valued measure system.

The Orlicz–Pettis theorem states that weak unconditional convergence andnorm unconditional convergence of a series are the same in every Banach space (c.f.[8]). Thus we have that

∑i E(Bi)x weakly unconditionally converges to E(B)x if

and only if∑

i E(Bi)x strongly unconditionally converges to E(B)x. So Definition3.2 is equivalent to saying that E is strongly countably additive, that is, if {Bi} isa disjoint countable collection of members of Σ with union B, then

E(B)x =∑i

E(Bi)x, ∀x ∈ X.

Definition 3.3. Let E be a B(X,Y )-valued measure on (Ω,Σ). Then the normof E is defined by

‖E‖ = supB∈Σ

‖E(B)‖.

We call E normalized if ‖E‖ = 1.

A B(X,Y )-valued measure E is always bounded; i.e.,

(3.1) supB∈Σ

‖E(B)‖ < +∞.

Indeed, for all x ∈ X and y∗ ∈ Y ∗, μx,y∗(B) := y∗(E(B)x) is a complex measure on(Ω,Σ). From complex measure theory (c.f. [25]), we know that μx,y∗ is bounded;i.e.,

supB∈Σ

|y∗(E(B)x)| < +∞.

By the Uniform Boundedness Principle, we get (3.1).Similar to the Hilbert space operator-alued measures, we introduce the follow-

ing definitions.

Definition 3.4. A B(X)-valued measure E on (Ω,Σ) is called:

(i) an operator-valued probability measure if E(Ω) = IX ,(ii) a projection-valued measure if E(B) is a projection on X for all B ∈ Σ,(iii) a spectral operator-valued measure if for all A,B ∈ Σ, E(A ∩B) = E(A) ·

E(B) (we will also use the term idempotent-valued measure to mean aspectral-valued measure.)


For general operator-valued measures we established the following dilation the-orem [18].

Theorem 3.5. Let E : Σ → B(X,Y ) be an operator-valued measure. Thenthere exist a Banach space Z, bounded linear operators S : Z → Y and T : X → Z,and a projection-valued probability measure F : Σ → B(Z) such that

E(B) = SF (B)T

for all B ∈ Σ.

We will call (F,Z, S, T ) in the above theorem a Banach space dilation system,and a Hilbert dilation system if Z can be taken as a Hilbert space. This theoremgeneralizes Naimark’s (Neumark’s) Dilation Theorem for positive operator-valuedmeasures. But even in the case that the underlying space is a Hilbert space thedilation space cannot always be taken to be a Hilbert space. Thus elements of thetheory of Banach spaces are essential in this work.

A key idea is the introduction of the elementary dilation space and and theminimal dilation norm.

Let X,Y be Banach spaces and (Ω,Σ, E,B(X,Y )) an operator-valued measuresystem. For any B ∈ Σ and x ∈ X, define

EB,x : Σ → Y, EB,x(A) = E(B ∩A)x, ∀A ∈ Σ.

Then it is easy to see that EB,x is a vector-valued measure on (Ω,Σ) of Y andEB,x ∈ MY

Σ .Let ME = span{EB,x : x ∈ X,B ∈ Σ}. We introduce some linear mappings on

the spaces X, Y and ME .For any {Ci}Ni=1 ⊂ C, {Bi}Ni=1 ⊂ Σ and {xi}Ni=1 ⊂ X, the mappings

S : ME → Y, S( N∑

i=1

CiEBi,xi

)=

N∑i=1

CiE(Bi)xi

T : X → ME , T (x) = EΩ,x

and

F (B) : ME → ME , F (B)( N∑

i=1

CiEBi,xi

)=

N∑i=1

CiEB∩Bi,xi, ∀B ∈ Σ

are well-defined and linear.

Definition 3.6. Let ME be the space induced by (Ω,Σ, E,B(X,Y )). Let ‖ · ‖be a norm on ME . Denote this normed space by ME,‖·‖ and its completion ME,‖·‖.

The norm on ME,‖·‖, with ‖ ·‖ := ‖ ·‖D given by a norming function D as discussedabove, is called a dilation norm of E if the following conditions are satisfied:

(i) The mapping SD : ME,D → Y defined on ME by

SD

( N∑i=1

CiEBi,xi

)=

N∑i=1

CiE(Bi)xi

is bounded.(ii) The mapping TD : X → ME,D defined by

TD(x) = EΩ,x

is bounded.


(iii) The mapping FD : Σ → B(ME,D) defined by

FD(B)( N∑

i=1

CiEBi,xi

)=

N∑i=1

CiEB∩Bi,xi

is an operator-valued measure,

where {Ci}Ni=1 ⊂ C, {xi}Ni=1 ⊂ X and {Bi}Ni=1 ⊂ Σ.

We call the Banach space ME,D the elementary dilation space of E and

(Ω,Σ, FD, B(ME,D), SD, TD)

the elementary dilation operator-valued measure system. The minimal dilationnorm ‖ · ‖α on ME is defined by∥∥∥ N∑

i=1

CiEBi,xi

∥∥∥α= sup

B∈Σ

∥∥∥ N∑i=1

CiE(B ∩Bi)xi

∥∥∥Y

for all∑N

i=1 CiEBi,xi∈ ME . Using this we show that every OVM has a projection

valued dilation to an elementary dilation space, and moreover, || · ||α is a minimalnorm on the elementary dilation space.

A corresponding dilation projection-valued measure system (Ω,Σ, F,B(Z), S, T )is said to be injective if

∑F (Bi)T (xi) = 0 whenever

∑EBi,xi

= 0.It is useful to note that all the elementary dilation spaces are Banach spaces of

functions.

Theorem 3.7. Let E : Σ → B(X,Y ) be an operator-valued measure and(F,Z, S, T ) be an injective Banach space dilation system. Then we have the fol-lowing:

(i) There exists an elementary Banach space dilation system (FD, ME,D, SD, TD)of E and a linear isometric embedding

U : ME,D → Z

such that

SD = SU, F (Ω)T = UTD, UFD(B) = F (B)U, ∀B ∈ Σ.

(ii) The norm ‖·‖α is indeed a dilation norm. Moreover, If D is a dilation norm

of E, then there exists a constant CD such that for any∑N

i=1 CiEBi,xi∈ ME,D,

supB∈Σ

∥∥∥ N∑i=1

CiE(B ∩Bi)xi

∥∥∥Y

≤ CD

∥∥∥ N∑i=1

CiEBi,xi

∥∥∥D,

where N > 0, {Ci}Ni=1 ⊂ C, {xi}Ni=1 ⊂ X and {Bi}Ni=1 ⊂ Σ. Consequently,

‖f‖α ≤ CD‖f‖D, ∀f ∈ ME .

Definition 3.8. Let E : Σ → B(X,Y ) be an operator-valued measure and(F,Z, S, T ) be a Banach space dilation system. Then (F,Z, S, T ) is called linearlyminimal if Z is the closed linear span of F (Σ)TX, where F (Σ)TX = {F (B)(Tx) :B ∈ Σ, x ∈ X}.


A projection valued measure can have a nontrivial linearly minimal dilation toanother projection valued measure. The following simple example illustrates this.It is also an example of a dilation projection-valued measure system which is notinjective and for which the conclusion of Theorem 3.7 is not true. This shows thatif we drop the “injectivity” in the hypothesis of Theorem 3.7 the conclusion neednot be true. However, a simple modification of the conclusion will be true (seeRemark 3.10).

Example 3.9. Let (Ω,Σ, μ) be a probability space and let ν be a finite measurethat dominates μ. Let X := L2(Ω, μ) and let Y := L2(Ω, ν). Let α be a boundedlinear functional on X that takes 1 at the function η = 1. Let Ω = Ωc

0 ∪ Ω0 be theHahn decomposition, where Ω0 is a measurable subset of Ω which is a null set forμ and which supports the singular part of ν with respect to μ. Regard L2(Ω, ν) asthe direct sum of L2(Ω, μ) and L2(Ω0, ν). Embed X into Y by T (f) = f⊕α(f)χΩ0

,where χΩ0

is the constant function 1 in L2(Ω0, ν). Since α is a linear functionalT is a linear map. In particular it maps the constant function 1 in X := L2(Ω, μ)to the constant function 1 in Y := L2(Ω, ν). Define a projection valued measureφ : Σ → B(X) by setting φ(B) = MχB

, the projection operator of multiplication bythe characteristic function of B. Do the same construction to define a projectionvalued measure Φ : Σ → B(Y ). Since TX contains the constant function 1 inL2(Ω, ν), the closed linear span of Φ(Σ)TX is Y .

Let S denote the mapping of Y := L2(Ω, ν) onto X := L2(Ω, μ) determined bythe function mapping f → f |Ω0

c . Then S has kernel L2(Ω0, ν).Then Φ is a dilation of φ for the dilation maps T and S, and the dilation is

linearly minimal because the closed linear span of Φ(Σ)TX is Y . The dilation isclearly non-injective, and the conclusion of Theorem 3.7 fails for it.

Remark 3.10. We have the following natural generalization of Theorem 3.7:Let all terms be as in the hypotheses of Theorem 3.7 except do not assume thatthe Banach space dilation system (F,Z, S, T ) is injective. First, obtain a reductionif necessary by restricting the range space of F so that the closure of the rangeof F times the range of T is all of Z. This makes the dilation linearly minimal.Example 3.9 shows that this reduction to linearly minimal is not alone sufficient togeneralize Theorem 3.7. Obtain a second reduction by replacing Z with its quotientby the kernel of S. Then the hypotheses of Theorem 3.7 are satisfied, so we canobtain a generalization of Theorem 3.7 by removing the injectivity requirement inthe hypothesis and inserting the restriction reduction followed by the quotient re-duction in the statement of the conclusion. In Example 3.9 the restriction reductionis unnecessary because the dilation is already linearly minimal, and the quotientreduction makes the reduced dilation equivalent to φ.

The point of this is that the minimal elementary norm dilation of this sectionis really a geometrically minimal dilation in the sense that any dilation, after asimple restriction reduction and a quotient reduction if necessary, is isometricallyisomorphic to an elementary dilation norm dilation. And the class of elementarydilation norm spaces are related in the sense that there is a minimal dilation normand a maximal dilation norm, and all dilation norms lie between the minimal andthe maximal norm on the elementary function space, and the actual dilation spaceis the completion of the elementary function space in one of the dilation norms. Soin this sense the minimal norm elementary dilation of an operator-valued measureis subordinate to all other dilations of the OVM.


While in general an operator-valued probability measure does not admit aHilbert space dilation, the dilation theory can be strengthened in the case thatit does admit a Hilbert space dilation:

Theorem 3.11. Let E : Σ → B(H) be an operator-valued probability measure.

If E has a Hilbert dilation system (E, H, S, T ), then there exists a correspondingHilbert dilation system (F,K, V ∗, V ) such that V : H → K is an isometric embed-ding.

This theorem turns out to have some interesting applications to framing induced-operator-valued measure dilation. In particular, it led to a complete characteriza-tion of framings whose induced operator valued measures are completely bounded.We include here a few sample examples with the following theorem:

Theorem 3.12. Let (xi, yi)i∈N be a non-zero framing for a Hilbert space H,and E be the operator-valued probability measure induced by (xi, yi)i∈N. Then wehave the following:

(i) E has a Hilbert dilation space K if and only if there exist αi, βi ∈ C, i ∈ Nwith αiβi = 1 such that {αixi}i∈N and {βiyi}i∈N both are the frames for the Hilbertspace H.

(ii) E is a completely bounded map if and only if {xi, yi}i∈N can be re-scaled todual frames.

(iii) If inf ‖xi‖ · ‖yi‖ > 0, then we can find αi, βi ∈ C, i ∈ N with αiβi = 1such that {αixi}i∈N and {βiyi}i∈N both are frames for the Hilbert space H. Hence,the operator-valued measure induced by {xi, yi}i∈N has a Hilbertian dilation.

For the existence of non-rescalable (to dual frame pairs) framings, we obtained thefollowing:

Theorem 3.13. There exists a framing for a Hilbert space such that its inducedoperator-valued measure is not completely bounded, and consequently it can not bere-scaled to obtain a framing that admits a Hilbert space dilation.

The second part of this theorem follows from the first part of Theorem 3.12(ii).

Remark 3.14. For the existence of such an example, the motivating exam-ple of framing constructed by Casazza, Han and Larson (Example 3.9 in [4]) cannot be dilated to an unconditional basis for a Hilbert space, although it can bedilated to an unconditional basis for a Banach space. We originally conjecturedthat this is an example that fails to induce a completely bounded operator-valuedmeasure. However it turns out that this framing can be re-scaled to a framing thatadmits a Hilbert space dilation , and consequently disproves our conjecture. Ourconstruction of the new example in Theorem 3.13 uses a non-completely boundedmap to construct a non-completely bounded OVM which yields the required fram-ing. This delimiting example shows that the dilation theory for framings developedin [4] gives a true generalization of Naimark’s Dilation Theorem for the discretecase. This is the example that led us to consider general (non-necessarily-discrete)operator-valued measures, and to the results of Chapter 2 that lead to the dilationtheory for general (not necessarily completely bounded) OVM’s that completelygeneralizes Naimark’s Dilation theorem in a Banach space setting, and which isnew even for Hilbert spaces.


Part (iii) of Theorem 3.12 provides us a sufficient condition under which aframing induced operator-valued measure has a Hilbert space dilation. This canbe applied to framings that have nice structures. For example, the following isan unexpected result for unitary system induced framings, where a unitary systemis a countable collection of unitary operators. This clearly applies to wavelet andGabor systems.

Corollary 3.15. Let U1 and U2 be unitary systems on a separable Hilbertspace H. If there exist x, y ∈ H such that {U1x,U2y} is a framing of H, then {U1x}and {U2y} both are frames for H.

There exist examples of sequences {xn} and {yn} in a Hilbert space H withthe following properties:

(i) x =∑

n〈x, xn〉yn hold for all x in a dense subset of H, and the convergenceis unconditionally.

(ii) inf ||xn|| · ||yn|| > 0.(iii) {xn, yn} is not a framing.

Example 3.16. Let H = L2[0, 1], and g(t) = t1/4, f(t) = 1/g(t). Definexn(t) = e2πintf(t) and yn(t) = e2πintg(t). Then it is easy to verify (i) and (ii). For(iii), we consider the convergence of the series∑

n∈Z

〈f, xn〉yn.

Note that ||〈f, xn〉yn||2 = |〈f, xn〉|2 · ||g||2 and {〈f, xn〉} is not in �2 (since f2 /∈L2[0, 1]). Thus,

∑n〈f, xn〉yn can not be convergent unconditionally. Therefore,

{xn, yn} is not a framing.

4. Dilations of Bounded Linear Maps

Inspired by the techniques used to build the dilation theory for general operator-valued measures we consider establishing a dilation theory for general linear maps.Historically the dilation theory has been extensively investigated in the context ofpositive, or completely bounded maps on C*-algebras, with Stinespring’s dilationtheorem as possibly one of the most notable results in this direction (c.f. [1,23]and the references therein).

Theorem 4.1. [Stinespring’s dilation theorem] Let A be a unital C∗-algebra,and let φ : A → B(H) be a completely positive map. Then there exists a Hilbertspace K, a unital ∗−homomorphism π : A → B(K), and a bounded operator V :H → K with ‖φ(1)‖ = ‖V ‖2 such that

φ(a) = V ∗π(a)V.

The following is also well known for commutative C∗-algebras:

Theorem 4.2 (cf. Theorem 3.11, [23]). Let B be a C∗-algebra, and let φ :C(Ω) → B be positive. Then φ is completely positive.

This result together with Theorem 4.1 implies that Stinespring’s dilation the-orem holds for positive maps when A is a unital commutative C∗-algebra.

A proof of Naimark’s dilation theorem by using Stinespring’s dilation theoremcan be sketched as follows: Let φ : A → B(H) be the natural extension of E to theC∗-algebra A generated by all the characteristic functions of measurable subsets


of Ω. Then φ is positive, and hence is completely positive by Theorem 4.2. ApplyStinespring’s dilation theorem to obtain a ∗−homomorphism π : A → B(K), and abounded, linear operator V : H → K such that φ(f) = V ∗π(f)V for all f in A. LetF be the B(K)−valued measure corresponding to π. Then it can be verified thatF has the desired properties.

Completely positive maps are completely bounded. In the other direction wehave Wittstock’s decomposition theorem [23]:

Proposition 4.3. Let A be a unital C∗-algebra, and let φ : A → B(H) be acompletely bounded map. Then φ is a linear combination of two completely positivemaps.

The following is a generalization of Stinespring’s representation theorem.

Theorem 4.4. Let A be a unital C∗-algebra, and let φ : A → B(H) be acompletely bounded map. Then there exists a Hilbert space K, a ∗−homomorphismπ : A → B(K), and bounded operators Vi : H → K, i = 1, 2, with ‖φ‖cb = ‖V1‖·‖V2‖such that

φ(a) = V ∗1 π(a)V2

for all a ∈ A. Moreover, if ‖φ‖cb = 1, then V1 and V2 may be taken to be isometries.

Now let Ω be a compact Hausdorff space, let E be a bounded, regular, operator-valued measure on Ω, and let φ : C(Ω) → B(H) be the bounded, linear mapassociated with E by integration as described in section 1.4.1. So for any f ∈ C(Ω),

〈φ(f)x, y〉 =∫Ω

f d μx,y,

where

μx,y(B) = 〈E(B)x, y〉.The OVM E is called completely bounded when φ is completely bounded.

Using Wittstock’s decomposition theorem, E is completely bounded if and only ifit can be expressed as a linear combination of positive operator-valued measures.

One of the important applications of our main dilation theorem is the dilationfor not necessarily cb-maps with appropriate continuity properties from a commu-tative von Neumann algebra into B(H). While the ultraweak topology on B(H) fora Hilbert space H is well-understood, we define the ultraweak topology on B(X)for a Banach space X through tensor products: Let X ⊗ Y be the tensor productof the Banach space X and Y. The projective norm on X ⊗ Y is defined by:

‖u‖∧ = inf{ n∑

i=1

‖xi‖‖yi‖ : u =n∑

i=1

xi ⊗ yi

}.

We will use X⊗∧Y to denote the tensor product X⊗Y endowed with the projectivenorm ‖ · ‖∧. Its completion will be denoted by X⊗Y. From [27] Section 2.2, for anyBanach spaces X and Y, we have the identification:

(X⊗Y )∗ = B(X,Y ∗).

Thus B(X,X∗∗) = (X⊗X∗)∗. Viewing X ⊆ X∗∗, we define the ultraweak topologyon B(X) to be the weak* topology induced by the predual X⊗X∗. We will alsouse the term normal to denote an ultraweakly continuous linear map.


Theorem 4.5. If A is a purely atomic abelian von Neumann algebra actingon a separable Hilbert space, then for every ultraweakly continuous linear mapφ : A → B(H), there exists a Banach space Z, an ultraweakly continuous uni-tal homomorphism π : A → B(Z), and bounded linear operators T : H → Z andS : Z → H such that

φ(a) = Sπ(a)T

for all a ∈ A.

The proof of this theorem uses some special properties of the minimal dilationsystem for the φ induced operator-valued measure on the space (N, 2N). Motivatedby some ideas used in the proof of the above theorem, we then obtained a universaldilation theorem for all bounded linear mappings between Banach algebras:

Theorem 4.6. Let A be a Banach algebra, and let φ : A → B(X) be a boundedlinear operator, where X is a Banach space. Then there exists a Banach space Z, abounded linear unital homomorphism π : A → B(Z), and bounded linear operatorsT : X → Z and S : Z → X such that

φ(a) = Sπ(a)T

for all a ∈ A.

Since this theorem is so general we would expect that there is a also purelyalgebraic dilation theorem for any linear transformations. This indeed is the case.

Proposition 4.7. If A is unital algebra, V a vector space, and φ : A → L(V )a linear map, then there exists a vector space W , a unital homomorphism π : A →L(V ), and linear maps T : V → W , S : W → V , such that

φ(·) = Sπ(·)T.

This result maybe well-known. However, we provide a short proof for interestedreaders.

Proof. For a ∈ A, x ∈ V , define αa,x ∈ L(A, V ) by

αa,x(·) := φ(·a)x.Let W := span{αa,x : a ∈ A, x ∈ V } ⊂ L(A, V ). Define π : A → L(W ) byπ(a)(αb,x) := αab,x. It is easy to see that π is a unital homomorphism. For x ∈ Vdefine T : V → L(A, V ) by Tx := αI,x = φ(·I)x = φ(·)x. Define S : W → Wby setting S(αa,x) := φ(a)x and extending linearly to W . If a ∈ A, x ∈ V arearbitrary, we have Sπ(a)Tx = Sπ(a)αI,x = Sαa,x = φ(a)x. Hence, φ = SπT. �

We note that the above proposition has been generalized by the second authorand F. Szafraniec [21] to the case where A is a unital semigroup.

Theorem 4.6 is a true generalization of our commutative theorem in an impor-tant special case, and generalizes some of our results for maps of commutative vonNeumann algebras to the case where the von Neumann algebra is non-commutative.

For the case when A is a von Neumann algebra acting on a separable Hilbertspace and φ is ultraweakly continuous (i.e., normal) we conjecture that the dilationspace Z can be taken to be separable and the dilation homomorphism π is alsoultraweakly continuous. While we are not able to confirm this conjecture we havethe following result. Here, SOT is the abbreviation of strong operator topology.


Theorem 4.8. Let K,H be Hilbert spaces, A ⊂ B(K) be a von Neumannalgebra, and φ : A → B(H) be a bounded linear operator which is ultraweakly-SOT

continuous on the unit ball BA of A. Then there exists a Banach space Z, a boundedlinear homeomorphism π : A → B(Z) which is SOT-SOT continuous on BA, andbounded linear operator T : H → Z and S : Z → H such that

φ(a) = Sπ(a)T

for all a ∈ A. If in addition that K,H are separable, then the Banach space Z canbe taken to be separable.

These results are apparently new for mappings of von Neumann algebras. Theygeneralize special cases of Stinespring’s Dilation Theorem. The standard discreteHilbert space frame theory is identified with the special case of our theory in whichthe domain algebra is abelian and purely atomic, the map is completely bounded,and the OVM is purely atomic and completely bounded with rank-1 atoms.

The universal dilation result has connections with Kadison’s similarity prob-lem for bounded homomorphisms between von Neumann algebras (see the Remark4.14). For example, if A belongs to one of the following classes: nuclear; A = B(H);A has no tracial states; A is commutative; II1-factor with Murry and von Neu-mann’s property Γ, then any non completely bounded map φ : A → B(H) cannever have a Hilbertian dilation (i.e. the dilation space Z can never be a Hilbertspace) since otherwise π : A → B(Z) would be similar to a *-homomorphism andhence completely bounded and so would be φ. On the other hand, if there existsa von Neumann algebra A and a non completely bounded map φ from A to B(H)that has a Hilbert space dilation: π : A → B(Z) (i.e., where Z is a Hilbert space),then π will be a counterexample to the Kadison’s similarity problem since in thiscase π is a homomorphisim that is not completely bounded and consequently cannot be similar to a *-homomorphisim.

5. Some Remarks and Problems

Remark 5.1. It is well known that there is a theory establishing a connectionbetween general bounded linear mappings from the C∗-algebra C(X) of continuousfunctions on a compact Hausdorf space X into B(H) and operator-valued measureson the sigma algebra of Borel subsets of X (c.f. [23]). If A is an abelian C∗-algebrathen A can be identified with C(X) for a topological space X and can also beidentified with C(βX) where βX is the Stone-Cech compactification of X. Thenthe support σ-algebra for the OVM is the sigma algebra of Borel subsets of βXwhich is enormous. However, in our generalized (commutative) framing theory Awill always be an abelian von Neumann algebra presented up front as L∞(Ω,Σ, μ),with Ω a topological space and Σ its algebra of Borel sets, and the maps on A intoB(H) are normal. In particular, to model the discrete frame and framing theoryΩ is a countable index set with the discrete topology (most often N), so Σ is itspower set, and μ is counting measure. So in this setting it is more natural to workdirectly with this presentation in developing dilation theory rather than passing toβΩ, and we took this approach in our investigation.

Remark 5.2. We feel that the connection we make with established discreteframe and framing theory is transparent, and then the OVM dilation theory forthe continuous case becomes a natural but nontrivial generalization of the theoryfor the discrete case that was inspired by framings. After doing this we attempted


to apply our techniques to the case where the domain algebra for a map is non-commutative. However, additional hypotheses are needed if dilations of maps are tohave strong continuity and structural properties. For a map between C*-algebrasit is well-known that there is a Hilbert space dilation if the map is completelybounded. (If the domain algebra is commutative this statement is an iff.) Even ifa map is not cb it has a Banach space dilation. We are interested in the continuityand structural properties a dilation can have. In the discrete abelian case, thedilation of a normal map can be taken to be normal and the dilation space canbe taken to be separable, and with suitable hypotheses this type of result can begeneralized to the noncommutative setting.

The following is a list of problems we think may be important for the generaldilation theory of operator-valued measures and bounded linear maps.

It was proven in [18] that if {i} is an atom in Σ and E is an operator-valuedframe on Σ, then the minimal dilation Fα has the property that the rank of Fα({i})is equal to the rank of E({i}). This leads to the following problem.

Problem 1. Is it always true that with an appropriate notion of rank functionfor an operator-valued measure, that r(Fα(B)) = r(E(B)) for every B ∈ Σ? Whatabout if a “rank” definition is defined by: r(B) = sup{rankE(A) : A ⊂ B,A ∈ Σ}?

Let (Ω,Σ, μ) be a probability space and let φ : L∞(μ) → B(H) be ultraweaklycontinuous. Then it naturally induces an operator-valued probablity measure

E(B) = φ(χB), ∀B ∈ Σ.

Problem 2. Let E : (Ω,Σ) → B(H) be an operator-valued measure. Is there anultraweakly continuous map φ : L∞(μ) → B(H) that induces E on (Ω,Σ)? If theanswer is negative, then determine necessary and sufficient conditions for E to beinduced by an ultraweakly continuous map?

As with Stinespring’s dilation theorem, if A and H in Theorem 4.1 are bothseparable then the dilated Banach space Z is also separable. However, the Banachalgebras we are interested in include von Neumann algebras and these are generallynot separable, and the linear maps φ : A → B(H) are often normal. So we posethe following two problems.

Problem 3. Let K,H be separable Hilbert spaces, let A ⊂ B(K) be a von Neu-mann algebra, and let φ : A → B(H) be a bounded linear map. When is there aseparable Banach space Z, a bounded linear unital homomorphism π : A → B(Z),and bounded linear operators T : H → Z and S : Z → H such that

φ(a) = Sπ(a)T

for all a ∈ A ?

Problem 4. Let A ⊂ B(K) be a von Neumann algebra, and φ : A → B(H) be anormal linear map. When can we dilate φ to a normal linear unital homomorphismπ : A → B(Z) for some (reflexive) Banach space Z?

Finally, concerning the Hilbert space dilations and Kadison’s Similarity Prob-lem, we are interested in the following questions:


Problem 5. Let A ⊂ B(K) be a von Neumann algebra, and let φ : A → B(H)be a bounded linear map. We know that φ has a Hilbert space dilation if it iscompletely bounded. Is there a non-completely bounded map that admits a Hilbertspace dilation? In particular, if φ(A) = At for any A ∈ ⊕∞

n=1Mn×n(C), then T isbounded but not completely bounded. What can we say about the dilation of φ?Does it admit a Hilbert space dilation?

Yes. An affirmative answer would yield a negative answer to the similarityproblem.

Problem 6. Let A ⊂ B(K) be a von Neumann algebra, and let φ : A → B(H) be abounded linear map. “Characterize” those maps that admit Hilbert space dilations,and “Characterize” those maps that admit reflexive Banach space dilations.

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[5] Man Duen Choi, Completely positive linear maps on complex matrices, Linear Algebra andAppl. 10 (1975), 285–290. MR0376726 (51 #12901)

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Department of Mathematics, University of Central Florida, Orlando, Florida


Department of Mathematics, Texas A&M University, College Station, Texas


Department of Mathematics, Tianjin University of Technology, Tianjin 300384,

People’s Republic of China


Department of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s

Republic of China and Department of Mathematics, Texas A&M University, College

Station, USA



Images of the continuous wavelet transform

Mahya Ghandehari and Keith F. Taylor

Abstract. A wavelet, in the generalized sense, is a vector in the Hilbertspace, Hπ , of a unitary representation, π, of a locally compact group, G, withthe property that the wavelet transform it defines is an isometry of Hπ intoL2(G). We study the image of this transform and how that image varies as thewavelet varies. We obtain a version of the Peter–Weyl Theorem for the classof groups for which the regular representation is a direct sum of irreduciblerepresentations.

1. Introduction

As the theory of wavelets emerged, it was recognized early, see [7], that thereconstruction formula for the continuous wavelet transform on R is a direct conse-quence of the abstract orthogonality relations for a square-integrable representationof a locally compact group [3] when applied to the translation and dilation repre-sentation of the group of all affine transformations of R. Here, we take the broadpoint of view, developed in [6], that a continuous wavelet theory may be usefullydeveloped whenever one has a unitary representation of some locally compact groupand a vector in the Hilbert space of that representation such that an appropriateanalog of the classical reconstruction formula exists. The theory of the continuousshearlet transform (see [12]) would be an example fitting within our concept of acontinuous wavelet transform. The details will be developed in section 3.

Our goal in this paper is to study, in a general setting, the image of a continuouswavelet transform as a subspace of L2(G) or of the Fourier algebra A(G), where G isthe underlying group, and how the images created by different wavelets interrelate.In particular, in Theorem 4.2 we show that the images of a linearly independentpair of wavelets intersect trivially. In Section 5, a version of the Peter–Weyl Theo-rem is established for [AR]-groups; that is, groups whose regular representation is adirect sum of irreducible representations. We conclude by formulating the conceptof a complete K-orthogonal set of wavelets for a square-integrable irreducible rep-resentation and exploring that concept for a particular kind of semi-direct productgroup.

2010 Mathematics Subject Classification. Primary 42C40; Secondary 43A65.Key words and phrases. Continuous wavelet transform, square integrable representation,

Fourier algebra.


55

56 MAHYA GHANDEHARI AND KEITH F. TAYLOR

2. General notations and definitions

Let G be a locally compact group equipped with left Haar integral∫G·dx.

The modular function Δ of G is a continuous homomorphism of G into R+, themultiplicative group of positive real numbers and satisfies Δ(y)

∫Gf(xy) dx =∫

Gf(x) dx whenever the integral on the right makes sense. We will also use that∫

Gf(x−1) dx =

∫Gf(x)Δ(x−1) dx.

Let π be a unitary representation of G in a Hilbert space Hπ. For vectors ξand η in Hπ, the continuous function

φπξ,η : G → C, x → 〈π(x)ξ, η〉

is called the coefficient function of G associated with the representation π andvectors ξ, η ∈ Hπ. One can integrate π to create a non-degenerate norm-decreasing∗-representation of the Banach ∗-algebra L1(G) in B(Hπ), the Banach algebra ofbounded linear operators on Hπ, via

〈π(f)ξ, η〉 =∫G

f(x)〈π(x)ξ, η〉dx,

for every f in L1(G) and vectors ξ and η in Hπ. We use the same symbol π todenote the ∗-representation and the associated unitary representation.

For a locally compact group G, the Fourier–Stieltjes algebra of G is the set ofall the coefficient functions of G, and is denoted by B(G). Clearly B(G) is a subsetof Cb(G), the algebra of bounded continuous functions on G. Eymard [4] provedthat B(G) is actually a subalgebra of Cb(G) and, moreover, it can be identified withthe Banach space dual of C∗(G), the group C*-algebra of G. Thus, for ϕ ∈ B(G),

‖ϕ‖B(G) = sup

{∫G

ϕ(x)f(x) dx : f ∈ L1(G), ‖f‖∗ ≤ 1

},

where, for f ∈ L1(G), ‖f‖∗ = sup{‖π(f)‖ : π is a representation of G}. TheFourier–Stieltjes algebra together with this dual norm turns out to be a Banachalgebra. The Fourier algebra of G, denoted by A(G), is the closed subalgebra ofthe Fourier–Stieltjes algebra generated by its compactly supported elements. Inthe special case of locally compact Abelian groups, one can identify the Fourierand Fourier–Stieltjes algebras with the L1-algebra and the measure algebra of thedual group respectively. One can refer to [5] for a detailed discussion on repre-sentation theory of locally compact groups, and [4] for the study of Fourier andFourier–Stieltjes algebras of locally compact groups.

Let π be a continuous unitary representation of G on a Hilbert space Hπ. LetAπ(G) denote the closed subspace of B(G) generated by the coefficient functionsof G associated with π, i.e.

Aπ(G) = SpanC{φπξ,η : ξ, η ∈ Hπ}

‖·‖B(G).

It is easy to see that Aπ(G) is a left and right translation-invariant closed subspaceof B(G). Conversely, by Theorem (3.17) of [1], any closed subspace of B(G) whichis left and right translation-invariant, is of the form Aπ(G) for some continuous uni-tary representation π. Moreover, the subspace Aπ(G) can be realized as a quotientof Hπ ⊗γ Hπ, the projective tensor product of Hπ and its conjugate Hπ, throughthe map P from Hπ⊗γHπ to Aπ(G) defined as

P : Hπ⊗γHπ → Aπ(G), P (ω)(x) = 〈ω, π(x)〉, ∀ω ∈ Hπ⊗γHπ, x ∈ G.

IMAGES OF THE CONTINUOUS WAVELET TRANSFORM 57

Here, we identify Hπ⊗γHπ with the trace class operators on Hπ, the predual ofB(Hπ).

In the special case where π is irreducible, the above map defines an isometrybetween Hπ⊗γHπ and Aπ(G) (See Theorem 2.2 and Remark 2.6 in [1]). We statea consequence of this as a proposition.

Proposition 2.1. Let π be an irreducible representation of a locally compactgroup G. Let ξ, η ∈ Hπ. Then ‖φπ

ξ,η‖Aπ(G) = ‖ξ‖‖η‖.

Proof. It is clear that φπξ,η = P (ξ ⊗ η). Therefore,

‖φπξ,η‖Aπ(G) = ‖P (ξ ⊗ η)‖B(G) = ‖ξ ⊗ η‖Hπ⊗γHπ

= ‖ξ‖‖η‖,as required. �

Let λG denote the left regular representation of G. The Hilbert space of λG isL2(G) and, for x ∈ G and f ∈ L2(G), λG(x)f(y) = f(x−1y), for almost all y ∈ G.

It was shown in [4] that AλG(G) = {φλG

f,g : f, g ∈ L2(G)} = A(G).

3. Wavelets and square-integrable representations

Let G be a locally compact group, π be a unitary representation of G on aHilbert space Hπ and η ∈ Hπ a nonzero vector. Define the linear map Vη : Hπ →Cb(G) by, for each ξ ∈ Hπ and x ∈ G,

Vη(ξ)(x) = 〈ξ, π(x)η〉.If the operator Vη forms an isometry of Hπ into L2(G) (that is, if the range of Vη

consists of square-integrable functions and ‖Vη(ξ)‖L2(G) = ‖ξ‖, for all ξ ∈ Hπ),then the vector η is called a wavelet for π. Thus, as an isometry, Vη preserves innerproducts. So 〈Vη(ξ), Vη(ξ

′)〉 = 〈ξ, ξ′〉, for all ξ, ξ′ ∈ Hπ. Writing out the innerproduct on the left hand side yields∫

G

〈ξ, π(x)η〉〈π(x)η, ξ′〉dx = 〈ξ, ξ′〉,

for all ξ, ξ′ ∈ Hπ. This leads to the reconstruction formula, for any ξ ∈ Hπ,

(3.1)

∫G

〈ξ, π(x)η〉π(x)ηdx = ξ,

weakly in Hπ. In fact η is a wavelet exactly when (3.1) holds and, in that case, Vη

is called a continuous wavelet transform.If the operator Vη is just a nonzero bounded operator from Hπ to L2(G) then

the vector η is called an admissible vector. The following propositions list somebasic and easily demonstrated properties of admissible vectors and wavelets. See[6] for a comprehensive introduction to this theory.

Proposition 3.1. Let η be a nonzero admissible vector for a unitary repre-sentation π of a locally compact group G.

(i) If π is irreducible, then η is a nonzero multiple of a wavelet for π.(ii) If π′ is a subrepresentation of π and Q is the orthogonal projection from

Hπ to Hπ′ , then Qη is either zero or an admissible vector for π′.

Proposition 3.2. Let η be a wavelet for a unitary representation π of a locallycompact group G. Then

(i) the vector η is a cyclic vector for π.


(ii) λG(x)Vη = Vηπ(x) for every x ∈ G.(iii) if π′ is a subrepresentation of π then Qη is a wavelet for π′, where Q is

the orthogonal projection from Hπ to Hπ′ .

Using the above properties, one sees that the operator Vη forms a unitary equiv-alence of π with a subrepresentation of λG whenever η is a wavelet for π. It is nothard to show that this is a sufficient condition when the unitary representation isirreducible. Namely, an irreducible unitary representation of G admits a waveletif and only if it is unitarily equivalent to a subrepresentation of λG. We refer thereader to [6] for more details. The study of wavelets for irreducible unitary repre-sentations connects naturally to square-integrable representations. An irreduciblerepresentation π is called a square-integrable representation if it admits a nonzerosquare-integrable coefficient function φπ

ξ1,ξ2for some ξ1, ξ2 ∈ Hπ. It has been shown

in [6] that for an irreducible representation π with a nonzero square-integrable co-efficient function φπ

ξ0,ξ0, the operator Vξ0 is a multiple of an isometry, thus ξ0 is a

multiple of a wavelet.In this article, we will use the following “orthogonality relations” for square-

integrable representations shown in [3]. Note that Δ denotes the modular functionon G.

Theorem 3.3. [Duflo–Moore [3]] Let π be a square-integrable irreducible rep-resentation of a locally compact group G. Then there is a unique densely definedself-adjoint and positive operator K on Hπ which satisfies the following conditions.

(i) For every x ∈ G, π(x)Kπ(x)−1 = Δ(x)−1K (semi-invariant with weightΔ−1).

(ii) 〈ξ, π(·)η〉 is square integrable if and only if η ∈ domK− 12 .

(iii) Let ξ, ξ′ ∈ Hπ and η, η′ ∈ domK− 12 . Then

〈〈ξ, π(·)η〉, 〈ξ′, π(·)η′〉〉L2(G) = 〈ξ, ξ′〉〈K− 12 η′,K− 1

2 η〉.

Corollary 3.4. Let π be an irreducible unitary representation, and η ∈ Hπ.Then, η is a wavelet if and only if η ∈ domK− 1

2 and ‖K− 12 η‖ = 1. Moreover, for

every x ∈ G, the vector√Δ(x)π(x)η is a wavelet as well.

Definition 3.5. The operator K of Theorem 3.3 is called the Duflo–Mooreoperator of π.

4. Inside Aπ(G) for an irreducible π

Throughout this section, let π be a square-integrable irreducible unitary repre-sentation of a locally compact group G, and η ∈ Hπ. Define Aη := Vη(Hπ). Thissection concerns decomposing Aπ(G) into blocks of the form Aη.

Lemma 4.1. Let π, η, and Aη be as above. Suppose η is a wavelet for π. ThenAη is a ‖ · ‖2-closed subspace of L2(G) which is left-invariant. The subspace Aη isnot right-invariant if G is non-unimodular. In addition, Aη is a ‖ · ‖B(G)-closedsubspace of Aπ(G), where π is the representation conjugate to π in the Hilbert spaceHπ.

Proof. It is clear that Aη is a ‖ · ‖2-closed subspace of L2(G), since η isa wavelet for π. Also, the subspace Aη is left-invariant by Proposition 3.2. Nowsuppose that G is non-unimodular, and let y be an element of G such that Δ(y) = 1.


Let ξ ∈ Hπ be a nonzero vector. Then for f = Vη(ξ) with fy denoting the righttranslation of f by y,

‖fy‖22 =

∫G

|f(xy)|2dx =

∫G

|f(x)|2Δ(y−1)dx = Δ(y−1)‖f‖22 = Δ(y−1)‖ξ‖2,

since η is a wavelet. Moreover, using Proposition 2.1,

‖fy‖B(G) = ‖Vπ(y)η(ξ)‖B(G) = ‖π(y)η‖‖ξ‖ = ‖η‖‖ξ‖.Now assume that fy is an element of Aη, i.e. there exists ξ′ in Hπ such thatfy = Vη(ξ

′). Then, ‖fy‖B(G) = ‖η‖‖ξ′‖. Hence, ‖ξ‖ = ‖ξ′‖, and ‖fy‖2 = ‖ξ′‖. Butthis is a contradiction with η being a wavelet.

To prove the last statement note that f = φπη,ξ. Since π is irreducible, so is π.

Therefore, by Proposition 2.1,

‖f‖B(G) = ‖φπη,ξ‖B(G) = ‖η ⊗ ξ‖Hπ⊗γHπ

= ‖η‖‖ξ‖.On the other hand, ‖f‖2 = ‖Vη(ξ)‖2 = ‖ξ‖. Thus,

‖f‖B(G) = ‖f‖2‖η‖.That is, the L2-norm and the Fourier–Sieltjes norm are equivalent on Aη. Hence,Aη is a ‖ · ‖B(G)-closed subspace of Aπ(G). �

Observe that for each x ∈ G, the subspace Aπ(x)η = {〈ξ, π(·)π(x)η〉 : ξ ∈ Hπ}is the right x-translation of Aη, and is a ‖ · ‖B(G)-closed subspace of Aπ(G). Notethat the proof of Lemma 4.1 implies that the subspaces Aπ(x)η and Aη intersecttrivially whenever Δ(x) = 1, if η is a wavelet. The following theorem generalizesthis fact, and shows that two subspaces Aη1

and Aη2, for admissible η1 and η2,

either coincide or intersect trivially.

Theorem 4.2. Let π be a square-integrable irreducible unitary representation ofa locally compact group G. Let η1 and η2 be admissible vectors in Hπ. Then eitherAη1

∩Aη2= {0}, or Aη1

∩Aη2= Aη1

= Aη2and the latter case happens if and only

if η1 = αη2 for some α ∈ C. If η1 and η2 are both wavelets and Aη1∩ Aη2

= {0},then η1 = αη2 for some α ∈ T

Proof. Assume that Aη1∩ Aη2

= {0}; that is, there exist nonzero vectors ξand ξ′ in Hπ such that 0 = f(·) = 〈ξ, π(·)η1〉 = 〈ξ′, π(·)η2〉. Note that

〈ξ, π(·)η1〉 = Vη1(ξ) and 〈ξ′, π(·)η2〉 = Vη2

(ξ′).

Hence, by the orthogonality relations stated in Theorem 3.3 we have

‖f‖22 = 〈Vη1(ξ), Vη1

(ξ)〉 = ‖K− 12 η1‖2‖ξ‖2,

‖f‖22 = 〈Vη2(ξ′), Vη2

(ξ′)〉 = ‖K− 12 η2‖2‖ξ′‖2,

‖f‖22 = 〈Vη1(ξ), Vη2

(ξ′)〉 = 〈K− 12 η2,K

− 12 η1〉〈ξ, ξ′〉.

Thus,

〈K− 12 η2,K

− 12 η1〉〈ξ, ξ′〉 = ‖K− 1

2 η1‖‖K− 12 η2‖‖ξ‖‖ξ′‖.

Since all of the above quantities must be nonzero, by the Cauchy–Schwarz inequal-ity, we have

|〈K− 12 η2,K

− 12 η1〉| = ‖K− 1

2 η1‖‖K− 12 η2‖ and |〈ξ, ξ′〉| = ‖ξ‖‖ξ′‖,


which implies that

K− 12 η2 = α1K

− 12 η1 and ξ′ = α2ξ

for scalars α1 and α2 in C\{0}. Recall that K− 12 is injective, so η2 = α1η1. Clearly,

each set Aη forms a vector subspace of L2(G). Hence,

Aη2= Aα1η1

= α1Aη1= Aη1

,

which proves the first statements of the theorem.Moreover, if η1 and η2 are wavelets, we have ‖K− 1

2 η1‖ = ‖K− 12 η2‖ = 1, which

implies that |α1| = 1. �

Corollary 4.3. Let π be a square-integrable irreducible unitary representationof a locally compact group G and let η be an admissible vector for π. Let x ∈ G besuch that Aπ(x)η ∩ Aη = {0}. Then π(x)η = αη, for some α ∈ T.

Proof. By Theorem 4.2, π(x)η = αη for some α ∈ C. But π(x) is a unitary,so |α| = 1. �

Theorem 4.4. Let π be a square-integrable irreducible unitary representationof a locally compact group G and let η be a wavelet for π. Then Σx∈GAπ(x)η is‖ · ‖B(G)-dense in Aπ(G)

Proof. Observe that Σx∈GAπ(x)η is a left and right translation invariant sub-

space of Aπ(G). Therefore, Σx∈GAπ(x)η‖·‖B(G)

is of the form Aσ(G) for a unitaryrepresentation σ of G. Since Aσ(G) ⊆ Aπ(G), the representation σ is a subrepresen-tation of π by Corollary (3.14) of [1]. This implies that Σx∈GAπ(x)η is ‖·‖B(G)-densein Aπ(G), as π is irreducible and has no proper subrepresentation. �

Example 4.5. Let G be the group of orientation preserving affine transforma-tions of the real line. Then G is the semidirect product R�R+, where R+ acts onR by multiplication. In [11], it has been shown that

A(G) = Aπ+(G)⊕�1 Aπ−(G),

where π± are inequivalent, irreducible unitary representations of G on the Hilbertspace L2(R∗

+, dt/t) defined by, for (b, a) ∈ G and ξ ∈ L2(R∗+, dt/t),

π±(b, a)ξ(t) := e∓2πibtξ(at),

for almost all t ∈ R∗+. Consider a continuous compactly supported function η on

R+ which is 1 on [ 12 , 1] and nonnegative everywhere else. It is known, see [10] fordetails, that η is a multiple of a wavelet. By normalizing if necessary, we assumethat η is a wavelet. Clearly, if π±(b, a)η = αη for some α ∈ T, then a = 1 andb = 0. Thus, by Corollary 4.3, Aπ±(x1)η ∩ Aπ±(x2)η = {0} whenever x1 = x2 in G.

Note that π+ and π− are unitarily equivalent, respectively, to the two irre-ducible subrepresentations of the classical wavelet representation ρ acting on L2(R),where

ρ(b, a)f(t) = a−1/2f

(t− b

a

),

for t ∈ R, (b, a) ∈ G, and f ∈ L2(R).

In fact, the phenomenon illustrated by Example 4.5 is somewhat general.


Theorem 4.6. Let π be a square-integrable irreducible unitary representation ofa locally compact group G and suppose that G has no nontrivial compact subgroup.Let η be any wavelet for π. Then Aπ(x1)η ∩Aπ(x2)η = {0} for any x1, x2 ∈ G, x1 =x2.

Proof. Let x ∈ G be such that Aπ(x)η ∩ Aη = {0}. Then, by Theorem 4.2,Aπ(xn)η = Aη for all n ∈ Z. Let K denote the closed subgroup of G generated byx. Since π is equivalent to a subrepresentation of the regular representation, Vηηvanishes at infinity. On the other hand, Corollary 4.3 and continuity implies that

|Vηη(y)| = |〈η, π(y)η〉| = ‖η‖2,

for all y ∈ K. Therefore K is compact and hence K = {e}, since G is compactfree. If x1, x2 ∈ G satisfy Aπ(x1)η ∩ Aπ(x2)η = {0}, then Aπ(x1)η = Aπ(x2)η, whichimplies Aπ(x1x

−12 )η = Aη. So x1 = x2. �

5. The Aη as subspaces of L2(G)

We continue with the assumption that π is a square-integrable irreducible rep-resentation of a locally compact group G. If η is a nonzero admissible vector forπ, then η is a scalar multiple of a wavelet η′ and Aη = Aη′ is a closed subspace ofL2(G). Let Kπ denote the smallest closed subspace of L2(G) that contains Aη forevery admissible vector η for π. Fix any wavelet ω for π. Since π is irreducible,

{π(x)ω : x ∈ G} is total in Hπ. Thus Kπ = 〈∪{Aπ(x)ω : x ∈ G}〉L2(G)

. There-

fore, Kπ is a closed subspace of L2(G) that is invariant under both left and righttranslations.

If G happens to be compact, any irreducible representation is finite dimensionaland square-integrable. In that case, let dπ denote the dimension of Hπ. Fromthe classical orthogonality relations, one sees that the operator K of Theorem 3.3is simply dπI, where I is the identity operator of Hπ. Let {ν1, · · · , νdπ

} be an

orthonormal basis of Hπ. For 1 ≤ j ≤ dπ, let ηj = d1/2π νj . So each ηj is a

wavelet for π. Moreover, the orthogonality relations also tell us that Aηj⊥ Aηk

if1 ≤ j = k ≤ dπ. Since the linear span of {η1, · · · , ηdπ

} is Hπ,

Kπ = ⊕dπj=1Aηj

.

Moreover, L2(G) = ⊕π∈GKπ. This is the essential content of the Peter–WeylTheorem.

With the appropriate interpretation, this generalizes to a class of non-compactgroups G, under the assumption of separability.

Definition 5.1. Let G be a locally compact group, π be a square-integrableirreducible representation of G and K the Duflo–Moore operator of π. A collection{ηj : j ∈ J} of vectors in domK−1/2 is called a complete K-orthogonal set of

wavelets for π if {ηj : j ∈ J} is total in Hπ and {K−1/2ηj : j ∈ J} is orthonormal.

If {ηj : j ∈ J} is a complete K-orthogonal set of wavelets for π then, byTheorem 3.3, the Aηj

are mutually orthogonal closed subspaces of Kπ whose unionsspan Kπ. Thus

Kπ = ⊕j∈JAηj.


Theorem 5.2. Let G be a separable locally compact group, π be a square-integrable irreducible representation of G and K the Duflo–Moore operator of π.There exists a countable set {ηj : j ∈ J} which is a complete K-orthogonal set ofwavelets for π. Moreover, if η is a fixed wavelet, each ηj can be constructed as afinite linear combination of {π(x)η : x ∈ G}.

Proof. Fix a wavelet η for π. Let {xj : j ∈ J ′} be a countable dense subset

of G. Then {π(xj)η : j ∈ J ′} is total in Hπ. Recall that domK−1/2 consists of

exactly the admissible vectors for π and that π(xj)η ∈ domK−1/2 for each j ∈ J ′.

Moreover, K−1/2 is injective on its domain. Perform the Gram–Schmidt process onthe countable set {K−1/2π(xj)η : j ∈ J ′} and pull the resulting linear combinations

back throughK−1/2 to produce a countable set {ηj : j ∈ J} of vectors in domK−1/2

which is total in Hπ and such that {K−1/2ηj : j ∈ J} is orthonormal. �

Remark 5.3. The above theorem appears as Theorem 2.33 in [6] without theassumption of separability. However, the sketch of the proof in [6] overlooks thefact that Gram–Schmidt requires the initial set of vectors to be countable. That iswhy we have included the argument here.

A locally compact group G is called an [AR]-group if the left regular repre-sentation, λG, is the direct sum of irreducible representations (see [16] and [17]).Let

Gr = {π ∈ G : π is equivalent to a subrepresentation of λG}.We use the symbol π for both an equivalence class of irreducible representationsand a particular member of that class. When λG is a direct sum of irreducibles, we

have left invariant closed subspaces Lπ, π ∈ Gr, of L2(G) such that λG restricted

to Lπ is equivalent to a multiple of π, for each π ∈ Gr, and L2(G) = ⊕π∈GrLπ. In

light of Theorem 5.2, Lπ = Kπ, for each π ∈ Gr. This amounts to a Peter–Weyltheory for separable [AR]-groups.

An example will demonstrate the concrete nature of the conditions the ηj ap-pearing in Theorem 5.2 must satisfy.

Example 5.4. Fix c ∈ R, c = 0. Let

Hc =

{(a 0b ac

): a, b ∈ R, a > 0

}act on R2 with the natural matrix action. Form the semidirect product

Gc = R2 �Hc = {[x, h] : x ∈ R2, h ∈ Hc},equipped with the group product [x, h][y, k] = [x + hy, hk], for [x, h], [y, k] ∈ G.When c = 1/2, this is the shearlet group [12]. For general c, this family of groupswas investigated in [15]. From [15], or using elementary Mackey theory [13], [14],

or [10], it is easy to show that Gc is an [AR]-group and Gc

r= {ρ+, ρ−}, were

ρ+ can be realized as follows. There is an analogous description of ρ− with theupper half plane replaced by the lower half plane. The Hilbert space of ρ+ is

Hρ+= {f ∈ L2(R2) : suppf ⊆ O+}, where O+ is the upper half plane and

ρ+

[(x1

x2

),

(a 0b ac

)]f

(y1y2

)=

1√ac+1

f

((y1 − x1)/a

y2−x2−a−1b(y1−x1)ac

),


for

(y1y2

)∈ R2,

[(x1

x2

),

(a 0b ac

)]∈ Gc, and f ∈ Hρ+

.

Admissibility conditions for ρ+ were worked out in [15]. This determines theDuflo–Moore operator, K, for this representation. Theorem 5.2 in this setting givesthe following method for decomposing the left regular representation of Gc into aninfinite multiple of ρ+ plus an infinite multiple of ρ−.

Construct {ηj : j ∈ J} as a total set in L2(O+, da db) such that {ηj : j ∈ J}is orthonormal in the weighted L2-space, L2

(O+, da db

ac

). For each j ∈ J , let wj ∈

L2(R2) satisfy wj = ηj . Then

Vwjf [x, h] =

∫R2

f(y)ρ+[x, h]wj(y) dy,

for [x, h] ∈ Gc, f ∈ Hρ+. If we let Awj

= VwjHρ+

, then λGcrestricted to Awj

isequivalent to ρ+, for each j ∈ J , and we have Kρ+

= ⊕j∈JAwj. Similarly for ρ−

and L2(Gc) = Kρ+⊕ Kρ− .

We conclude by formulating the construction of Example 5.4 in a more generalsetting. Let G be a locally compact group of the form A�H, where A is an abelianlocally compact group and H is a σ-compact locally compact group acting on A

via (h, a) → h · a, for h ∈ H, a ∈ A. Then H acts on the dual group A by, for

h ∈ H,χ ∈ A, (h · χ)(a) = χ(h−1 · a), for all a ∈ A. Further, assume that there

exists an open free H-orbit O in A. Then for fixed ω ∈ O, the map h → h−1 · ωis a homeomorphism of H onto O. See [9] and Sections 7.2 and 7.3 of [10] for atreatment of this situation.

Let δ denote the homomorphism of H into R∗+ such that, for any integrable

function g on A and any h ∈ H, we have δ(h)∫Ag(h ·a) da =

∫Ag(a) da. There is a

square-integrable irreducible representation πO of G associated with O which canbe realized as follows. The Hilbert space of πO is L2(O,m), where the measure m

on O is the restriction of the Haar measure of A, and, for (a, h) ∈ G, ξ ∈ L2(O,m),

πO(a, h)ξ(χ) = δ(h)1/2χ(a)ξ(h−1 · χ),for all χ ∈ O (Proposition 7.17, [10]).

Thus, there are two relevant measures on the orbit O, the Haar measure of Arestricted toO and the left Haar measure onH, moved toO via the homeomorphismh → h−1 · ω. Let μ denote the latter measure. Then

(5.1)

∫Oϕ(χ) dμ(χ) =

∫H

ϕ(h−1 · ω) dh,

for any ϕ ∈ Cc(O). Note that the right hand side of (5.1) is independent of thechoice of ω ∈ O. We use L2(O, μ) to denote the L2-space when μ is the measureon O.

The computation in the proof of Theorem 7.19 of [10], after adjusting thenotation, shows that, for ξ, η ∈ L2(O,m),

(5.2) ‖Vηξ‖22 =

∫O|ξ(χ)|2dχ

∫O|η(χ)|2dμ(χ).

This implies that ‖K−1/2η‖2 =∫O |η(χ)|2dμ(χ) and, via polarization, that

(5.3) 〈K−1/2η1,K−1/2η2〉 =

∫Oη1(χ)η2(χ)dμ(χ) =

∫H

η1(h−1 · ω)η2(h−1 · ω)dh,


for all admissible η1 and η2 in L2(O,m). Note that the set of admissible vectors inL2(O,m) is exactly L2(O,m)∩L2(O, μ) and this intersection makes sense becausethe two measures in question are mutually absolutely continuous. Observe that acompleteK-orthogonal set of wavelets for πO is a collection {ηj : j ∈ J} of functionsin L2(O,m) ∩ L2(O, μ) which is total in L2(O,m) and orthonormal in L2(O, μ).

Remark 5.5. In all the examples known to the authors where a σ-compactlocally compact group H acts on an abelian locally compact group A in such a

manner that there exists an open free H-orbit in A, the union of all of the open

free H-orbits is co-null in A and, as a result, A�H is an [AR]-group.

References

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[2] David Bernier and Keith F. Taylor, Wavelets from square-integrable representations, SIAMJ. Math. Anal. 27 (1996), no. 2, 594–608, DOI 10.1137/S0036141093256265. MR1377491(97h:22004)

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CRC Press, Boca Raton, FL, 1995. MR1397028 (98c:43001)[6] Hartmut Fuhr, Abstract harmonic analysis of continuous wavelet transforms, Lecture Notes

in Mathematics, vol. 1863, Springer-Verlag, Berlin, 2005. MR2130226 (2006m:43003)[7] A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group

representations. I. General results, J. Math. Phys. 26 (1985), no. 10, 2473–2479, DOI10.1063/1.526761. MR803788 (86k:22013)

[8] E. Hewitt and K.A. Ross: Abstract harmonic analysis. I, (Berlin: Springer, 1963).[9] Eberhard Kaniuth and Keith F. Taylor, Minimal projections in L1-algebras and open points

in the dual spaces of semi-direct product groups, J. London Math. Soc. (2) 53 (1996), no. 1,141–157, DOI 10.1112/jlms/53.1.141. MR1362692 (97d:43002)

[10] Eberhard Kaniuth and Keith F. Taylor, Induced representations of locally compact groups,Cambridge Tracts in Mathematics, vol. 197, Cambridge University Press, Cambridge, 2013.MR3012851

[11] Idriss Khalil, Sur l’analyse harmonique du groupe affine de la droite (French), Studia Math.51 (1974), 139–167. MR0350330 (50 #2823)

[12] Gitta Kutyniok and Demetrio Labate, Resolution of the wavefront set using continuous shear-lets, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2719–2754, DOI 10.1090/S0002-9947-08-04700-4. MR2471937 (2010b:42043)

[13] George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2)

55 (1952), 101–139. MR0044536 (13,434a)[14] George W. Mackey, The theory of unitary group representations, University of Chicago Press,

Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslagerof lectures given at the University of Chicago, Chicago, Ill., 1955; Chicago Lectures in Math-ematics. MR0396826 (53 #686)

[15] Eckart Schulz and Keith F. Taylor, Extensions of the Heisenberg group and wavelet analysisin the plane, Spline functions and the theory of wavelets (Montreal, PQ, 1996), CRM Proc.Lecture Notes, vol. 18, Amer. Math. Soc., Providence, RI, 1999, pp. 217–225. MR1676245(99m:42053)

[16] Keith F. Taylor, Geometry of the Fourier algebras and locally compact groups with atomicunitary representations, Math. Ann. 262 (1983), no. 2, 183–190, DOI 10.1007/BF01455310.MR690194 (84h:43020)

[17] Keith F. Taylor, Groups with atomic regular representation, Representations, wavelets, andframes, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2008, pp. 33–45, DOI10.1007/978-0-8176-4683-7 3. MR2459312 (2009j:22004)


Department of Mathematics and Statistics, University of Saskatchewan, Saska-

toon, SK, S7N 5E6, Canada


Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H

4J1, Canada



Decompositions of generalized wavelet representations

Bradley Currey, Azita Mayeli, and Vignon Oussa

Abstract. Let N be a simply connected, connected nilpotent Lie group which

admits a uniform subgroup Γ. Let α be an automorphism of N defined byα (expX) = expAX. We assume that the linear action of A is diagonalizableand we do not assume that N is commutative. Let W be a unitary waveletrepresentation of the semi-direct product group

⟨∪j∈Zα

j (Γ)⟩� 〈α〉 defined

by W (γ, 1) = f(γ−1x

)and W (1, α) = |detA|1/2 f (αx) . We obtain a de-

composition of W into a direct integral of unitary representations. Moreover,we provide an explicit unitary operator intertwining the representations, aprecise description of the representations occurring, the measure used in thedirect integral decomposition and the support of the measure. We also studythe irreducibility of the fiber representations occurring in the direct integraldecomposition in various settings. We prove that in the case where A is an ex-pansive automorphism then the decomposition of W is in fact a direct integralof unitary irreducible representations each occurring with infinite multiplici-ties if and only if N is non-commutative. This work naturally extends resultsobtained by H. Lim, J. Packer and K. Taylor who obtained a direct integral

decomposition of W in the case where N is commutative and the matrix A isexpansive, i.e. all eigenvalues have absolute values larger than one.

1. Introduction

The main purpose of this paper is to present an abstract harmonic analysis ap-proach to the theory of wavelets on both commutative and non commutative groups.Although in the past few years, there has been extensive work done to extend theconcept of wavelets to non-commutative domains [2–5,8,11] this theory is not en-tirely well-understood. We recall that, in its classical definition, a wavelet systemis an orthonormal basis generated by a combination of integral shifts and dyadicdilations of a single function in L2 (R) . Although the set of operators involved ina wavelet system does not form a group, it generates a group which is isomorphicto a subgroup of the one-dimensional affine group also known as the ax+ b group.To be more precise, let Γ2 = {m2n ∈ Q : m,n ∈ Z} and let ϕ : Z → Aut (Γ2) bedefined such that ϕ (m) γ = 2−mγ for γ ∈ Γ2 and m ∈ Z. We define the unitaryoperators D, and T such that

Df (t) =√2f (2t) , and Tγf (t) = f (t− γ) .

2010 Mathematics Subject Classification. Primary 22E25.Key words and phrases. wavelets, nilpotent, Lie groups, Direct integral, decomposition.


67

68 B. CURREY, A. MAYELI, AND V. OUSSA

An orthonormal wavelet is a unit vector ψ ∈ L2 (R) such that {DnTkψ : n ∈ Z, k ∈ Z}forms an orthonormal basis for L2 (R) . It is not too hard to see that the subgroupof U

(L2 (R)

)which is generated by the operators D and T is isomorphic to Γ2�ϕZ

via the faithful representation

W : (γ, n) −→ TγDn.

Moreover, Γ2 �ϕ Z is a finitely generated metabelian solvable group generated bytwo elements and Γ2 �ϕ Z has the following finite presentation:

〈a,m : mam−1 = a2〉.The representation W of this group was termed the wavelet representation by L-H.Lim, J. Packer and K. Taylor in [10]. In [10], the authors obtained a direct integraldecomposition of W into its irreducible components which are some monomialrepresentations parametrized by a wavelet set. More precisely, they show that Wis equivalent to ∫ ⊕

E

IndΓ2�ϕZ

Γ2(χt) dt.

The representations occurring in the direct integral decomposition are irreducible,and E is a subset of R which up to a null set tiles the real line by both dyadicdilations and integral translations. Their results were actually obtained for themore general case where

D,Tk : L2 (Rn) → L2 (Rn)

such that

Df (t) = det (A)1/2

f (At) , Tkf (t) = f (t− k) ,

A is an expansive matrix (all eigenvalues have absolute value greater than one) inM (n,Z) ∩GL (n,Q) and k ∈ Zn.

The present work undertakes a thorough investigation of the case where Rn isreplaced by a simply connected, connected nilpotent Lie group N . In this work,we do not assume that N is commutative. If N is commutative, then N = Rn

and the main purpose of this paper is to generalize the results obtained in [10].However, in this more general setting, it is not straightforward to define a waveletrepresentation acting in L2 (N) . In order to summarize our results, we must firsttake care of some technical issues.

Let n be an algebra of m × m nilpotent real matrices and set N = exp n.Then n is a nilpotent Lie subalgebra of gl(m,R), N is a closed, simply connectednilpotent subgroup of GL(m,R) and the matrix exponential exp : n → N is abijection. A basis {X1, X2, . . . , Xn} for n can be chosen in such a way that ni =spanR{X1, X2, . . . , Xi} is an ideal in n. Such a basis is called a strong Malcevbasis, and from now on, we fix such a basis for n. Thus n is identified with Rn,and we let dX denote the Lebesgue measure on n. The mapping X → expX is ahomeomorphism, as is the mapping

(t1, t2, . . . , tn) → exp t1X1 exp t2X2 · · · exp tnXn.

Both coordinate systems induce the same bi-invariant, Borel measure μ on N asthe push forward of Lebesgue measure.

Let N be a connected nilpotent Lie group. A discrete subgroup Γ is a uniformsubgroup if N/Γ is compact. Since nilpotent Lie groups are unimodular, thenit can be shown that N/Γ has a finite volume if N/Γ is compact. Not every

DECOMPOSITIONS OF GENERALIZED WAVELET REPRESENTATIONS 69

simply connected nilpotent Lie group admits a uniform subgroup. A necessary andsufficient condition for existence of a uniform subgroup is that there is a strongMalcev basis for which the associated structure constants are rational numbers. Inthis case such a basis can be chosen so that

Γ = expZX1 expZX2 · · · · expZXn

and henceforth we assume that this is the case.Each element x ∈ N defines a unitary translation operator Tx on L2(N) by

(Txf)(y) = f(x−1y).

In order to define dilations in L2(N), we fix a Lie algebra automorphism A ∈ GL(n).Via the chosen basis for n, A is given by an invertible n × n real matrix and wewrite evaluation of A on elements of n multiplicatively. Now, we will assume thatA defines an automorphism α of N by α(expX) = expAX such that

AXk = akXk, ak ∈ R.

It follows from our assumption that A is a diagonal matrix with respect to the fixedstrong Malcev basis for the Lie algebra n. For any integrable function f on N ,∫N

f(αx)| detA|dμ(x) =∫n

f(expAX)| detA|dX =

∫n

f(expX)dX =

∫N

f(x)dμ(x).

Define the unitary operator D on L2(N) by

Df(x) = | detA| 12 f(αx).

Let G(D,T,Γ) denote the subgroup of the group U(L2(N)) of unitary operators onL2(N), generated by the operators D and Tγ , γ ∈ Γ. Then G(D,T,Γ) is the imageof a unitary representation of the group G = Γα � H, where H = 〈α〉 is the freeabelian group generated by α, and Γα is the subgroup of N generated by

∪m∈Zαm(Γ).

G is a subgroup of the semi-direct product F = N �H with operation

(x1, αm1) (x2, α

m2) =(x1α

−m1x2, αm1+m2

), (x1, α

m1) , (x2, αm2) ∈ F.

The mapping V : F → U(L2 (N)

)defined by

V (x, αm) = TxDmα

is a unitary representation of F acting in L2(N), and the wavelet representation isthe restriction of V to G. Write W = V |G.

The present work is organized as follows: The second section contains somestandard preliminary work. In the third section, we provide a direct integral de-composition of the representation W into smaller components. We describe theunitary representations occurring, the measure used in the direct integral and thesupport of the measure as well. We show that W is equivalent to a measurable fieldof unitary representations over a set which tiles the unitary dual of N by dilation.In the fourth section, we deal with the irreducibility of the representations occur-ring in the direct integral decomposition. We prove that in the case where A is anexpansive automorphism then the decomposition of W is in fact a direct integralof unitary irreducible representations, each occurring with infinite multiplicities ifand only if N is non-commutative. We also discuss some surprising results derivedfrom Bekka and Driutti’s work [1]. More precisely, we show that there are instanceswhere the irreducibility of representations occurring in the decomposition of W is


completely independent of the nature of the dilation action coming from H. That is,there are examples (for non-commutative N) where the irreducibility (or reducibil-ity) of the representations occurring in the decomposition of W only depends on thestructure constants of the Lie algebra n. Several examples are presented throughoutthe paper to help the reader follow the stream of ideas.

2. Preliminaries

Recall that the Fourier transform f → f on L1(Rn) satisfies

Txf(λ) = e2πi〈x,λ〉f(λ)

and the set Rn = {x → e2πi〈x,λ〉 : λ ∈ Rn} of exponentials is precisely the set ofcontinuous unitary homomorphisms of Rn into C∗. To define the group Fouriertransform for the present class of simply connected nilpotent groups N , we denoteby N the space of equivalent classes of strongly continuous, irreducible unitaryrepresentations of N , where π1 : N → U(H1) and π2 : N → U(H2) are equivalentif there is a unitary operator U : H1 → H2 such that U ◦ π1(x) = π2(x) ◦ U holdsfor all x ∈ N .

Given an irreducible unitary representation π : N → U (H) and f ∈ L1(N), we

define the operator f(π) on H by

f(π) =

∫N

f(x)π(x)dμ(x)

where the integral is taken in the weak sense. One can show that f(π) is trace-class,and by the translation invariance of μ, we have

Txf(π) = π(x) ◦ f(π)holds for all x ∈ N .

Just as the Fourier transform on L1(Rn) ∩ L2(Rn) extends to a unitary iso-

morphism of L2(Rn) with L2(Rn), the group Fourier transform provides a similarisomorphism by which we study the wavelet representation on L2(N). Although

the topology of Rn is exactly the same at that of Rn, the topological structureof N is not even Hausdorff if N is not commutative. In fact there is a canonicalhomeomorphism κ : n∗/Ad∗(N) → N , where Ad∗ : N → GL(n∗) is the coadjointrepresentation of N acting on the linear dual n∗ of n. In this case, it is natural toparametrize a conull subset of N by an explicit subset Λ of n∗ which is a cross-section for almost all of the coadjoint orbits. We make the following somewhattechnical digression in order to describe the explicit Plancherel isomorphism. See[6] for more details and the original sources.

For a setX we denote by |X| the number of elements in the setX. Any coadjointorbit has the structure of a symplectic manifold, and hence is even dimensional.Let 2d be the maximal dimension of the coadjoint orbits. One has the following:

a) subsets j ⊂ e ⊂ {1, 2, . . . , n} such that |e| = 2|j| = 2d,

j = {j1 < j2 < · · · < jd}b) an Ad∗(N)-invariant Zariski open subset Ω whose orbits have dimension

2d and such that Λ = M ∩ Ω is a cross-section for the coadjoint orbits inΩ, where

M = {� ∈ n∗ : �(Xj) = 0, for all j ∈ e}


c) for each λ ∈ Λ, an analytic subgroup P (λ) of N such thatc1) (t1, t2, . . . , td) → exp t1Xj1 exp t2Xj2 · · · exp tdXjd · P (λ) is a homeo-

morphism of Rd with N/P (λ),c2) χλ(expY ) = e2πiλ(Y ) defines a unitary character of P (λ),c3) the unitary representation πλ of N induced from P (λ) by χλ is irre-

ducible and associated with λ under the canonical mapping κ.

The subalgebra p(λ) = log(P (λ)) is defined by

(2.1) p(λ) =

n∑i=1

ni(λ)

where

(2.2) ni(λ) = {Y ∈ ni : λ[X,Y ] = 0 for all X ∈ ni}.Now, let q : N → N/P (λ) be the canonical quotient map. For each λ ∈ Λ, theHilbert space for the induced representation πλ is the completion Hλ of the spaceof complex-valued functions g on N satisfying the following:

(1) the image of the support of g through the quotient map q is compact(2) g(xp) = χλ(p)

−1g(x), x ∈ N, p ∈ P (λ)(3)

∫Rd |g(exp t1Xj1 exp t2Xj2 · · · exp tdXjd)|2dt < ∞.

The induced representation πλ is just the action of left translation: πλ(y)g(x) =g(y−1x), and the mapping U : Hλ → L2(Rd) defined by

Ug(t1, t2, . . . , td) = g(exp t1Xj1 exp t2Xj2 · · · exp tdXjd)

is an isomorphism of Hilbert spaces.

Remark 1. If N is abelian, then e = ∅, d = 0, and P (λ) = N for all λ. In

this case the dual N of N = Rn is just the set Rn defined above.

For f ∈ L1 (N) ∩ L2 (N) , we put f (λ) = f (πλ) and the map f → f extendsto a unitary operator

P : L2 (N) →∫ ⊕

Λ

HS(L2

(Rd))

|P (λ)| dλ

where P is a non-vanishing polynomial function on Λ and HS = L2(Rd)⊗L2(Rd).Next, we consider the map

P1 : L2 (N) →∫ ⊕

Λ

HS(L2

(Rd))

dλ

such thatP1f =

(f (λ)

√|P (λ)|

)λ∈Λ

.

Then it is also clear that the map P1 is a unitary map obtained by modifying thePlancherel transform P.

We shall now present a few examples of nilpotent Lie groups, their duals andthe associated Plancherel measures.

Example 2. The Heisenberg group N has as a basis for its Lie algebra {Z, Y,X}where [X,Y ] = Z and all other brackets vanish. Putting X1 = Z,X2 = Y,X3 = X,we see that the structure constants for this basis are rational and hence n hasa rational structure. If we identify N with R3 via coordinates n = (z, y, x) =


exp zZ exp yY expxX, then the set Γ of integer points in N is a uniform dis-crete subgroup of N and the above basis is strongly based in Γ. Now, in this casee = {2, 3}, and j = {3}. Explicitly,

Ω = {� ∈ n∗ : �(Z) = 0}

and

Λ = {λZ∗ : λ = 0} � R \ {0}.

Note also that |P(�)| = |�(Z)| in this example.

Example 3. Let N be the upper triangular group of 4× 4 matrices. A typicalelement of the Lie algebra n is of the form⎡⎢⎢⎣

0 u1 y1 z0 0 u2 y20 0 0 u3

0 0 0 0

⎤⎥⎥⎦ .

In fact, n is spanned by the basis

{Z, Y1, Y2, U1, U2, U3}

with the following non-trivial Lie brackets

[U1, U2] = Y1, [U3, U2] = −Y2

[U1, Y2] = Z, [U3, Y1] = −Z.

Put

X1 = Z,X2 = Y1, X3 = Y2, X4 = U1, X5 = U2, X6 = U3.

We define

Γ =

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

1 k4 k2 k10 1 k5 k30 0 1 k60 0 0 1

⎤⎥⎥⎦ : ki ∈ Z

⎫⎪⎪⎬⎪⎪⎭ .

It is easy to see that Γ is a discrete uniform subgroup of N and the basis givenabove is strongly based in Γ. Next, let e = {2, 3, 4, 6} . Then

Ω = {λ ∈ n∗ : λ (Z) = 0}

and

Λ = {λ ∈ Ω : λ (Y1) = λ (Y2) = λ (X1) = λ (X3) = 0} � R∗ × R.

Finally, the Plancherel measure is given by |λ1|2 dλ1dλ2 on R∗ × R.


Example 4. Let N be a nilpotent Lie subgroup of GL (10,R) such that a typicalelement of N is of the form⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 x1 x2 x3 −y1 0 −y2 − y3 z10 1 0 x2 x3 x2 −y2 −y1 − y3 0 z20 0 1 x3 x1 x1 −y3 −y2 −y1 z30 0 0 1 0 0 0 0 0 1

2y1

0 0 0 0 1 0 0 0 0 12y2

0 0 0 0 0 1 0 0 0 12y3

0 0 0 0 0 0 1 0 0 12x1

0 0 0 0 0 0 0 1 0 12x2

0 0 0 0 0 0 0 0 1 12x3

0 0 0 0 0 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

The Lie algebra of N is spanned by

{Z1, Z2, Z3, Y1, Y2, Y3, X1, X2, X3}with non-trivial Lie brackets

[X1, Y1] = Z1, [X1, Y2] = Z2, [X1, Y3] = Z3

[X2, Y1] = Z2, [X2, Y2] = Z3, [X2, Y3] = Z2

[X3, Y1] = Z3, [X3, Y2] = Z1, [X3, Y3] = Z1.

Therefore,

Λ =

⎧⎪⎪⎨⎪⎪⎩ λ ∈ n∗ : det

⎡⎣ λ (Z1) λ (Z2) λ (Z3)λ (Z2) λ (Z3) λ (Z2)λ (Z3) λ (Z1) λ (Z1)

⎤⎦ = 0, λ (Xi) = λ (Yi) = 0

1 ≤ i ≤ 3

⎫⎪⎪⎬⎪⎪⎭and the Plancherel measure is equivalent to

|(λ1 − λ3) (λ2 − λ3) (λ1 + λ2 + λ3)| dλ1dλ2dλ3

defined over a Zariski open subset of R3. For each λ ∈ Λ the corresponding irre-ducible representation πλ is realized as acting on L2

(R3)as follows (see [13] for

more details)

πλ (exp (t1X1 + t2X2 + t3X3)) f (x1, x2, x3) = f (x1 − t1, x2 − t2, x3 − t3) ,

πλ (exp (t1Y1 + t2Y2 + t3Y3)) f (x1, x2, x3)

= e

⎛⎜⎜⎝−2πi

⟨⎡⎢⎢⎣

λ1t1 + λ2t2 + λ3t3λ2t1 + λ2t3 + λ3t2λ1t2 + λ1t3 + λ3t1

⎤⎥⎥⎦,

⎡⎢⎢⎣

x1

x2

x3

⎤⎥⎥⎦⟩⎞

⎟⎟⎠f(x1, x2, x3),

and

πλ (exp (t1Z1 + t2Z2 + t3Z3)) f (x1, x2, x3)

= e

⎛⎜⎜⎝2πi

⟨⎡⎢⎢⎣

λ1

λ2

λ3

⎤⎥⎥⎦,

⎡⎢⎢⎣

t1t2t3

⎤⎥⎥⎦⟩⎞

⎟⎟⎠f (x1, x2, x3) .


To see more examples, we refer the reader to the book by Corwin and Greenleaf [6]which contains several other explicit examples.

3. Direct Integral Decompositions

In order to define a unitary dilation D on the Fourier transform side, we considerthe action of the dilation group on N . Let π be an irreducible representation of Nand let αm ∈ H. Define αm · π by

αm · π = π ◦ αm.

Then αm · π is irreducible, and may or may not be isomorphic with π. Thus wehave an action of H on N that may well be non-trivial.

At the same time H acts naturally on n∗ by αm ·λ(X) = λ(AmX), and we writeαm · λ = Amλ. Furthermore, since it is assumed that A is a diagonal matrix withrespect to the fixed strong Malcev basis of the Lie algebra n, then it is clear thatΛ is H-invariant, and we claim that the parametrization λ → πλ is H-equivariantwith respect to the action on N and Λ. To see this, we observe that the definition(2.1) shows that

P (Aλ) = α−1(P (λ)).

Next, we define a unitary representation C of the group H such that C : H →U(Hλ) and given g ∈ Hλ, C(α)g(x) = g(α(x))| detA|1/2. Thus, for p ∈ P (Aλ), x ∈N , C(α)g(xp) = χAλ(p)

−1C(α)g(x), and one easily checks that for each α ∈ H,the map C(α) : Hλ → HAλ is a unitary isomorphism. Moreover, for each m ∈ Z,

(3.1) C(αm)πλ(·) = πAmλ(·)C(αm).

Finally, identifying the Hilbert space Hλ with L2(Rd), given g ∈ L2(Rd), we havethe following:

C(α)g(t1, t2, . . . , td) = g(aj1t1, aj2t2, . . . , ajdtd) |aj1aj2 · · · ajd |1/2.

Let I be a countable set which is parameterizing an orthornormal basis forL2(Rd). Fix such orthonormal basis

B = {bκ : κ ∈ I}for L2

(Rd). It is worth noticing that in the case where N is abelian, then d =

0 and it is understood that the representation C is simply the one-dimensionaltrivial representation. Moreover, if N is commutative then L2

(Rd)= C and I is a

singleton. We recall that

AXk = akXk

for some real numbers ak. H acts trivially on Λ if and only if ak = 1 for all k ∈ e.So, we say that H acts non-trivially on Λ if and only if there exists some indexk ∈ e such that ak is not equal to 1.

Remark 5. Let G be a group acting on a set X. We say that the action iseffective if gx = x for all x in X implies that g is the identity in G. Therefore, ifH acts non-trivially on Λ then it must be the case that H acts effectively on the setΛ.

A measurable subset E of Λ is called a dilation tiling of Λ if AjE ∩ AmE = ∅for j = m, and ∪m∈ZA

mE is conull in Λ. We have the following:


Proposition 6. Suppose that H acts non-trivially on Λ, and let E be a dilationtiling of Λ. Then

V �∫ ⊕

E

⊕κ∈IIndN�HN (πλ) dλ.

Proof. We define a representation V of the group F as V (·) = P1V (·)P−11 .

Since P1 is a unitary map then clearly V and V are equivalent representations.

Moreover, V is acting in the Hilbert space∫ ⊕Λ

L2(Rd)⊗ L2(Rd)dλ. This proof will

be structured as follows. First, we will show that L2(N) can be decomposed intoa direct sum of multiplicity-free spaces which are stable under the action of therepresentation V. Second, we will obtain a decomposition of the restriction of Von each multiplicity-free subspace. Fix κ0 ∈ I. Let v = vκ0 : Λ → L2

(Rd)be a

measurable vector-valued function defined such that

v (λ) =∑j∈Z

C(α−j

)bκ0

1A−jE (λ) .

We write v (λ) = vλ. Now, we define the multiplicity-free Hilbert space

Hκ0={λ → uλ ⊗ vκ0

λ ∈ L2(Λ,HS

(L2

(Rd))

, dλ): uλ ∈ L2

(Rd)}

.

We would like to show that Hκ0is V -invariant space. Let g ∈ Hκ0

such thatg (λ) = uλ ⊗ vκ0

λ . First, it is easy to see that

V (x) g (λ) = πλ (x)uλ ⊗ vκ0

λ ∈ L2(Rd)⊗ vκ0

λ .

Next, let δ be defined such that d(α−mx) = δ(m)dx. For ease of notation write

V (m) = V (αm), αm ∈ H. Then

V (m) g (λ) = C (αm) ◦ g(A−mλ

)◦ C (αm)

−1δ (m)

1/2

= C (αm) uA−mλ ⊗ C (αm) vκ0

A−mλδ (m)1/2

= δ (m)1/2 C (αm) uA−mλ ⊗ C (αm) vκ0

A−mλ.

Since

vκ0

A−mλ =∑j∈Z

C(α−j

)bκ0

1A−jE

(A−mλ

)=∑j∈Z

C(α−j

)bκ0

1Am−jE (λ)

=∑s∈Z

C(α−s−m

)bκ0

1A−sE (λ)

=∑s∈Z

C(α−m

)C(α−s

)bκ0

1A−sE (λ) ,

then

V (m) g (λ) = δ (m)1/2 C (αm)uA−mλ ⊗(∑

s∈Z

C (αm)C(α−m

)C(α−s

)bκ0

1A−sE (λ)

)

= δ (m)1/2

C (αm)uA−mλ ⊗(∑

s∈Z

C(α−s

)bκ0

1A−sE (λ)

)= δ (m)1/2 C (αm)uA−mλ ⊗ v (λ) .


Thus, V (m) g (λ) ∈ L2(Rd)⊗ vκ0

λ . This shows that indeed, Hκ0is V -invariant.

Now, we define the unitary map

Φ = Φκ0: Hκ0

→ L2(E × Z, L2

(Rd), dλ

)�∫ ⊕

E

l2(Z, L2

(Rd))

dλ

such that for g ∈ Hκ0, we write g (λ) = ug

λ ⊗ vκ0

λ and

Φg (λ) =(C (α)

−jugAjλ |detA|j/2

)j∈Z

.

With some straightforward computations, we obtain

ΦV (x) g (λ) =(C (α)−j πAjλ (x)u

gAjλ |detA|j/2

)j∈Z

and

ΦV (m) g (λ) =(C (α)

m−jugAj−mλ |detA|

j−m2

)j∈Z

.

Let ρλ � IndN�HN (πλ) be realized as acting in l2

(Z, L2

(Rd))

. Then,

ρλ (x)Φg (λ) =(πλ

(αjx

)C (α)

−jugAjλ |detA|

j2

)j∈Z

=(C (α)−j πAjλ (x)u

gAjλ |detA|

j2

)j∈Z

= ΦV (x) g (λ) .

Similarly, it is easy to see that

ρλ (m)Φg (λ) = ΦV (m) g (λ) .

Thus, the restriction of V to the Hilbert space Hκ0is equivalent to∫ ⊕

E

IndN�HN (πλ) dλ.

Finally, we obtain

V � V �∫ ⊕

E

⊕κ∈IIndN�HN (πλ) dλ.

This concludes the proof. �

Lemma 7. If the action of H is trivial on Λ, then |detA| = 1.

Proof. Let v = R-span {Xi : i ∈ e}, and let λ ∈ Λ. We endow v with thebi-linear form ω defined by ω (X,Y ) = λ ([X,Y ]) . This bi-linear form is non-degenerate, and the vector space v together with the non-degenerate bilinear form ωhas the structure of a symplectic vector space (see Lemma 27 [11]). Next, since A isa diagonal matrix then v is A-invariant and the restriction of A to v is a symplectictransformation. Therefore |detA|v| = 1 and |detA| = |detA|v| |detA|n�v| = 1. �

Let E1 ⊆ Rd such that {Am1 E1 : m ∈ Z} is a measurable partition of Rd and A1

is the restriction of A to the vector space R-span {Xj1 , · · · , Xjd} . Fix an orthonor-mal basis for L2(E1). More precisely, {bj : j ∈ J} is a fixed orthonormal basis forL2(E1) and the set J is a parametrizing set for this orthonormal basis. The set Jwill be important for the following proposition.


Proposition 8. Assume that N is not commutative and that H acts triviallyon Λ. Then

V �∫ ⊕

Λ

∫ ⊕

T

⊕κ∈Jπλ,σ dσdλ

where πλ,σ(x) = πλ(x) for x ∈ Γα and πλ,σ(α) = C(α)χσ(α) and χσ is a characterof H.

Proof. We aim to construct an intertwining unitary operator for the repre-sentations described above. We recall that the representation V is equivalent to

V which is acting in∫ ⊕Λ

L2(Rd)⊗ L2

(Rd)dλ. To simplify our proof, we will only

consider rank-one operators; since their linear span is dense in the Hilbert space

of Hilbert–Schmidt operators. We recall that V = P1 ◦ V ◦ P−11 and we write

V =∫ ⊕Λ

Vλ dλ such that

Vλ (x) g (λ) = πλ (x)ugλ ⊗ vλ

for g (λ) = ugλ ⊗ vλ. It is also fairly easy to see that Vλ (α

m) g (λ) = C (αm) ◦ g (λ) ◦C (αm)−1 . We define a unitary map P1 : L2

(Rd)→

∫ ⊕E1

l2 (Z) dt where

P1f (t,m) =(f (Am

1 t) |detA1|m/2)m∈Z

.

We also define a unitary map: P2 :∫ ⊕E1

l2 (Z) dt →∫ ⊕E1

∫ ⊕T

C dσdt such that

P2

({(atk)k∈Z

}t∈E1

)(s, σ) =

∑k∈Z

aske2πikσ (Fourier transform of (ask)k∈Z .)

Now, we define the unitary map

Q : L2(Λ,HS

(L2

(Rd))

, dλ)−−−−−→

∫ ⊕

Λ

L2(Rd)⊗(∫ ⊕

E1

∫ ⊕

T

C dσdt

)dλ

as follows:

Q({uλ ⊗ vλ}λ∈Λ

)=({uλ ⊗ P2P1 (vλ)}λ∈Λ

).

We write Q =∫ ⊕Λ

Qλ dλ such that

QλVλ (x) (uλ ⊗ vλ) = Qλ ((πλ (x)uλ)⊗ vλ)

= πλ (x)Qλ (uλ ⊗ vλ)

and

QλVλ (m) (uλ ⊗ vλ) = Qλ (C (αm) uλ ⊗ C (αm) vλ)

= C (αm)uλ ⊗ P2P1C (αm) vλ

= C (αm)uλ ⊗ χσ (m)P2P1vλ.


The last equality above is justified by the following computations:

P2P1C (αm) vλ (t, σ) =∑k∈Z

(P1C (αm) vλ)tk e

2πikσ

=∑k∈Z

vλ(Ak−m

1 t)|detA1|

k−m2 e2πikσ

=∑l∈Z

vλ(Al

1t)|detA1|

l2 e2πi(l+m)σ

= e2πimσ∑l∈Z

vλ(Al

1t)|detA1|

l2 e2πilσ

= e2πimσP2P1vλ (t, σ) .

Let {bk : k ∈ J} be an orthonormal basis for L2 (E1) . Define the unitary map

Z :

∫ ⊕

E1

∫ ⊕

T

C dσdt →∫ ⊕

T

⊕k∈JC dσ

such thatZf (σ) = (〈f (·, σ) , bk〉)k∈J .

Via the map Z, we identify the Hilbert space∫ ⊕

Λ

L2(Rd)⊗(∫ ⊕

E1

∫ ⊕

T

C dσdt

)dλ

with ∫ ⊕

Λ

L2(Rd)⊗(∫ ⊕

T

⊕k∈JC dσ

)dλ

which is then identified with∫ ⊕

Λ

∫ ⊕

T

L2(Rd)⊗ (⊕k∈JC) dσdλ.

Via the identifications described above, it follows immediately that the quasi-regularrepresentation V is unitarily equivalent to the representation∫ ⊕

Λ

∫ ⊕

T

⊕k∈Jπλ,σ dσdλ.

This concludes the proof. �Proposition 9. If N = Rn is commutative and H acts trivially on Λ then

V �∫ ⊕

Λ

πλ dλ

where πλ

(x, α0

)= e2πi〈x,λ〉.

Proof. Since N is commutative, then all eigenvalues of the matrix A are

equal to one if H acts trivially on Λ = Rn. Via the Plancherel transform, the

representation T of N is unitarly equivalent to∫ ⊕Rn χλ dλ. Since the representation

D of H acts trivially on Λ, then it follows that V �∫ ⊕Λ

πλ dλ. �

The following lemma shows that a decomposition of V yields immediately adecomposition of W .


Lemma 10. Let N be a locally compact group, H a group of automorphismsof N , and let N0 be a subgroup of N that is normalized by H. Let π be a unitaryrepresentation of N . Then

IndN�HN (π)|N0

� IndN0�HN0

(π|N0).

Proof. Put τ = IndN�HN (π)|N , and τ0 = IndN�H

N (π)|N0which are acting in

the Hilbert spaces Hτ and Hτ0 respectively. Since elements of Hτ and Hτ0 areboth determined by their values on H , then the restriction map g → g|N0�H is anisomorphism of Hτ with Hτ0 . Given g ∈ Hτ , then for x ∈ N0, h ∈ H since N0 is nor-malized by H, τ (xh)g(k) = g(h−1x−1k)δ(h)−1/2 = g(h−1k(k−1x−1k))δ(h)−1/2 =π(k−1xk)g(h−1k) = τ0(xh)g(k). �

From the preceding results, the decomposition of the wavelet representationnow follows.

Theorem 3.1.1. Suppose that H acts non-trivially on Λ, then

W �∫ ⊕

E

(⊕κ∈IInd

Γα�HΓα

(πλ|Γα))dλ.

2. If H acts trivially on Λ, and if N is not commutative then the wavelet represen-tation is decomposed into a direct integral of representations as follows

W �∫ ⊕

Λ

∫ ⊕

T

(⊕κ∈Jπλ,σ|G) dσdλ.

3. If H acts trivially on Λ and if N is commutative then

W �∫ ⊕

Λ

πλ|Γ�H dλ.

We remark that the sets I and J are infinite sets if and only if N is not com-mutative. Thus in both decompositions above, if N is not commutative, the fiberrepresentations are always decomposable into direct sums of equivalent representa-tions. However, it is generally not true that the representations occurring in thedirect sums are irreducible. In two examples below, we consider dilations on thethree-dimensional Heisenberg group. For the first example, the representations oc-curring in the decomposition are direct sums of reducible representations. In thesecond example, the fiber representations occurring in the direct integral decompo-sitions are direct sums of equivalent irreducible representations. It then becomesobvious that sometimes the irreducibility of the representations occurring in thedirect sums depends on some properties of the action of the automorphism α onN. Surprisingly, we will see that there are also instances where the irreducibility ofthe representations occurring in the direct sums only depends on the Lie bracketstructure of n.


Example 11. Let N be as in Example 2: its Lie algebra is given by n =(X1, X2, X3)R such that [X3, X2] = X1. Let α ∈ Aut(N) be defined by α(expX1) =exp 2X1, α(expX2) = exp 2X2, α(expX3) = expX3. As seen in Example 2, wehave

Λ = {λX∗1 : λ = 0},

which we identify with R \ {0}, and for each λ ∈ Λ, Aλ = 2λ. If

Γ = expZX1 expZX2 expZX3

then

Γα ={exp

(2jk1X1

)exp

(2jk2X2

)exp (k3X3) : k1, k2, k3, j ∈ Z, j < 0

}.

Let E be a dilation tiling for Λ (for example, the Shannon set (−1,−1/2] ∪[1/2, 1).) Then

W �∫ ⊕

E

⊕k∈I IndΓα�HΓα

(πλ|Γα) dλ.

Moreover, the representation πλ|Γαacts on L2 (R) as follows:

πλ (expx1X1) f (t) = e2πiλx1f (t)

πλ (expx2X2) f (t) = e−2πiλx2tf (t)

πλ (expx3X3) f (t) = f (t− x3) .

Notice that IndΓα�HΓα

(πλ|Γα) is a reducible representation of Γα since for q < 1, the

linear span of the set (πλ|Γα) (Γα)χ[0,q] is not dense in L2 (R) .

It is easily seen that the closure of Γα in N is the group

N0 = {expx1X1 expx2X2 exp kX3 : k ∈ Z, x1, x2 ∈ R}.

Put P = P (λ) = exp (RX1 + RX2). Since expZX3 acts freely on P then σλ =

IndN0

P (χλ) is irreducible by Mackey theory, as is σtλ := σλ(exp−tX3·exp tX3), t ∈ R.

Now by inducing in stages, πλ � IndNN0(σλ), and a standard calculation shows then

that

πλ|N0�∫ ⊕

[0,1)

σtλ dt.

Since Γα is dense in N0, then σtλ|Γα

is also irreducible, so

πλ|Γα�∫ ⊕

[0,1)

σtλ|Γα

dt

is an irreducible decomposition of πλ|Γα. Finally, since H acts freely on Γα, then

IndΓα�HΓα

(σtλ|Γα

) is an irreducible representation of the wavelet group, and we ob-tain the irreducible decomposition

W �∫ ⊕

E


(πλ|Γα) dλ �

∫ ⊕

E

⊕k∈I

∫ ⊕

[0,1)

IndΓα�HΓα

(σtλ|Γα

)dt dλ.


Example 12. Again let N be the Heisenberg group with Lie algebra given asin the preceding. Now, let α ∈ Aut (N) be defined by

α (expX1) = expX1,

α (expX2) = expX2

2,

α (expX3) = exp(2X3)

and put

Γ = expZX1 expZX2 expZX3.

Then

W �∫ ⊕

R∗

∫ ⊕

T

⊕k∈J πλ,σ|G dλ dσ.

Moreover, πλ,σ|G is a representation of G acting in L2 (R) as follows

πλ,σ|G (expx1X1) f (t) = e2πiλx1f (t)

πλ,σ|G (expx2X2) f (t) = e−2πiλx2tf (t)

πλ,σ|G (expx3X3) f (t) = f (t− x3)

πλ,σ|G (α) = e2πiσ√2f (2t) .

Now it is easy to show that the group Γα generated by the sets αm(Γ) is in factdense in N . It follows that the representation πλ restricted to Γα is irreducible.Since πλ,σ|G is an extension of π|Γα

, then it is also irreducible. The above is anirreducible decomposition. We refer to Lemma 13 and Proposition 14 for a generalproof of this claim.

4. Irreducibility of the Fiber Representations

In this section, we would like to obtain some conditions on the irreducibility ofthe restrictions of irreducible representations of F to G. We recall that a matrix isan expansive matrix if and only if all its eigenvalues have absolute values strictlygreater than one.

Lemma 13. If A is expansive then Γα is dense in N in the subspace topologyof N.

Proof. Let us assume that A is expansive. It is enough to show that⋃k∈Z

Ak (ZX1 + · · ·+ ZXn)

is dense in n. Indeed if c : n → N is defined by

c

(n∑

i=1

xiXi

)= exp (x1X1) · · · exp (xnXn)

then

c

(⋃k∈Z

Ak (ZX1 + · · ·+ ZXn)

)= Γα.


For each eigenvalue ai of A,{Zaki : k ∈ Z

}is dense in R. Thus, given

x = c

(n∑

i=1

xiXi

)∈ N,

we have l1, · · · , ln,m1, · · · ,mn ∈ Z,mi > 0 such that for every ε > 0,

∣∣∣∣ liamii

− xi

∣∣∣∣ < ε.

Put m = maxi mi and let ji = liam−mii . Then for each i,

∣∣∣ jiami

− xi

∣∣∣ < ε since

liamii

=jia

−m+mii

amii

=jiami

.

So with k = −m,

∥∥∥∥∥Ak (j1X1 + · · ·+ jnXn)−n∑

i=1

xiXi

∥∥∥∥∥max−norm

< ε.

The above norm is the max norm obtained by naturally identifying the Lie algebran with the vector space Rn. �

Proposition 14. If A is expansive then πλ|Γαis irreducible.

Proof. If A is expansive then Γα is dense in N in the subspace topology. Nowlet f be an arbitrary element of L2

(Rd).Given a non-zero vector g, for any ε > 0, by

the irreducibility of πλ there exist {xk : 1 ≤ k ≤ m} ⊂ N and {ck : 1 ≤ k ≤ m} ⊂ Csuch that ∥∥∥∥∥

m∑k=1

ckπλ (xk) g − f

∥∥∥∥∥L2(Rd)

<ε

2.

Now, since Γα is dense in N and since πλ is a continuous representation, then thereexists {γk : 1 ≤ k ≤ m} ⊂ Γα such that

‖πλ (xk) g − πλ (γk) g‖ <ε

2m |ck|.


Now,∥∥∥∥∥m∑

k=1

ckπλ (γk) g − f

∥∥∥∥∥L2(Rd)

=

∥∥∥∥∥m∑

k=1

ckπλ (γk) g −m∑

k=1

ckπλ (xk) g +m∑

k=1

ckπλ (xk) g − f

∥∥∥∥∥L2(Rd)

≤∥∥∥∥∥

m∑k=1

ckπλ (γk) g −m∑

k=1

ckπλ (xk) g

∥∥∥∥∥L2(Rd)

+

∥∥∥∥∥m∑

k=1

ckπλ (xk) g − f

∥∥∥∥∥L2(Rd)

≤m∑

k=1

|ck| ‖πλ (γk) g − πλ (xk) g‖L2(Rd) +

∥∥∥∥∥m∑

k=1

ckπλ (xk) g − f

∥∥∥∥∥L2(Rd)

≤m∑

k=1

|ck|ε

2m |ck|+

ε

2

≤ ε

2+

ε

2= ε.

Thus, the linear span of πλ (Γα) g is dense in L2(Rd), and it follows that πλ|Γα

isirreducible. �

Example 15. Let N be a nilpotent Lie group with Lie algebra spanned by{Z1, Z2, X1, X2, Y } such that the only non-trivial Lie brackets are [Xi, Y ] = Zi.Now, let e = {3, 5} . Then

Ω =

{λ1Z

∗1 + λ2Z

∗2 + β1X

∗1 + β2X

∗2 + γY ∗ ∈ n∗ : λ1 = 0

(λ1, λ2, β1, β2, γ) ∈ R5

}and the unitary dual of N is parametrized by

Λ ={λ1Z

∗1 + λ2Z

∗2 + β2X

∗2 ∈ Ω : (λ1, λ2, β2) ∈ R3

}� R∗ × R2.

Now for each λ ∈ Λ, πλ is realized as acting in L2 (R) as follows:

πλ (expx1X1) f (y) = e2πix1yλ1f (y)

πλ (expx2X2) f (y) = e2πix2(β2+yλ2)f (y)

πλ (exp sY ) f (y) = f (y − s)

πλ (exp zkZk) f (y) = e2πizkλkf (y) .

Define

Γ = exp (ZZ1) exp (ZZ2) exp (ZX1) exp (ZX2) exp (ZY ) .

Now, let α ∈ Aut (N) such that

α (exp (z1Z1) exp (z2Z2) exp (x1X1) exp (x2X2) exp (yY ))

= exp (4z1Z1) exp (4z2Z2) exp (2x1X1) exp (2x2X2) exp (2yY ) .

Then

W �∫ ⊕

E


(πλ|Γα) dλ

and IndΓα�HΓα

(πλ|Γα) is irreducible for each λ ∈ E.


4.1. Bekka–Driutti Condition for Irreducibility. In this subsection, wewill recall a result of Bekka and Driutti (see [1].) Let

{X1, · · · , Xm, · · · , Xn}be a strong Malcev basis of n strongly based on Γ passing through [n, n] such thatdim [n, n] = m. Let p : n → [n, n] be the canonical projection. A subalgebra h isnot contained in a proper rational ideal of n if and only if p (h) is not contained ina proper rational subspace of n/ [n, n] . In fact, this is the case if and only if thereexists

X =

n∑i=1

xiXi ∈ h

such that dimQ {xm+1, · · · , xn} = n−m. Moreover, the restriction πλ|Γ of πλ to Γis irreducible if and only if the radical (see 2.2) n (λ) is not contained in a properrational ideal of n. Since Γ ⊆ Γα, then the following holds:

Proposition 16. If the radical n (λ) is not contained in a proper rational idealof n then πλ|Γα

is irreducible.

Example 17. Let N be the freely generated two step nilpotent Lie group withthree generators. The Lie algebra of N is spanned by {Z23, Z13, Z12, Z3, Z2, Z1} suchthat the only non-trivial Lie brackets are defined as follows: [Zi, Zj ] = Zij for i < j.Now let

Γ = expZZ23 expZZ13 expZZ12 expZZ3 expZZ2 expZZ1.

Thus, the basis above is a strong Malcev basis strongly based on Γ and is passingthrough the ideal [n, n] = z. It is not too hard to see that the radical correspondingto λ is:

n (λ) = z⊕ R (λ (Z23)Z1 − λ (Z13)Z2 + λ (Z12)Z3) .

It is clear that n (λ) is not contained in a proper rational ideal if

dimQ (λ (Z23) ,−λ (Z13) , λ (Z12)) = 3.

Next, define α such that

α (expZk) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

exp (2Zk) if k = 23exp (2Zk) if k = 13exp (Zk) if k = 12exp (2Zk) if k = 3exp (Zk) if k = 2exp (Zk) if k = 1

.

Then,

W �∫ ⊕

E


(πλ|Γα) dλ

whereE =

{At (±Z∗

23 + β1Z∗13 + β2Z

∗12 + β3Z

∗1 ) : t ∈ [0, 1) , βk ∈ R

}and almost every representation πλ|Γα

is irreducible.

Acknowledgment

Brad Currey and Vignon Oussa would like to thank the Graduate Center ofthe City University of New York for their hospitality. We also thank the referee forproviding constructive comments and help in improving the paper.


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[10] Lek-Heng Lim, Judith A. Packer, and Keith F. Taylor, A direct integral decomposition of thewavelet representation, Proc. Amer. Math. Soc. 129 (2001), no. 10, 3057–3067 (electronic),DOI 10.1090/S0002-9939-01-05928-7. MR1840112 (2002c:47146)

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[12] V. Oussa, Bandlimited Spaces on Some 2-step Nilpotent Lie Groups With One ParsevalFrame Generator, to appear in Rocky Mountain Journal of Mathematics

[13] V. Oussa, Sinc Type Functions on a Class of Nilpotent Lie groups, (2013), to appear inAdvances in Pure and Applied Mathematics

Department of Mathematics and Computer Science, St. Louis University, St. Louis,

Missouri 63103


Mathematics Department, Queensborough C. College, City University of New York,

Bayside, New York 11362


Department of Mathematics, Bridgewater State University, Bridgewater, Mas-

sachusetts 02324



Exponential splines of complex order

Peter Massopust

Abstract. We extend the concept of exponential B-spline to complex orders.This extension contains as special cases the class of exponential splines and alsothe class of polynomial B-splines of complex order. We derive a time domainrepresentation of a complex exponential B-spline depending on a single pa-rameter and establish a connection to fractional differential operators definedon Lizorkin spaces. Moreover, we prove that complex exponential splines giverise to multiresolution analyses of L2(R) and define wavelet bases for L2(R).

1. Preliminaries on Exponential Splines

Exponential splines are used to model phenomena that follow differential sys-tems of the form x = Ax, where A is a constant matrix. For such equations thecoordinates of the the solutions are linear combinations of functions of the typeeax and xneax, a ∈ R. In approximation theory exponential splines are modelingdata that exhibit sudden growth or decay and for which polynomials are ill-suitedbecause of their oscillatory behavior. Some of the mathematical issues regarding ex-ponential splines in the theory of interpolation and approximation can be found inthe following references: [ADM91,DM87,DM89,McC91,SU86,SU89,Spa69,UB05,ZRR00].

Another approach to exponential splines is based on certain classes of lineardifferential operators with constant coefficients. The original ideas of such an ap-proach can be found in, for instance, [Mic76,UB05] and in exposition in [Mas10].The classical polynomial splines s of order n, n ∈ N, can be interpreted as (distri-butional) solutions to equations of the form

(1) Dns =

n∑�=1

c� δ(· − �), c� ∈ R.

where D denotes the (distributional) derivative and δ the Dirac delta distribution.A well known class of splines of central importance is the class of polynomial B-splines Bn, defined as the n-fold convolution of the characteristic function of theunit interval. Polynomial B-splines satisfy the above equation and they lay thefoundations for further generalizations.

2010 Mathematics Subject Classification. Primary 41A15, 65D07; Secondary 26A33, 46F25.Key words and phrases. Polynomial B-splines, exponential splines, exponential B-splines,

complex B-splines, complex exponential B-splines, scaling function, multiresolution analysis, Rieszbasis, wavelet, Lizorkin space, fractional differential and integral operator.


87

88 PETER MASSOPUST

Equation (1) is a special case of the more general expression

(2) Lnf :=

n∏j=1

(D + ajI)f =

n∑�=1

c� δ(· − �), aj , c� ∈ R.

where I denotes the identity operator. Solutions to (2) are then called exponen-tial splines and they reduce to polynomial splines in case all aj = 0. For laterpurposes, we record a particular identity involving a special case of (2): If, for allj ∈ {1, . . . , n}, aj =: a ∈ R, then

(3) (D + aI)n(e−a(•)f) = e−a(•)Dnf, n ∈ N.

One can define the differential operator on the left-hand side in this manner andshow that this definition is equivalent to the usual definition involving the binomialtheorem for linear differential operators with constant coefficients:

(D + aI)n =

n∑k=0

(n

k

)akDn−k.

However, for our later purposes of replacing the integer n by a complex number z,such a finite expression is not available and we need to resort to (3) as the basicidentity.

In this paper, we extend the concept of exponential spline to include complexorders in the defining equation (2). For this purpose, we need to extend the dif-ferential operator Ln to a fractional differential operator Lz of complex order zdefined on an appropriate function space. We obtain the generalization of expo-nential splines to complex order via exponential B-splines. To this end, we brieflyreview polynomial and exponential B-splines, and revisit the definition of polyno-mial B-splines of complex order. The former is done in Section 2 and the latterin Section 3. In Section 3, we also introduce the fractional derivative operatorsand function spaces needed for the generalization. Exponential splines of complexorder depending on one parameter are then defined in Section 4. For this purpose,we first introduce exponential B-splines of complex order, for short complex expo-nential B-splines, in the Fourier domain and discuss some of their properties. Inparticular, we derive a time domain representation and show that this new classof splines defines multiresolution analyses of and wavelet bases for L2(R). At thispoint, we also establish the connection to fractional differential operators of com-plex order defined on Lizorkin spaces. A brief discussion of how to incorporatemore than one parameter into the definition of a complex exponential B-spline andthe derivation of an explicit formula in the time domain for complex exponentialB-splines depending on two parameters concludes this section. The last sectionsummarizes the results and describes future work.

2. Brief Review of Polynomial and Exponential B-Splines

Based on the interpretation (2) one defines, analogously to the introduction ofB-splines, exponential B-splines as convolution products of exponential functionsea(·) restricted to [0, 1]. In this section, we briefly review the definitions of polyno-mial and exponential B-splines. For more details, we refer the interested reader tothe vast literature on spline theory.

EXPONENTIAL SPLINES OF COMPLEX ORDER 89

2.1. Polynomial B-Splines. Let n ∈ N. The nth order classical Curry-Schoenberg (polynomial) B-splines [CS47] are defined as the n-fold convolutionproduct of the characteristic function χ = χ[0,1] of the unit interval:

Bn :=n∗

j=1χ.

Equivalently, one may define Bn in the Fourier domain as

F(Bn)(ω) =: Bn(ω) :=

∫R

Bn(x) e−iωx dx =

(1− e−iω

iω

)n

,

where F denotes the Fourier–Plancherel transform on L2(R).Polynomial B-splines generate a discrete family of approximation/interpolation

functions with increasing smoothness:

Bn ∈ C n−2, n ∈ N.

and possess a natural multiscale structure via knot insertions. (Here, C−1 is in-terpreted as the space of piece-wise continuous functions.) In addition, polynomialB-splines generate approximation spaces and satisfy several recursion relations thatallow fast and efficient computations within these spaces.

2.2. Exponential B-Splines. Exponential B-splines of order n ∈ N are de-fined as n-fold convolution products of exponential functions of the form ea(·) re-stricted to the interval [0, 1]. More precisely, let n ∈ N and a := (a1, . . . , an) ∈ Rn,with at least one aj = 0. Then the exponential B-splines of order n for the n-tupleof parameters a is given by

(4) Ean :=

n∗j=1

(eaj(•)χ

).

This class of splines shares several properties with the classical polynomial B-splines, but there are also significant differences that makes them useful for differentpurposes. In [CM12], a useful explicit formula for these functions was derived andthose cases characterized for which the integer translates of an exponential B-splineform a partition of unity up to a multiplicative factor, i.e.,∑

k∈Z

Ean(x− k) = C, x ∈ R,

for some nonzero constant C.Moreover, series expansions for functions in L2(R) in terms of shifted and modu-

lated versions of exponential B-splines were derived, and dual pairs of Gabor framesbased on exponential B-splines constructed. We note that exponential B-splines alsohave been used to construct multiresolution analyses and obtain wavelet expansions.(See, for instance, [UB05,LY11].) In addition, it is shown in [CG14] that expo-nential splines play an important role in setting up a one-to-one correspondencebetween dual pairs of Gabor frames and dual pairs of wavelet frames.

3. Polynomial Splines of Complex Order

Now, we like to extend the concept of cardinal polynomial B-splines to ordersother than n ∈ N. Such an extension to real orders was investigated in [UB00,Zhe82]. In [UB05], these splines were named fractional B-splines. Their extension

90 PETER MASSOPUST

to complex orders was undertaken in [FBU06]. The resulting class of cardinal B-splines of complex order or, for short, complex B-splines, Bz : R → C are definedin the Fourier domain by

(5) F(Bz)(ω) =: Bz(ω) :=

∫R

Bz(t)e−iωt dt :=

(1− e−iω

iω

)z

,

for z ∈ C>1 := {ζ ∈ C : Re ζ > 1}. At the origin, there exists a continuous

continuation satisfying Bz(0) = 1. Note that as { 1−e−iω

iω | ω ∈ R} ∩ {y ∈ R | y <0} = ∅, complex B-splines reside on the main branch of the complex logarithm andare thus well-defined.

The motivation behind the definition of complex B-splines is the need for asingle-band frequency analysis. For some applications, e.g., for phase retrieval tasks,complex-valued analysis bases are needed since real-valued bases can only provide asymmetric spectrum. Complex B-splines combine the advantages of spline approx-imation with an approximate one-sided frequency analysis. In fact, the spectrum

|Bz(ω)| has the following form. Let

(6) Ω(ω) :=1− e−iω

iω.

Then the spectrum consists of the spectrum of a real-valued B-spline, combinedwith a modulating and a damping factor:

|Bz(ω)| = |BRe z(ω)|e−iIm z ln |Ω(ω)|eIm z arg Ω(ω).

The presence of the imaginary part Im z causes the frequency components on thenegative and positive real axis to be enhanced with different signs. This has theeffect of shifting the frequency spectrum towards the negative or positive frequencyside, depending on the sign of Im z. The corresponding bases can be interpreted asapproximate single-band filters [FBU06].

For the purposes of this article, we summarize some of the most important prop-erties of complex B-splines. Complex B-splines have a time-domain representationof the form

(7) Bz(x) =1

Γ(z)

∞∑k=0

(−1)k(zk

)(x− k)z−1

+ ,

where the equality holds point-wise for all x ∈ R and in the L2(R)–norm [FBU06].Here, the complex valued binomial is defined by(

z

k

):=

Γ(z + 1)

Γ(k + 1)Γ(z − k + 1),

where Γ : C \ Z−0 → C denotes the Euler Gamma function, and

xz+ :=

{xz = ez lnx, x > 0;

0, x ≤ 0,

is the complex-valued truncated power function. Formula (7) can be verified byFourier inversion of (5).

Equation (7) shows that Bz is a piecewise polynomial of complex degree z − 1and that its support is, in general, not compact. It was shown in [FBU06] thatBz belongs to the Sobolev spaces W s(L2(R)) for Re z > s+ 1

2 (with respect to the


L2-norm and with weight (1 + |x|2)s). The smoothness of the Fourier transform ofBz implies fast decay in the time domain:

Bz(x) ∈ O(x−m), for m < Re z + 1, as |x| → ∞.

If z, z1, z2 ∈ C>1, then the convolution relation

Bz1 ∗Bz2 = Bz1+z2

and the recursion relation

Bz(x) =x

z − 1Bz−1(x) +

z − x

z − 1Bz−1(x− 1)

holds. Complex B-splines generate a continuous family of approximation/interpo-lation functions in the sense that they are elements of (inhomogenous) Holder spaces[UB00]:

Bz ∈ Cs, s := Re z − 1, z ∈ C>1.

In addition, complex B-splines are scaling functions, i.e., they satisfy a two-scalerefinement equation, generate multiresolution analyses and wavelets, and relatedifference and differential operators. For more details and other properties of thisnew class of splines, we refer the interested reader to [FM09b,FsMS13,FMU12,FM11,FM09a,FM08,MF10,MF07,Mas10,Mas12,Mas09].

Based on the more general definition [KMPS76,Mas10,UB05] of polynomialsplines s : [a, b] → R of order n as the solution of a differential equation of the form

Dns =n∑

�=0

c�δx− �, c� ∈ R,

where D denotes the distributional derivative and δxνthe Dirac distribution at

� ∈ Z, one can define a spline of complex order z in a similar manner [FM11]. Forthis purpose, we denote by S(R) the Schwartz space of rapidly decreasing functionson R, and introduce the Lizorkin space

Ψ := {ψ ∈ S(R) : Dmψ(0) = 0, ∀m ∈ N} ,and its restriction to the nonnegative real axis:

Ψ+ := {f ∈ Ψ : supp f ⊆ [0,∞)}.Let C+ := {z ∈ C : Re z > 0} and define a kernel function Kz : R → C by

Kz(x) :=xz−1+

Γ(z).

For an f ∈ Ψ+, define a fractional derivative operator Dz of complex order z onΨ+ by

(8) Dzf := (Dnf) ∗Kn−z︸︷︷︸(Caputo)

= Dn(f ∗Kn−z)︸︷︷︸(Riemann–Liouville)

, n = Re z!.

where ∗ denotes the convolution on Ψ+. Note that, since we are defining thefractional derivative operator on the Lizorkin space Ψ+, the Caputo and Riemann-Liouville fractional derivatives coincide.

Similarly, a fractional integral operatorD−z of complex order z on Ψ+ is definedby

D−zf := f ∗Kz.

92 PETER MASSOPUST

It follows from the definitions of Dz and D−z that f ∈ Ψ+ implies D±zf ∈ Ψ+

and that all derivatives of D±zf vanish at x = 0. The convolution-based definitionof the fractional derivative and integral operator of functions f ∈ Ψ+ also ensuresthat {Dz : z ∈ C} is a semi-group in the sense that

(9) Dz+ζ = DzDζ = DζDz = Dζ+z,

for all z, ζ ∈ C. (See, for instance, [Pod99].)For our later purposes, we need to define fractional derivative and integral

operators D±z on the dual space Ψ′+ of Ψ+. To this end, we may regard the locally

integrable function Kz, Re z > −1, as an element of Ψ′+ by setting

〈Kz, ϕ〉 =∫ ∞

0

Kz(t)ϕ(t)dt, ∀ϕ ∈ Ψ+.

Here, 〈•, •〉 denotes the canonical pairing between Ψ+ and Ψ′+. In passing, we like

to mention that the function z → 〈Kz, ϕ〉, ϕ ∈ Ψ+, is holomorphic for all z ∈ C\N0.

Remark 3.1. The function Kz may also be defined for general z ∈ C viaHadamard’s partie finie and represents then a pseudo-function. For more details,we refer the interested reader to [DL00,GS59], or [Zem87].

Note that for f, g ∈ Ψ′+ the convolution exists on Ψ′

+ and is defined in theusual way [SKM87] by

(10) 〈f ∗ g, ϕ〉 := 〈(f × g)(x, y), ϕ(x+ y)〉 = 〈f(x), 〈g(y), ϕ(x+ y)〉〉, ϕ ∈ Ψ+.

The pair (Ψ′+, ∗) is a convolution algebra with the Dirac delta distribution δ as its

unit element. Thus, we can extend the operators D±z to Ψ′+ in the following way.

Let z ∈ C+, let ϕ ∈ Ψ+ be a test function and f ∈ Ψ′+. Then the fractional

derivative operator Dz on Ψ′+ is defined by

〈Dzf, ϕ〉 := 〈(Dnf) ∗Kn−z, ϕ〉, n = Re z!,

and the fractional integral operator D−z by 〈D−zf, ϕ〉 := 〈f ∗ Kz, ϕ〉. The semi-group properties (3) also hold for f ∈ Ψ′

+. (For a proof, see [Pod99] or [GS59].)In [Pod99,SKM87,GS59] it was shown that the z-th derivative of a truncated

power function is given

(11) Dz

[(x− k)z−1

+

Γ(z)

]= δ(x− k), k < x ∈ [0,∞).

Thus, by the semi-group properties of Dz, one obtains D−zδ(• − k) =(•−k)z−1

+

Γ(z) .

Now, we are ready to define a spline of complex order z [FM11]: Let z ∈ C+

and let {ak : k ∈ N0} ∈ �∞(R). A solution of the equation

(12) Dzf =

∞∑k=0

ak δ(• − k)

is called a spline of complex order z.It can be shown [FM11] that the complex B-spline

Bz(x) =1

Γ(z)

∞∑k=0

(−1)k(z

k

)(x− k)z−1

+ , z ∈ C>1.


is a solution of Equation (12) with

ak = (−1)k(z

k

),

and is thus a nontrivial example of a spline of complex order.

4. Exponential Splines of Complex Order

In this section, we to extend the concept of exponential B-spline to incorporatecomplex orders while maintaining the favorable properties of the classical exponen-tial splines.

4.1. Definition and basic properties. To this end, we take the Fouriertransform of an exponential function of the form e−ax, a ∈ R, and define in completeanalogy to (5), an exponential B-spline of complex order z ∈ C>1 for a ∈ R (forshort, complex exponential B-spline) in the Fourier domain by

(13) Eaz (ω) :=

(1− e−(a+iω)

a+ iω

)z

.

We set

Ω(ω, a) :=1− e−(a+iω)

a+ iω,

and note that trivially Ω(ω, 0) = Ω(ω) and, therefore, E0z = Bz; see (6).

The function Ω(•, a) only well-defined for a ≥ 0. We may verify this as follows.The real part and imaginary parts of Ω(•, a) are explicitly given by

ReΩ(ω, a) := f(ω, a) =e−aω sinω − e−aa cosω + a

a2 + ω2,

ImΩ(ω, a) := g(ω, a) =ae−a sinω + e−aω cosω − ω

a2 + ω2.

If g(ω, a) = 0 then a sinω = eaω − ω cosω. Therefore,

f(ω, a) =e−a

a(a2 + ω2)

(aω sinω − a2 cosω + a2ea

)=

e−a

a(a2 + ω2)

(eaω2 − ω2 cosω − a2 cosω + a2ea

)=

e−a

a(ea − cosω) ≥ 0, only if a ≥ 0.

Thus, the graph of Ω(ω, a) does not cross the negative x-axis; see Figure 1 forexamples reflecting the different choices for a.

In particular, this implies that the infinite series

Ω(ω, a)z =∞∑�=0

(z

�

)(−1)�

e−(a+iω)�

(a+ iω)z

converges absolutely for all ω ∈ R.As a side result, which is summarized in the following proposition, we obtain

the asymptotic behavior of Ω(•, a) as a → ∞.

94 PETER MASSOPUST

Figure 1. The graph of Ω(•, a): a = 2 (left) and a = −1,−2 (right).

Proposition 4.1. The real and imaginary parts of Ω(•, a) satisfy the identity(f(ω, a)− 1

2a

)2

+ g(ω, a)2 =1

4a2+

e−a(−ω sin(ω)

a + e−a − cos(ω))

a2 + ω2, ω ∈ R.

Furthermore, the curve C(a) := {(f(ω, a), g(ω, a)) : ω ∈ R} approaches the circle

K(a) :

(x− 1

2a

)2

+ y2 =1

4a2,

in the sense that

|C(a)−K(a)| ∈ e−2aO(a−2) + e−aO(a−2), a " 1.

Proof. The first statement is a straight-forward algebraic verification, and

the second statement follows from the linearization ofe−a(−ω sin(ω)

a +e−a−cos(ω))a2+ω2 . �

Remark 4.2. For real z > 0, the function Ω(ω)z and its time domain represen-tation were already investigated in [Wes74] in connection with fractional powersof operators and later also in [UB00] in the context of extending Schoenberg’spolynomial splines to real orders. In the former, a proof that this function is inL1(0,∞) was given using arguments from summability theory (cf. Lemma 2 in[Wes74]), and in the latter the same result was shown but with a different proof.In addition, it was proved in [UB00] that for real z, Ω(ω)z ∈ L2(R) for z > 1/2(using our notation). (Cf. Theorem 3.2 in [UB00].)

To obtain a relationship between Ω(•, a) and Ω, we require a lemma whosestraightforward proof is omitted.

Lemma 4.3. For all x ∈ R, we have the following inequalities between cos andcosh.

1− cosx

x2≤ 1

2≤ coshx− 1

x2.

Our next goal is to obtain inequalities relating Ω(•, a) to Ω. To this end, noticethat

|Ω(ω, a)|2 =

∣∣∣∣1− e−(a+iω)

a+ iω

∣∣∣∣2 =2e−a(cosh a− cosω)

a2 + ω2.(14)


Employing the statement in Lemma (4.3), we see that

cosh a− 1

a2≥ 1

2≥ 1− cosω

ω2, ∀ a ∈ R; ∀ω ∈ R

The latter inequality is equivalent to the following expressions:

ω2(cosh a− 1) ≥ a2(1− cosω) ⇐⇒ ω2 cosh a− ω2 ≥ a2 − a2 cosω

⇐⇒ ω2 cosh a− ω2 cosω ≥ a2 + ω2 − a2 cosω − ω2 cosω

⇐⇒ cosh a− cosω

a2 + ω2≥ 1− cosω

ω2

Therefore, (14) implies

|Ω(ω, a)|2 =2e−a(cosh a− cosω)

a2 + ω2≥ e−a 2(1− cosω)

ω2= e−a |Ω(ω)|2.

A straightforward computation using again the inequalities in Lemma 4.3 showsthat

|Ω(ω, a)| ≤ 1− e−a

a, ∀ a > 0; ∀ω ∈ R.

As the right-hand side of the above inequality is bounded above by 1, we obtain anupper bound for |Ω(ω, a)| in the form

|Ω(ω, a)| ≤ 1 + |Ω(ω)|.These two results are summarized in the next proposition.

Proposition 4.4. For all a > 0 and all ω ∈ R, the following inequalities hold:

(15) e−a/2|Ω(ω)| ≤ |Ω(ω, a)| ≤ 1 + |Ω(ω)|.

Next, we use the inequalities in the above proposition to establish lower and

upper bounds for Eaz in terms of Bz.

Proposition 4.5. For all z ∈ C>1 and for all a > 0, we have that

(16) e−aRe z/2−2π|Im z| |Bz| ≤ |Eaz | ≤ 1 + 2Re ze2π|Im z||Bz|.

Proof. Let z ∈ C>1 and a > 0. Then, the following estimates hold

|Bz| = |Ωz| = |Ω|Re z e−Im zArg Ω ≤ eaRe z/2|Ω(•, a)|Re z e−Im zArg Ω

= eaRe z/2|Ω(•, a)|Re z e−Im zArg Ω(•,a) eIm z(ArgΩ(•,a)−Arg Ω)

≤ eaRe z/2|Eaz | e2π|Im z|,

implying the lower bound. To verify the upper bound, note that

|Eaz | = |Ω(•, a)z| = |Ω(•, a)|Re z e−Im zArg Ω(•,a) ≤ (1 + |Ω|)Re z e−Im zArgΩ(•,a)

≤ 1 + 2Re z|Ω|Re ze−Im zArg Ω eIm z(ArgΩ−Arg Ω(•,a))

≤ 1 + 2Re z e2π|Im z| |Bz|.Above, we used the fact that (1 + x)p ≤ 1 + 2pxp, for 0 ≤ x ≤ 1 and p ≥ 1. �

The upper bound in (16) together with the arguments employed in [UB00,Theorem 3.1] and [FBU06, 5.1] immediately yield the next result.

Proposition 4.6. The complex exponential B-spline Eaz , a ≥ 0, is an element

of L2(R) for Re z > 12 and of the Sobolev spaces W s(L2(R)) for Re z > s+ 1

2 .

96 PETER MASSOPUST

We finish this subsection by mentioning that the complex exponential B-splineEa

z has a frequency spectrum analogous to that for complex B-splines consisting ofa modulating and a damping factor:

|Eaz (ω)| = |Ea

Re z(ω)|e−iIm z ln |Ω(ω,a)|eIm z arg Ω(ω,a).

Hence, complex exponential splines combine the advantages of exponential splinesas described at the beginning of Section 2 with those of complex B-splines as men-tioned in Section 3.

In Figure (2), the graphs of Eaz for z = 2 + k/4 + i, k = 0, 1, 2, 3, 4, and a = 1

are displayed.

Figure 2. The graphs of E1z for z = 2 + k/4 + i, k = 0, 1, 2, 3, 4:

Real part (upper left), imaginary part (upper right), and modulus(lower middle).

4.2. Time domain representation. Next, we derive the time domain rep-resentation for a complex exponential B-spline Ea

z . For this purpose, we introducethe (backward) exponential difference operator ∇a acting of functions f : R → Rvia

∇af := f − e−af(• − 1), a ∈ R+0 .

For an n ∈ N, the n-fold exponential difference operator is then given by ∇na :=

∇a(∇n−1a ) with ∇1

a := ∇a. A straightforward calculation yields an explicit formulafor ∇n

a :

(17) ∇naf =

∞∑�=0

(n

�

)(−1)�e−�af(• − �).


In the above expression, we replaced the usual upper limit of summation n by ∞.This does not alter the value of the sum since for � > n the binomial coefficientsare identically equal to zero.

Based on the expression (17), we define a (backward) exponential differenceoperator of complex order ∇z

a as follows.

∇zaf :=

∞∑�=0

(z

�

)(−1)�e−�af(• − �), z ∈ C>1.

Theorem 4.7. Let z ∈ C>1. Then the complex exponential B-Spline Eaz pos-

sesses a time domain representation of the form

(18) Eza(x) =

1

Γ(z)

∞∑�=0

(z

�

)(−1)�e−�ae

−a(x−�)+ (x− �)z−1

+ ,

where e(•)+ := χ[0,∞) e

(•) and x+ := max{x, 0}. The sum converges both point-wise

in R and in the L2–sense.

Proof. For z ∈ C>1 and a > 0, we consider the Fourier transform (in thesense of tempered distributions) of the function ∇z

ae−ax+ xz−1

+ .

1

Γ(z)(∇z

ae−ax+ xz−1

+ )∧ =1

Γ(z)

∞∑�=0

(z

�

)(−1)�e−�a

∫R

e−a(x−�)+ (x− �)z−1

+ e−iωxdx

=1

Γ(z)

∞∑�=0

(z

�

)(−1)�e−�a

∫ ∞

�

e−a(x−�)(x− �)z−1 e−iωxdx

=1

Γ(z)

∞∑�=0

(z

�

)(−1)�e−�a

∫ ∞

0

e−axxz−1 e−iω(x+�)dx

=1

Γ(z)

∞∑�=0

(z

�

)(−1)�

∫ ∞

0

xz−1 e−(a+iω)(x+�)dx,

where the interchange of sum and integral is allowed by the Fubini–Tonelli Theoremusing the fact that the sum over the binomial coefficients is bounded. (See, forinstance, [AS65] for the asymptotic behavior of the Gamma function.)

Using the substitution (a + iω)x → t, the integral on the left becomes theGamma function up to a multiplicative factor:

e−(a+iω)�

(a+ iω)zΓ(z).

Thus,

1

Γ(z)(∇z

ae−ax+ xz−1

+ )∧ =∞∑�=0

(z

�

)(−1)�

e−(a+iω)�

(a+ iω)z= Ω(ω, a)z.

Employing a standard density argument, we deduce the validity of the above equal-ity for both the L1(R)– and L2(R)–topology. �

It follows directly from the time representations (7) and (18) for complex B-splines, respectively, complex exponential B-splines that

|Eaz (x)| ≤ e−ax

+ |Bz(x)|, ∀x ∈ R+0 ; ∀ a ∈ R+

0 .

98 PETER MASSOPUST

Complex exponential B-splines also satisfy a partition of unity property up to amultiplicative constant:∫

R

Eaz (x)dx = Ea

z (0) =

(1− e−a

a

)z

= 0.

The graphs of some complex exponential B-splines are shown in Figure 3.

Figure 3. The graphs of E1.3z for z = 3+k/4+ i, k = 0, 1, 2, 3, 4]:

Real part (upper left), imaginary part (upper right), and modulus(lower middle).

4.3. Multiresolution and Riesz bases. In this subsection, we investigatemultiscale and approximation properties of the complex exponential B-splines. To

this end, we consider the relationship between Eaz (•) and E2a

z (2 •). (The case ofreal z > 1 was first considered in [UB05] and then also in [Mas10].)

Under the assumptions z ∈ C>1 and a > 0, the following holds:

Ω(2ω, 2a) =1− e−(2a+2iω)

2a+ 2iω=

(1 + e−(a+iω))(1− e−(a+iω))

2(a+ iω)

=

(1 + e−(a+iω)

2

)Ω(ω, a).

This then implies that

E2az (2ω) =

(1 + e−(a+iω)

2

)z

Eaz (ω) =: 2H0(ω, a) E

az (ω).

Therefore, the low pass filter H0(ω, a) is given by

H0(ω, a) =1

2z−1

∞∑k=0

(z

k

)e−(a+iω)k,


from which we immediately derive a two-scale relation between complex exponentialB-splines:

(19) E2az (x) =

1

2z

∞∑�=0

(z

k

)e−ak Ea

z (2x− k).

Denote by T : L2(R) → L2(R) the unitary translation operator defined byTf := f(• − 1).

Proposition 4.8. Let z ∈ C>1 and a ≥ 0. Then the system {T kEaz : k ∈ Z}

is a Riesz sequence in L2(R).

Proof. It suffices to show that there exist constants 0 < A ≤ B < ∞ so that

A ≤∑k∈Z

∣∣∣Eaz (ω + 2πk)

∣∣∣2 ≤ B.

To this end, we use the fact that the complex B-splines form a Riesz sequence ofL2(R) [FBU06, Theorem 9], and employ Proposition 4.5. �

Corollary 4.9. Suppose that z ∈ C>1 and a ≥ 0. Let

V a0 := span {T kEa

z : k ∈ Z}L2(R)

.

Then {T kEaz : k ∈ Z} is a Riesz basis for V a

0 .

For j ∈ Z, we define

V 2jaj := span {E2ja

z (2j • −k) : k ∈ Z}L2(R)

.

Then

V 2jaj ⊂ V 2j+1a

j+1 , ∀ j ∈ Z.

To establish that the ladder of subspaces {V 2jaj : j ∈ Z} forms a multiresolution

analysis of L2(R), we use Theorem 2.13 in [Woj97]. We have already shown thatassumptions (i) (existence of a Riesz basis for V a

0 ) and (ii) (existence of a two-scale

relation) in this theorem hold. The third assumption requires that Eaz is continuous

at the origin and Eaz (0) = 0. Both requirements are immediate from (13). Hence,

we arrive at the following result.

Theorem 4.10. Assume that a ∈ R+0 and z ∈ C>1. Let ϕ

2jaz;j,k := E2ja

z (2j •−k).Then the spaces

V 2jaj := span {ϕ2ja

z;j,k : k ∈ Z}L2(R)

, j ∈ Z

form a dyadic multiresolution analysis of L2(R) with scaling function ϕaz;0,0 = Ea

z .

Denote the wavelet associated with Eaz by θaz , and the autocorrelation function

of θaz by

Raz(ω) :=

∑k∈Z

|θaz (ω + k)|2.

Then ψaz := θaz/

√Ra

z is an orthonormal wavelet, i.e., 〈ψaz , Tkψa

z 〉 = δk0, ∀ k ∈ Z.

100 PETER MASSOPUST

4.4. Connection to fractional differential operators. Our next goal is torelate complex exponential B-splines to fractional differential operators of type (8)considered in the previous section. For this purpose, we assume throughout thissubsection that a ≥ 0 and z ∈ C>1.

As Eaz is in L1

loc, we see that Eaz is in the Lizorkin dual Ψ′

+. Therefore, DzEaz

exists and we can define the following operator. (See also the comment at the endof Section 2.)

Definition 4.11. Let the fractional differential operator (D+aI)z : Ψ′+ → Ψ′

+

be defined by

(20) (D + aI)z(e−a(•)f) := e−a(•)Dzf.

The next results show that the fractional differential operator (20) satisfiesproperties similar to those of the associated differential operator of positive integerorder.

Proposition 4.12. For the fractional differential operator (D + aI)z definedin (20), the following statements hold.

(i) As f ≡ 1 ∈ Ψ′+, the function e−a(•) ∈ ker(D + aI)z.

(ii) The complex monomials (•)z−1 are in Ψ′+ implying that (•)z−1 e−a(•) ∈

ker(D + aI)z.

Moreover,

(21) (D + aI)zEaz =

∞∑�=0

[(z

�

)(−1)�e−�a

]δ(• − �).

Proof. In order to verify statements (i) and (ii), note that 〈Dz1, ϕ〉 = 0 and〈Dz(•)z−1, ϕ〉 = Γ(z)〈δ, ϕ〉 = Γ(z)ϕ(0) = 0, for all ϕ ∈ Ψ+. The conclusions nowfollow from (20).

Equation (21) is a consequence of (i), (ii), as well as definition (20) of theoperator (D + aI)z, and the fact that Bz satisfies (12). �

Equation (21) suggests a more general definition of exponential spline of com-plex order.

Definition 4.13. An exponential spline of complex order z ∈ C>1 correspond-ing to a ∈ R+ is any solution of the fractional differential equation

(22) (D + aI)zf =∞∑�=0

c� δ(• − �),

for some �∞-sequence {c� : � ∈ N}.

Clearly, the complex exponential spline Eaz is a nontrivial solution of (22). The

coefficients c� are bounded since∣∣∣∣∣∞∑�=0

c�

∣∣∣∣∣ =∣∣∣∣∣∞∑�=0

(z

�

)(−1)�e−�a

∣∣∣∣∣ ≤∣∣∣∣∣∞∑�=0

(z

�

)∣∣∣∣∣ ≤ c e|z−1|,

for some constant c > 0. (See the proof of Theorem 3 in [FBU06] for the finalinequality.)


4.5. Generalities. So far, we have only considered complex exponential B-splines that depend on one parameter a ∈ R+. Now we will briefly look at themore general setting based on (4). To this end, let a := (a1, . . . , an) ∈ (R+

0 )n be

an n-tuple of parameters with at least one aj = 0.

Let z := (z1, . . . , zn) ∈ Cn>1 :=

n×j=1

C>1 and define

Eaz :=

n∗j=1

Eajzj ,

or, equivalently,

Eaz :=

n∏j=1

(1− e−(aj+iω)

aj + iω

)zj

.

As above, we have that∫R

Eaz (x)dx =

n∏j=1

(1− e−aj

aj

)zj

= 0.

For illustrative purposes, let us consider the case n := 2, and set a := a1,b := a2, z := z1 and ζ := z2. Using the time domain representation of Ea

z andEb

ζ , we can compute the time domain representation of the complex exponential

B-spline E(a,b)(z,ζ). The result suggests that there are connections to the theory of

special functions.By Mertens’ Theorem [Har49], we can write the double product Ea

z (y)Ebζ(x−y)

in the following form:

Eaz (y)E

bζ(x− y) =

1

Γ(z)Γ(ζ)

∞∑k=0

k∑�=0

(z

�

)(−1)�e−�ae

−a(y−�)+ (y − �)z−1

+

×(

ζ

k − �

)(−1)k−�e−(k−�)be

−b(x−y−(k−�))+ (x− y − (k − �))ζ−1

+

=1

Γ(z)Γ(ζ)

∞∑k=0

k∑�=0

(z

�

)(ζ

k − �

)(−1)ke−�a−(k−�)be

−a(y−�)+ e

−a(y−�)+

× (y − �)z−1+ [(x− k)− (y − �)]ζ−1

+

Thus,

E(a,b)(z,ζ)(x) = (Ea

z ∗ Ebζ)(x) =

∫R

Eaz (y)E

bζ(x− y)dy

= (Σ)

∫R

H(y − �)e−a(y−�)H((x− k)− (y − �))e−b((x−k)−(y−�))

× (y − �)z−1+ [(x− k)− (y − �)]ζ−1

+ dy.

Here, we put all non-variable quantities into the expression (Σ) and used the Heav-iside function H.

102 PETER MASSOPUST

Recognizing that x − k ≥ y − � ≥ 0 holds, the integral in the last line above,can be written as

=

∫ x−k+�

�

e−a(y−�)eb(y−�)e−b(x−k)(y − �)z−a[(x− k)− (y − �)]ζ−1dy

= e−b(x−k)

∫ x−k+�

�

e(a−b)(y−�)(y − �)z−a[(x− k)− (y − �)]ζ−1dy

= e−b(x−k)

∫ x−k

0

e−(a−b)ηηz−1[(x− k)− η]ζ−1dη,

where we used the substitution y − � → η. After the substitution η → τ/(x −k), the last integral becomes the integral representation of Kummer’s confluenthypergeometric M–function [AS65]:∫ x−k

0

e−(a−b)ηηz−1[(x− k)− η]ζ−1dη =

(x− k)z+ζ−1Γ(z)Γ(ζ)

Γ(z + ζ)M(z, z + ζ;−(a− b)(x− k)).

Combining all terms, we arrive at an explicit formula for E(a,b)(z,ζ):

E(a,b)(z,ζ)(x) =

1

Γ(z + ζ)

∞∑k=0

[k∑

�=0

(z

�

)(ζ

k − �

)e−�(a−b)

](−1)k e−bx

×M(z, z + ζ;−(a− b)(x− k)) (x− k)z+ζ−1.(23)

Realizing that the expression in brackets is equal to [AS65](ζ

k

)2F1(−k,−z, 1− k + ζ; e−(a−b)),

we may write (23) also as

E(a,b)(z,ζ)(x) =

1

Γ(z + ζ)

∞∑k=0

(ζ

k

)(−1)k e−bx

2F1(−k,−z, 1− k + ζ; e−(a−b))

×M(z, z + ζ;−(a− b)(x− k)) (x− k)z+ζ−1.

Notice that the above equation is a sampling procedure involving Kummer’s M -function (and Gauß’s 2F1 hypergeometric function).

5. Summary and Further Work

We extended the concept of exponential B-spline to complex orders z ∈ C>1.This extension contains as a special case the class of polynomial splines of complexorder. The new class of complex exponential B-splines generates multiresolutionanalyses of and wavelet bases for L2(R), and relates to fractional differential oper-ators defined on Lizorkin spaces and their duals.

Explicit formulas for the time domain representation of complex exponentialB-spline depending on one parameter and two parameters were derived. In thelatter case, there seem to be connection to the theory of special functions as theKummer M -function appears in the representation.

An approximation-theoretic investigation of complex exponential B-splines forseveral parameters needs to be initiated and numerical schemes for the associated


approximation spaces developed. Connections to fractional differential operators ofthe form

Lza :=

n∏i=1

(D + aiI)zi ,

where a = (a1, . . . , an) ∈ (R+)n and z = (z1, . . . , zn) ∈ Cn>1, need to be established.

Moreover, the time domain representation for complex exponential B-splines de-pending on an n-tuple a ∈ (R+)n of parameters has to be derived. Finally, therelation to special functions is worthwhile an investigation.

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Fluids 33 (2000), 429–452.


Institute of Biomathematics and Biometry, Helmholtz Zentrum Munchen, Ingolstadter

Landstrasse 1, 85764 Neuherberg, Germany – and – Zentrum Mathematik, Lehrstuhl M6,

Technische Universtitat Munchen, Boltzmannstrasse 3, 85747 Garching b. Munchen,

Germany

E-mail address: [email protected]; [email protected]


Local translations associated to spectral sets

Dorin Ervin Dutkay and John Haussermann

Abstract. In connection to the Fuglede conjecture, we study groups of localtranslations associated to spectral sets, i.e., measurable sets in R or Z that havean orthogonal basis of exponential functions. We investigate the connectionsbetween the groups of local translations on Z and on R and present someexamples for low cardinality. We present some relations between the group oflocal translations and tilings.

1. Introduction

In his study of commuting self-adjoint extensions of partial differential opera-tors, Fuglede proposed the following conjecture [Fug74]:

Conjecture 1.1. Denote by eλ the function eλ(x) = e2πiλ·x, for λ, x ∈ Rd.Let Ω be a Lebesgue measurable subset of Rd of finite positive measure. There existsa set Λ such that {eλ : λ ∈ Λ} is an orthogonal basis in L2(Ω) if and only if Ω tilesRd by translations.

Definition 1.2. Let Ω be a Lebesgue measurable subset of Rd of finite, positivemeasure. We say that Ω is a spectral set if there exists a set Λ in Rd such that{eλ : λ ∈ Λ} is an orthogonal basis in L2(Ω). In this case, Λ is called a spectrumfor Ω.

We say that Ω tiles Rd by translations if there exists a set T in Rd such thatthe sets Ω + t, t ∈ T form a partition of Rd, up to measure zero.

Terrence Tao [Tao04] has proved that spectral-tile implication in the Fugledeconjecture is false in dimensions d ≥ 5 and later both directions were disproved indimensions d ≥ 3, see [KM06a]. At the moment the conjecture is still open in bothdirections for dimensions 1 and 2. In this paper, we will only focus on dimensiond = 1.

Recent investigations have shown that the Fuglede can be reduced to analogousstatements in Z, see [DL13].

Definition 1.3. Let A be a finite subset of Z with |A| = N . We say that Ais spectral if there exists a set Γ in R such that {eγ : γ ∈ Γ} is an orthogonal basis

2010 Mathematics Subject Classification. Primary 42A16, 05B45, 15B34.Key words and phrases. Spectrum, tile, Hadamard matrix, Fuglede conjecture, local

translations.


107

108 DORIN ERVIN DUTKAY AND JOHN HAUSSERMANN

in l2(A), equivalently, the matrix

(1.1)1√N

(e2πiγa

)γ∈Γ,a∈A

is unitary. This matrix is called the Hadamard matrix associated to the pair (A,Γ).We say that A tiles Z by translations if there exists a set T in Z such that the

sets A+ t, t ∈ T forms a partition of Z.

The Fuglede conjecture for Z can be formulated as follows

Conjecture 1.4. A finite subset A of Z is spectral if and only if it tiles Z bytranslations.

As shown in [DL13] the tile-spectral implications for R and Z are equivalent:

Theorem 1.5. Every bounded tile in R is spectral if and only if every tile in Zis spectral.

It is not clear if the spectral-tile implications for R and Z are equivalent. Itis known that, if every spectral set in R is a tile, then every spectral set in Z is atile in Z. It was shown in [DL13], that the reverse also holds under some extraassumptions.

Theorem 1.6. Suppose every bounded spectral set Ω in R, of Lebesgue measure|Ω| = 1, has a rational spectrum, Λ ⊂ Q. Then the spectral-tile implications for Rand for Z are equivalent, i.e., every bounded spectral set in R is a tile if and onlyif every spectral set in Z is a tile.

All known examples of spectral sets of Lebesgue measure 1 have a rationalspectrum. There is another, stronger variation of the spectral-tile implication in Zwhich is equivalent to the spectral-tile implication in R.

Definition 1.7. We say that a set Λ has period p if Λ + p = Λ. The smallestpositive p with this property is called the minimal period.

Theorem 1.8. [DL13] The following statements are equivalent

(i) Every bounded spectral set in R is a tile.(ii) For every finite union of intervals Ω = A + [0, 1] with A ⊂ Z, if Λ is a

spectrum of Ω with minimal period 1N , then Ω tiles R with a tiling set

T ⊂ NZ.

We should note here that for a finite set A in Z, the set A+ [0, 1] is spectral inR if and only if A is spectral in Z. Also, if Γ is a spectrum for A then Λ := Γ+Z isa spectrum of A+ [0, 1], therefore Λ has period 1 and the minimal period will haveto be of the form 1

N . See [DL13] for details.The spectral property of a set, in either R or Z, can be characterized by the

existence of a certain unitary group of local translations. We will describe this inthe next section and present some properties of these groups. In Theorem 2.11 wegive a characterization of spectral sets in Z in terms of the existence of groups oflocal translations or of a local translation matrix. In Proposition 2.14 we establisha formula that connects the local translation matrix for a spectral set A in Z andthe group of local translations for the spectral set A + [0, 1] in R. Proposition2.16 shows how one can look for tiling sets for the spectral set A using the localtranslation matrix. Proposition 2.18 shows that the rationality of the spectrum ischaracterized by the periodicity of the group of local translations.

LOCAL TRANSLATIONS ASSOCIATED TO SPECTRAL SETS 109

2. Local translations

In this section we introduce the unitary groups of local translations associatedto spectral sets. These are one-parameter groups of unitary operators on L2(Ω),for subsets Ω of R, or on l2(A), for subsets of Z, which act as translations onΩ or A whenever such translations are possible. The existence of such groupswas already noticed earlier by Fuglede [Fug74] and [Ped87]. They were furtherstudied in [DJ12b]. The idea is the following: the existence of an orthonormal basis{eλ : λ ∈ Λ} allows the construction of the Fourier transform from L2(Ω) to l2(Λ).On l2(Λ) one has the unitary group of modulation operators, i.e., multiplicationby et, or the diagonal matrix with entries e2πitλ, λ ∈ Λ. Conjugating via theFourier transform we obtain the unitary group of local translations. For furtherinformation on local translations and connections to self-adjoint extensions andscattering theory, see [JPT12a,JPT12d,JPT12b,JPT12c].

Definition 2.1. Let Ω be a nonempty, bounded Borel subset of R. A unitarygroup of local translations on Ω is a strongly continuous one parameter unitarygroup U(t) on L2(Ω) with the property that for any f ∈ L2(Ω) and any t ∈ R,

(2.1) (U(t)f)(x) = f(x+ t) for a.e x ∈ Ω ∩ (Ω− t)

If Ω is spectral with spectrum Λ, we define the Fourier transform F : L2(Ω) →l2(Λ)

(2.2) Ff =

(⟨f ,

1√|Ω|

eλ

⟩)λ∈Λ

.

We define the unitary group of local translations associated to Λ by(2.3)

UΛ(t) = F−1UΛ(t)F where UΛ(t)(aλ)λ∈Λ = (e2πiλtaλ)λ∈Λ, ((aλ) ∈ l2(Λ)).

Proposition 2.2. With the notations in Definition 2.1

(2.4) UΛ(t)eλ = eλ(t)eλ, (t ∈ R, λ ∈ Λ)

Thus, the functions eλ are the eigenvectors for the operators U(t) corresponding tothe eigenvalues e2πiλt with multiplicity one.

Proof. Clearly Feλ =√

|Ω|δλ for all λ ∈ Λ, where δλ(x) = 0 for x = λ, x ∈ Λand δλ(λ) = 1. The rest follows from a simple computation. �

Theorem 2.3. Let Ω be a nonempty bounded Borel subset of R. Assume that Ωis spectral with spectrum Λ. Let UΛ be the associated unitary group as in Definition2.1. Then U := UΛ is a unitary group of local translations.

In the particular case when Ω is a finite union of intervals the converse alsoholds:

Theorem 2.4. [DJ12b] The disjoint union of intervals Ω = ∪ni=1(αi, βi) is

spectral if and only if there exists a strongly continuous one-parameter unitary group(U(t))t∈R on L2(Ω) with the property that, for all t ∈ R and f ∈ L2(Ω):

(2.5) (U(t)f)(x) = f(x+ t), for almost every x ∈ Ω ∩ (Ω− t).

Moreover, given the unitary group U , the spectrum Λ of the self-adjoint infinitesimalgenerator D of the group U(t) = e2πitD (as in Stone’s theorem), is a spectrum forΩ.


Remark 2.5. As shown in [DJ12a], and actually even in the motivation ofFuglede [Fug74] for his studies of spectral sets, the self-adjoint operator D appear-ing in Theorem 2.4 are self-adjoint extensions of the differential operator 1

2πiddx on

L2(Ω).

Example 2.6. The simplest example of a spectral set is Ω = [0, 1] with spec-trum Λ = Z. In this case, the group of local translations is

(U(t)f)(x) = f((x+ t)mod 1), (f ∈ L2[0, 1], x, t ∈ [0, 1]).

This can be checked by verifying that U(t)en = en(t)en, t ∈ R, n ∈ Z.

Proposition 2.7. Let Ω = ∪ni=1(αi, βi) be a disjoint union of intervals which

is a spectral set with spectrum Λ. Let E be a nonempty Lebesgue measurable subsetof Ω and t0 ∈ R such that E + t0 ⊂ Ω. Then

(2.6) U(−t0)χE = χE+t0 .

Proof. Since E + t0 is contained in Ω ∩ (Ω + t0), by Theorem 2.4, we havethat, for almost every x ∈ E + t0, (U(−t0)χE)(x) = χE(x− t0) = χE+t0(x). Thus,if g := U(−t0)χE , then g(x) = χE+t0(x) for a.e., x ∈ E + t0. On the other hand,since U(−t0) is unitary, we have that ‖g‖2L2 = ‖χE‖L2 = μ(E). But

μ(E) = ‖g‖2L2 =

∫E+t0

|g(x)|2 dx+∫Ω\(Ω+t0)

|g(x)|2 dx = μ(E+t0)+

∫Ω\(Ω+t0)

|g(x)|2 dx,

so g(x) = 0 for a.e. x ∈ Ω \ (E + t0). �

Next, we focus on spectral subsets of Z and define the one-parameter unitarygroup of local translations in an analogous way. As we will see, in this case, theparameter can be restricted from R to Z and thus the unitary group of local trans-lations is determined by a local translation unitary matrix.

Definition 2.8. Let A be a finite subset of Z. A group of local translationson A is a continuous one-parameter unitary group U(t), t ∈ R, on l2(A) with theproperty that

(2.7) U(a− a′)δa = δa′ , (a, a′ ∈ A)

A unitary matrix B on l2(A) is called a local translation matrix if

(2.8) Ba−a′δa = δa′ , (a, a′ ∈ A)

If A is a spectral subset of Z with spectrum Γ and |A| = N , we define the Fouriertransform from l2(A) to l2(Γ) by the matrix:

(2.9) F =1√N

(e−2πiλa

)λ∈Γ,a∈A

.

Let DΓ(t) be the diagonal matrix with entries e2πiλt, λ ∈ Γ. We define the groupof local translations on A associated to Γ by

(2.10) UΓ(t) := F−1DΓ(t)F , (t ∈ R)

The local translation matrix associated to Γ is B = UΓ(1).


Proposition 2.9. With the notations as in Definition 2.8,

(2.11) UΓ(t)eλ = eλ(t)eλ, Beλ = e2πiλeλ, (λ ∈ Γ)

Thus, the vectors eλ in l2(A) are the eigenvectors of B corresponding to the eigen-values e2πiλ of multiplicity one.

The matrix entries of UΓ(t) are(2.12)

UΓ(t)aa′ =1

N

∑λ∈Γ

e2πi(a−a′+t)λ, Baa′ =1

N

∑λ∈Γ

e2πi(a−a′+1)λ, (a, a′ ∈ A, t ∈ R).

Proof. We have Feλ =√Nδλ, for all λ ∈ Γ. The rest follows from an easy

computation. �Theorem 2.10. Let A be a spectral subset of Z with spectrum Γ and let UΓ

be the unitary group associated to Γ as in Definition 2.8. Then UΓ is a group oflocal translations on Γ, i.e., equation (2.7) is satisfied. Also B := UΓ(1) is a localtranslation matrix.

Proof. We have Fδa = (e−2πiλa)λ∈Γ. Then DΓ(a−a′)Fδa = (e−2πiλa′)λ∈Γ =

Fδa′ . Hence UΓ(a− a′)δa = δa′ . �The converse holds also in the case of subsets of Z, i.e., the existence of a group

of local translations, or of a local translation matrix guarantees that A is spectral.

Theorem 2.11. Let A be a finite subset of Z. The following statements areequivalent:

(i) A is spectral.(ii) There exists a unitary group of local translations U(t), t ∈ R, on A.(iii) There exists a local translation matrix B on A.

The correspondence from (i) to (ii) is given by U = UΓ where Γ is a spectrum forA. The correspondence from (ii) to (iii) is given by B = U(1). The correspondencefrom (iii) to (i) is given by: if {e2πiλ : λ ∈ Γ} is the spectrum of B then Γ is aspectrum for A.

Proof. The implications (i)⇒(ii)⇒(iii) were proved above. We focus on (iii)⇒(i).Let {e2πiλ : λ ∈ Γ} be the spectrum of the unitary matrix B, the eigenvalues re-peated according to multiplicity and let {vλ : λ ∈ Γ} be an orthonormal basis ofcorresponding eigenvectors. Let Pλ be the orthogonal projection onto vλ. Then

Bm =∑λ∈Γ

e2πiλmPλ, (m ∈ Z).

We have, from (2.8),

δa′ =∑λ

e2πiλ(a−a′)Pλδa

so Pλδa′ = e2πiλ(a−a′)Pλδa for all λ ∈ Γ which implies that e2πiaλPλδa does notdepend on a, so it is equal to c(λ)vλ for some c(λ) ∈ C. Then Pλδa = e−2πiaλc(λ)vλfor all λ ∈ Γ and

δa =∑λ

c(λ)e−2πiλavλ,

so ∑λ

c(λ)e−2πiλavλ(a′) = δaa′ , (a, a′ ∈ A)


Consider the matrices S = (c(λ)e−2πiλa)a∈A,λ∈Γ and T = (vλ(a))λ∈Γ,a∈A. Theprevious equation implies that ST = I and since T is unitary, we get that S is also.But then the columns have unit norm so

1 =∑a∈A

|c(λ)|2

and this implies that |c(λ)| = 1√N. The fact that the rows are orthonormal means

that1

N

∑λ∈Γ

e2πi(a−a′)λ = δaa′ .

But this means, first, that all the λ’s are distinct and that Γ is a spectrum forA. �

Remark 2.12. Given the group of local translations U(t), t ∈ R, the localtranslation matrix is given by B = U(1). Conversely, given the local translationmatrix B, this defines U on Z in a unique way U(n) = Bn, n ∈ Z. However, thereare many ways to interpolate this to obtain a local translation group depending onthe real parameter t. One can pick some choices for Γ such that {e2πiλ : λ ∈ Γ} isthe spectrum of B. Then consider the spectral decomposition

B =∑λ∈Γ

e2πiλPλ.

DefineU(t) =

∑λ∈Γ

e2πiλtPλ, (t ∈ R).

Note that U(t) depends on the choice of Γ. Any two such choices Γ, Γ′ are congruentmodulo Z, and therefore the corresponding groups UΓ(t) and UΓ′(t) coincide fort ∈ Z.

Example 2.13. The simplest example of a spectral set in Z is A = {0, 1, . . . , N−1} with spectrum Γ = {0, 1

N , . . . , N−1N }. The local translation matrix associated to

Γ is the permutation matrix:

B =

⎛⎜⎜⎜⎜⎜⎜⎝

0 1 . . . 0

0 0. . . 0

......

......

0 0 . . . 11 0 . . . 0

⎞⎟⎟⎟⎟⎟⎟⎠ .

To see this, it is enough to check that, for k = 0, . . . , N − 1,

B

⎛⎜⎝ e kN(0)...

e kN(N − 1)

⎞⎟⎠ = e2πikN

⎛⎜⎝ e kN(0)...

e kN(N − 1)

⎞⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝e k

N(1)

e kN(2)...

e kN(N − 1)

e kN(0)

⎞⎟⎟⎟⎟⎟⎟⎠ .

In the next proposition we link the two concepts for Z and for R: if A is aspectral set in Z, with spectrum Γ, then Ω = A + [0, 1] is a spectral set in R withspectrum Λ = Γ + Z. The local group of local translations for Ω and Λ can beexpressed in terms of the local translation matrix associated to A and Γ.


Proposition 2.14. Let A = {a0, . . . , aN−1} be a spectral set in Z, with spec-trum Γ = {λ0, . . . , λN−1}. Then the set Ω = A+[0, 1] is spectral in R with spectrumΛ = Γ + [0, 1]. Define the matrix of the Fourier transform from l2(A) to l2(Γ):

(2.13) F = FA,Γ =1√N

(e−2πiλjak

)N−1

j,k=0,

and let DΓ be the N ×N diagonal matrix with entries e2πiλj , j = 0, . . . , N − 1.

(2.14) B = F∗DΓFThe group of local translations (UΛ(t))t∈R associated to the spectrum Λ of Ω is givenby(2.15)⎛⎜⎝ (UΛ(t)f)(x+ a0)

...(UΛ(t)f)(x+ aN−1)

⎞⎟⎠ = B�x+t�

⎛⎜⎝ f({x+ t}+ a0)...

f({x+ t}+ aN−1)

⎞⎟⎠ , (f ∈ L2(Ω), x ∈ [0, 1], t ∈ R).

where %·& and {·} represent the integer and the fractional parts, respectively.

Proof. The fact that Λ is a spectrum for Ω can be found, for example, in[KM06b]. The formula for UΛ appears in a slightly different form in [DJ12b], butwe can check it here directly in a different way: it is enough to prove that

(2.16) UΛ(t)eλ = eλ(t)eλ for all λ ∈ Λ

Thus, we have to substitute f = eλi+n in the right-hand side of (2.15), with λi ∈ Γ,n ∈ Z. For the computation, we will use the following relation:

(2.17) F 1√N

⎛⎜⎝ e2πiλia0

...e2πiλiaN−1

⎞⎟⎠ = δi.

Indeed, we have, for j = 0, . . . , N − 1,

1

N

N−1∑k=0

e−2πiλjake2πiλiak =1

N

N−1∑k=0

e2πi(λi−λj)ak = δij ,

because Γ is a spectrum for A.Then, for m ∈ Z,

(2.18) Bm


...e2πiλiaN−1

⎞⎟⎠ = e2πiλim


...e2πiλiaN−1

⎞⎟⎠We have, for x ∈ [0, 1] and t ∈ R:

B�x+t�

⎛⎜⎝ e2πi({x+t}+a0)(λi+n)

...e2πi({x+t}+aN−1)(λi+n)

⎞⎟⎠ = e2πi{x+t}(λi+n)B�x+t�

⎛⎜⎝ e2πia0λi

...e2πiaN−1λi

⎞⎟⎠

= e2πi{x+t}(λi+n)e2πi�x+t�λi


...e2πiaN−1λi

⎞⎟⎠


= e2πi(λi+n)(x+t)


...e2πiaN−1λi

⎞⎟⎠ = e2πi(λi+n)t

⎛⎜⎝ e2πi(x+a0)(λi+n)

...

e2πi(x+aN−1)(λi+n)

⎞⎟⎠ .

This proves (2.16). �

In the following we present some connections between the local matrix B andpossible tilings for the set A. We define a set ΘB as the set of powers of the matrixB which have a canonical vector as a column, with 1 not on the diagonal.

Definition 2.15. Let A be a spectral subset of Z with spectrum Γ. Let B bethe associated local translation matrix. Define(2.19)ΘB := {m ∈ Z : Bm has a column a equal to the canonical vector δa′ for some a = a′}

= {m ∈ Z : Bmδa = δa′ for some a = a′ in A}.

Proposition 2.16. Let A be a spectral subset of Z with spectrum Γ with |A| =N . Assume 0 ∈ Γ. Assume in addition that the smallest lattice that contains Γ isrdZ for some mutually prime integers r, d ≥ 1. For a subset T of Z the followingstatements are equivalent:

(i) T ⊕A = Zd, in the sense that T ⊕A is a complete set of representativesmodulo d and every element x in T + A can be represented in a uniqueway as x = t+a with t ∈ T and a ∈ A. In this case A tiles Z by T ⊕ dZ.

(ii) (T − T ) ∩ΘB = {0} and |T ||A| = d.

Proof. First, we present ΘB in a more explicit form. By Proposition 2.9 wehave

(Bm)aa′ =1

N

∑λ∈Γ

e2πi(a−a′+m)λ.

Since we want (Bm)aa′ to be 1 for some a = a′, we must have equality in thetriangle inequality

|(Bm)aa′ | ≤ 1

N

∑λ∈Γ

1 = 1,

and since 0 ∈ Γ this implies that e2πi(a−a′+m)λ = 1 so (a−a′+m)λ ∈ Z for all λ ∈ Γ.Since the smallest lattice that contains Γ is r

dZ, we obtain that (a− a′ +m) rd ∈ Zwhich means that m ≡ a′ − amod d. The converse also holds: if a′ ≡ a+mmod dthen Bm has a 1 on position aa′. Thus,

(2.20) ΘB = {m ∈ Z : m ≡ a′ − amod d for some a = a′ ∈ A}

(i)⇒(ii). Suppose there exists t = t′ in T such that t − t′ ∈ ΘB . Then thereexist a = a′ in A such that a′ − a ≡ t − t′ mod d. Then a′ + t′ ≡ a + tmod d, acontradiction. Also if A⊕ T = Zd, then |A||T | = d.

(ii)⇒(i). It is enough to prove that (A − A) ∩ (T − T ) = {0}mod d, becausethis implies that the map from A × T to (A + T )mod d, (a, t) → a + tmod d isinjective, and the condition |A||T | = d implies that it has to be bijective.

Suppose not. Then there exist a = a′ in A t = t′ in T such that a + t ≡a′ + t′ mod d. Then t− t′ ≡ a′ − amod d. Therefore t− t′ ∈ ΘB which contradictsthe hypothesis. �


Corollary 2.17. If the local translation matrix B is

B =

⎛⎜⎜⎜⎜⎜⎜⎝

0 1 . . . 0

0 0. . . 0

......

......

0 0 . . . 11 0 . . . 0

⎞⎟⎟⎟⎟⎟⎟⎠ ,

then A tiles Z by NZ.

Proof. We have ΘB = {1, . . . , N − 1} + NZ so one can take T = {0} inProposition 2.16. �

Another piece of information that is contained in the local translation matrixis the rationality of the spectrum:

Proposition 2.18. Let A be a spectral set in Z with spectrum Γ and localtranslation matrix B. Let d ∈ Z, d ≥ 1. Then Γ ⊂ 1

dZ if and only if Bd = I.The spectrum Γ is rational if and only if the group of local translations UΓ has aninteger period, i.e., there exists p ∈ Z, p ≥ 1 such that UΓ(t+ p) = UΓ(t), t ∈ R.

Proof. If Γ ⊂ 1dZ, using equation (2.10), with t = d, we have that DΓ(d) = I

so Bd = I. Conversely, if Bd = I then DΓ(d) = I and therefore dλ ∈ Z for allλ ∈ Γ. The second statement follows from the first. �

3. Examples

In this section we study the local translation groups associated to spectral setsof low cardinality N = 2, 3, 4, 5. Such sets were described in [DH12]. We recallhere the results:

Definition 3.1. The standard N ×N Hadamard matrix is

(3.1)1√N

(e2πi

jkN

)N−1

j,k=0.

We say that an N ×N matrix is equivalent to the standard Hadamard matrix if itcan be obtained from it by permutations of rows and columns.

Let A and L be two subsets of Z and R ∈ Z, R ≥ 1. We say that (A,L) is aHadamard pair with scaling factor R if 1

RL is a spectrum for A.

The Hadamard pairs with Hadamard matrix equivalent to the standard oneare described in the next theorem from [DH12].

Theorem 3.2. Let A ⊂ Z have N elements and spectrum Γ. Assume 0 is inA and Γ. Suppose the Hadamard matrix associated to (A,Γ) is equivalent to thestandard N by N Hadamard matrix. Then A has the form A = dA0 where d isan integer and A0 is a complete set of residues modulo N with gcd(A0) = 1. Inthis case any such spectrum Γ has the form Γ = 1

RfL0 where f and R are integers,L0 is a complete set of residues modulo N with greatest common divisor one, andR = NS where S divides df and df

S is mutually prime with N . The converse alsoholds.

Since for N = 2, 3, 5 our Hadamard matrices are equivalent to the standard one(see [Haa97,TZ06]) the next corollary follows:


Corollary 3.3. A set A ⊂ Z with |A| = N = 2, 3, or 5, where 0 ∈ A isspectral if and only if A = NkA0 where k is a positive integer and A0 is a completeset of residues modulo N .

For cardinality N = 4 the situation is more complex, but the Hadamard pairsof size 4 can be also classified and the next result from [DH12] contains the details.

Theorem 3.4. Let A be spectral with spectrum Γ and size N = 4. Assume 0is in both sets. Then there exists a set of integers L, containing 0, and an integerscaling factor R so that Γ = 1

RL.(A,L) is a Hadamard pair (each containing 0) of integers of size N = 4, with

scaling factor R, if and only if R = 2C+M+a+1d, A = 2C{0, 2ac1, c2, c2+2ac3}, andL = 2M{0, n1, n1+2an2, 2

an3}, where ci and ni are all odd, a is a positive integer,C and M are non-negative integers, and d divides c1n, c3n, n2c, and n3c, where cis the greatest common divisor of the ck’s and similarly for n.

The next proposition helps us simplify our study:

Proposition 3.5. Let A be a spectral set in Z with spectrum Γ, local translationgroup UΓ and local translation matrix B. Let d ∈ Z d ≥ 1. Then dA is spectralwith spectrum 1

dΓ. The local translation group U 1dΓ

and the local translation matrix

B 1dΓ

are related to the corresponding ones for A and Γ by

(3.2) U 1dΓ

(t) = UΓ(t

d), (t ∈ R), Bd

1dΓ

= BΓ

Proof. Everything follows from (2.10) by a simple calculation. �

N=2. We can take A = 2c{0, a0}, with c ∈ Z, c ≥ 0 and a0 odd, andΓ = {0, γ1 = g

2c+1 } with g odd. The matrix of the Fourier transform is

F =1√2

(1 11 −1

).

By equation (2.10), we can compute the local translation matrix B and the localtranslation group UΓ:

(3.3) B =1

2

(1 11 −1

)(1 00 eπiγ1

)(1 11 −1

).

Here γ1 depends on the non-zero element of A, called a1. Let a1 = 2ca0, where a0is odd. Then γ1 = g

2c+1 , where g is odd. Multiplying, we obtain

(3.4) B =1

2

(1 + eπiγ1 1− eπiγ1

1− eπiγ1 1 + eπiγ1

).

We also have

(3.5) UΓ(t) =1

2

(1 + etπiγ1 1− etπiγ1

1− etπiγ1 1 + etπiγ1

).

Note that when c = 0,

B =

(0 11 0

).

In this case ΘB = Z.


N=3. We can take A = 3c{0, 3j1 + 1, 3j2 + 2} and Γ = {0, γ1 = 3j3+13c+1 , γ2 =

3j4+23c+1 }. The matrix of the Fourier transform is, with ω = e

2πi3 :

F =1√3

⎛⎝1 1 11 ω ω1 ω ω

⎞⎠ .

We compute the group of local translations:

(3.6) UΓ(t) =1

3

⎛⎝1 1 11 ω ω1 ω ω

⎞⎠⎛⎝1 0 00 e2πiγ1t 00 0 e2πiγ2t

⎞⎠⎛⎝1 1 11 ω ω1 ω ω

⎞⎠ .

Multiplying, we obtain

UΓ(t) =1

3

⎛⎝

1 + e2πiγ1t + e2πiγ2t 1 + ωe2πiγ1t + ωe2πiγ2t 1 + ωe2πiγ1t + ωe2πiγ2t

1 + ωe2πiγ1t + ωe2πiγ2t 1 + e2πiγ1t + e2πiγ2t 1 + ωe2πiγ1t + ωe2πiγ2t

1 + ωe2πiγ1t + ωe2πiγ2t 1 + ωe2πiγ1t + ωe2πiγ2t 1 + e2πiγ1t + e2πiγ2t

⎞⎠

Note that, when c = 0,

B = UΓ(1) =

⎛⎝0 1 00 0 11 0 0

⎞⎠ .


N=4. We take a simple case to obtain some nice symmetry, so we will ignore,after some rescaling, the common factor. So take A = {0, 2ac1, c2, c2 + 2ac3},Γ = 1

2a+1 {0, 2an1, n2, n2 + 2an3} as in Theorem 3.4. The matrix of the Fouriertransform is

(3.7)1

2

⎛⎜⎜⎝1 1 1 11 1 −1 −11 −1 ρ −ρ1 −1 −ρ ρ

⎞⎟⎟⎠ ,

where ρ = exp(−πic2n2

2a

). We compute integers powers of the spectral matrix B:

(3.8)

Bk =1

4

⎛⎜⎜⎝1 1 1 11 1 −1 −11 −1 ρ −ρ1 −1 −ρ ρ

⎞⎟⎟⎠∗⎛⎜⎜⎝

1 0 0 00 (−1)k 0 00 0 zk 00 0 0 (−z)k

⎞⎟⎟⎠⎛⎜⎜⎝1 1 1 11 1 −1 −11 −1 ρ −ρ1 −1 −ρ ρ

⎞⎟⎟⎠ ,

where z = exp(πin2

2a

).

We obtain for odd k,

(3.9) Bk =1

2

⎛⎜⎜⎝0 0 1 + zkρ 1− zkρ0 0 1− zkρ 1 + zkρ

1 + zkρ 1− zkρ 0 01− zkρ 1 + zkρ 0 0

⎞⎟⎟⎠ .

We obtain for even k,

(3.10) Bk =1

2

⎛⎜⎜⎝1 + zk 1− zk 0 01− zk 1 + zk 0 0

0 0 1 + zk 1− zk

0 0 1− zk 1 + zk

⎞⎟⎟⎠ .


We compute ΘB . We have k ∈ ΘB if and only if one of the following situationsoccurs: zk = −1, zkρ = ±1, zkρ = ±1. This means that kn2

2a = 2m + 1 orkn2

2a ± c2n2

2a = m for some m ∈ Z. Since n2 is odd, this implies that

ΘB = {2a(2m+ 1) : m ∈ Z} ∪ {2am± c2 : m ∈ Z}.Then, one can easily see that T := {0, 2, 4, . . . , 2a − 2} satisfies the conditions

in Propositon 2.16, and therefore A⊕ T = Z2a+1 .

Example 3.6. Let A = {0, 1, 4, 5} and Γ = 18{0, 1, 4, 5}. We illustrate how the

group of local translations acts on an indicator function.

1

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=0.1

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t= −1.00101

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−2.0020

1

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−3.00301

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−4.00401

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−5.0050

1

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−6.00601

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−7.00701

0.8

0.6

0.4

0.2

−1 0 1 2 3y

4 5 6 7

t=−8.0000

Figure 1. Local translations for A = {0, 1, 4, 5} and Γ = 18{0, 1, 4, 5}.

The indicator function f is the one in the first picture, for t = 0. We usenegative values for t to move the function to the right. We show here the absolutevalue of U(t)f . Note that for t ≈ −1,−4 and −5, since the the interval [0, 1] ismoved into the intervals [1, 2], [4, 5] [5, 6], which are contained in A + [0, 1], aspredicted by the theory, e.g., Proposition 2.7, the group U(t) really acts a simpletranslation.


For t ≈ 2, the interval [0, 1] + 2 is no longer contained in A+ [0, 1]. The localtranslation U(−2) splits the indicator function into 2 pieces, supported on [0, 1] and[4, 5]. Similarly for t ≈ −3,−6,−7. Since Γ is contained in 1

8Z, the group of localtranslations has period 8. We see this in the last picture U(−8)f = f .

N=5. For simplicity, by rescaling we can ignore the common factors in A andΓ so we take A = {0, a1, a2, a3, a4} with aj ≡ jmod 5 and Γ = 1

5{0, γ1, γ2, γ3, γ4}with γj ≡ jmod 5. Then the matrix of the Fourier transform is

F =1√5

(e2π

jk5

)4

j,k=0.

The local translation matrix is

B =

⎛⎜⎜⎜⎜⎝0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 0

⎞⎟⎟⎟⎟⎠ .


Acknowledgements. This work was partially supported by a grant from theSimons Foundation (#228539 to Dorin Dutkay).

References

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[DJ12b] Dorin Ervin Dutkay and Palle E. T. Jorgensen, Spectral measures and Cuntz algebras,Math. Comp. 81 (2012), no. 280, 2275–2301, DOI 10.1090/S0025-5718-2012-02589-0.MR2945156

[DL13] Dorin Ervin Dutkay and Chun-Kit Lai, Some reductions of the spectral set conjec-ture to integers, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 1, 123–135, DOI10.1017/S0305004113000558. MR3144214

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[JPT12a] Palle Jorgensen, Steen Pedersen, and Feng Tian, Translation representations andscattering by two intervals, J. Math. Phys. 53 (2012), no. 5, 053505, 49, DOI10.1063/1.4709770. MR2964262

[JPT12b] Palle E. T. Jorgensen, Steen Pedersen, and Feng Tian, Momentum operators in twointervals: spectra and phase transition, Complex Anal. Oper. Theory 7 (2013), no. 6,1735–1773, DOI 10.1007/s11785-012-0234-x. MR3129890

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Department of Mathematics, University of Central Florida, 4000 Central Florida

Blvd., P.O. Box 161364, Orlando, Florida 32816-1364


Department of Mathematics, University of Central Florida, 4000 Central Florida

Blvd., P.O. Box 161364, Orlando, Florida 32816-1364



Additive spectra of the 14Cantor measure

Palle E. T. Jorgensen, Keri A. Kornelson, and Karen L. Shuman

Abstract. In this paper, we add to the characterization of the Fourier spectrafor Bernoulli convolution measures. These measures are supported on Cantorsubsets of the line. We prove that performing an odd additive translationto half the canonical spectrum for the 1

4Cantor measure always yields an

alternate spectrum. We call this set an additive spectrum. The proof worksby connecting the additive set to a spectrum formed by odd multiplicativescaling.

1. Introduction

Traditional Cantor sets are generated by iterations of an operation of down-scaling by fractions which are powers of a fixed positive integer. For each iterationin the process, we leave gaps. For example, the best-known ternary Cantor set isformed by scaling down by 1

3 and leaving a single gap in each step. An associatedCantor measure μ is then obtained by the same sort of iteration of scales, and, ateach step, a renormalization. In accordance with classical harmonic analysis, thesemeasures may be seen to be infinite Bernoulli convolutions.

Our present analysis is motivated by earlier work, beginning with [JP98]. Weconsider recursive down-scaling by 1

2n for n ∈ N and leave a single gap at eachiteration-step. It was shown in [JP98] that the associated Cantor measures μ 1

2n

have the property that L2(μ 12n) possesses orthogonal Fourier bases of complex ex-

ponentials (i.e., Fourier ONBs). More recently, it was shown in [Dai12] that thescales 1

2n are the only values that generate measures with Fourier bases.Given a fixed Cantor measure μ, a corresponding set of frequencies Γ of ex-

ponents in an ONB is said to be a spectrum for μ. For example, in the case ofrecursive scaling by powers of 1

4 ; i.e., n = 2, a possible spectrum Γ for L2(μ) hasthe form Γ as shown below in Equation (2.4). A spectrum for a Cantor measureturns out to be a lacunary (in the sense of Szolem Mandelbrojt) set of integers orhalf integers. We direct the interested reader to [Kah85] regarding lacunary seriesand their Riesz products.

2010 Mathematics Subject Classification. Primary 42B05, 28A80, 28C10, 47A10.Key words and phrases. Cantor set, fractal, measure, Bernoulli convolution, spectrum, op-

erator, isometry, unitary.The second author was supported in part by grant #244718 from The Simons Foundation.The third author was supported in part by the Grinnell College Committee for the Support

of Faculty Scholarship.


121

122 P.E.T. JORGENSEN, K.A. KORNELSON, AND K.L. SHUMAN

When n and μ are fixed, we now become concerned with the possible variety ofspectra. Given Γ some canonical choice of spectrum for μ, then one possible way toconstruct a new Fourier spectrum for L2(μ) is to scale by an odd positive integerp to form a set pΓ. While for some values of p this scaling produces a spectrum, itis known that other values of p do not yield spectra.

This particular question is intrinsically multiplicative: Since μ is an infiniteBernoulli convolution, the ONB questions involve consideration of infinite productsof the Riesz type. Despite this intuition, we show here (Theorem 4.3) that thereis a connection between this multiplicative construction and a construction of newONBs with an additive operation. We are then able to produce even more examplesof these additive-construction spectra.

2. Background

Throughout this paper, we consider the Hilbert space L2(μ 14) where μ 1

4is the 1

4 -

Bernoulli convolution measure. This measure has a rich history, dating back to workof Wintner and Erdos [Win35,Erd39,Erd40]. More recently, Hutchinson [Hut81]developed a construction of Bernoulli measures via iterated function systems (IFSs).The measure μ 1

4is supported on a Cantor subset X 1

4of R which entails scaling by 1

4 .

In 1998, Jorgensen and Pedersen [JP98] discovered that the Hilbert space L2(μ 14)

contains a Fourier basis — an orthonormal basis of exponential functions — andhence allows for a Fourier analysis.

For ease of notation, throughout this paper we will write et for the functione2πit· and for a discrete set Γ we will write E(Γ) for the collection of exponentials{eγ : γ ∈ Γ}.

There is a self-similarity inherent in the 14 -Bernoulli convolution

(2.1)

∫X 1

4

f dμ 14=

1

2

∫X 1

4

f(14(x+ 1)

)dμ 1

4(x) +

1

2

∫X 1

4

f(14(x− 1)

)dμ 1

4(x)

which yields an infinite product formulation for μ 14:

(2.2) μ 14(t) =

∫X 1

4

e2πitx dμ 14(x) =

∞∏k=1

cos(2πt4k

).

Exponential functions eγ and eξ are orthogonal when

〈eγ , eξ〉 = μ 14(γ − ξ) = 0.

A collection of exponential functions E(Γ) indexed by the discrete set Γ is anorthonormal basis for L2(μ 1

4) exactly when the function

(2.3) cΓ(t) :=∑γ∈Γ

|〈et, eγ〉|2 =∑γ∈Γ

∞∏k=1

cos2(2π(t− γ)

4k

)is the constant function 1. We call the function cΓ the spectral function for the setΓ.

The Fourier basis for μ 14constructed in [JP98] is the set {e2πiγ· : γ ∈ Γ},

where

(2.4) Γ =

{m∑i=0

ai4i : m finite, ai ∈ {0, 1}

}= {0, 1, 4, 5, 16, 17, 20 . . .}.

ADDITIVE SPECTRA OF THE 14 CANTOR MEASURE 123

If E(Γ) is an orthonormal basis (ONB) for L2(μ 14), we say that Γ is a spectrum

for μ 14. It is straightforward to show that if Γ is a spectrum for μ 1

4and p is an odd

integer, then E(pΓ) is an orthogonal collection of exponential functions. In manycases, we find that E(pΓ) is actually another ONB [�LW02,DJ12,JKS11]. This israther surprising, or at least very different behavior from the usual Fourier analysison an interval with respect to Lebesgue measure.

We often refer to the spectrum in Equation (2.4) as the canonical spectrumfor L2(μ 1

4), while other spectra for the same measure space can be called alternate

spectra.

3. Isometries

In this section, we describe two naturally occurring isometries on L2(μ 14) which

are defined via their action on the canonical Fourier basis E(Γ). Observe fromEquation (2.4) that Γ satisfies the invariance equation

Γ = 4Γ ( (4Γ + 1),

where ( denotes the disjoint union. We then define

S0 : eγ → e4γ(3.1)

S1 : eγ → e4γ+1 for all γ ∈ Γ.

Since S0 and S1 map the ONB elements into a proper subset of the ONB, theyare proper isometries. Therefore, for i = 0, 1 we have S∗

i Si = I and SiS∗i is a

projection onto the range of the respective operator. The adjoints of S0, S1 arereadily computed (see [JKS12] for details):

(3.2) S∗0eγ =

{e γ

4when γ ∈ 4Γ

0 otherwise

and

(3.3) S∗1eγ =

{e γ−1

4when γ ∈ 1 + 4Γ

0 otherwise.

It is shown in [JKS13, Section 2] that the definitions of S0 and S1 extend toall en for n ∈ Z; i.e.,

(3.4) S0 : en → e4n and S1 : en → e4n+1 ∀n ∈ Z.

For every integer N > 1, there is a C∗-algebra with N generators called theCuntz algebra, which we denote by ON [Cun77]. We will describe representationsof O2 which are generated by two isometries on L2(μ 1

4) satisfying the conditions

below.

Definition 3.1. We say that isometry operators T0, T1 on L2(μ 14) satisfy Cuntz

relations if

(1) T0T∗0 + T1T

∗1 = I,

(2) T ∗i Tj = δi,jI for i, j = 0, 1.

When these relations hold, {T0, T1} generate a representation of the Cuntz algebraO2.


From [BJ99,JKS12], we know that S0 and S1 defined in Equation (3.1) satisfythe Cuntz relations for k = 2, hence yield a representation of the Cuntz algebraO2 (in fact, an irreducible representation) within the algebra of bounded operatorsB(L2(μ 1

4)).

4. Spectral function decompositions

As we mentioned above, given a spectrum Γ, the frequencies pΓ, for p an oddinteger, generate an orthonormal collection of exponential functions in L2(μ 1

4).

Given Γ from Equation (2.4), one question of interest is the characterization of theodd integers p for which the scaled spectrum pΓ generates an ONB. As a means ofexploring this question, we let Up be the operator

(4.1) Up : eγ → epγ ∀γ ∈ Γ.

Since Up maps an ONB to an orthonormal collection, Up is an isometry and isunitary if and only if E(pΓ) is an ONB.

The following lemmas provide useful relationships between the isometries S0,S1, and Up.

Lemma 4.1. Let S0 and S1 be the isometry operators from Equation (3.1). Ifρ is a ∗-automorphism on B(L2(μ 1

4)), then the operator

W = ρ(S0)S∗0 + ρ(S1)S

∗1

is unitary.

Proof. Assume ρ is a ∗-automorphism. The Cuntz relations on S0 and S1

give

WW ∗ =(ρ(S0)S

∗0 + ρ(S1)S

∗1

)(ρ(S0)S

∗0 + ρ(S1)S

∗1

)∗

=(ρ(S0)S

∗0 + ρ(S1)S

∗1

)(S0ρ(S

∗0) + S1ρ(S

∗1))

= ρ(S0)S∗0S0ρ(S

∗0) + ρ(S0)S

∗0S1ρ(S

∗1)

+ρ(S1)S∗1S0ρ(S

∗0) + ρ(S1)S

∗1S1ρ(S

∗1)

= ρ(S0)ρ(S∗0) + ρ(S1)ρ(S

∗1)

= ρ(S0S∗0 + S1S

∗1 )

= ρ(I) = I

A similar computation proves that W ∗W = I, hence W is unitary. �

Lemma 4.2. Let Mk be the multiplication operator Mkf = ekf . Given p ∈ Nsuch that Up is unitary, we define the map α(X) = UpXU∗

p on B(L2(μ 14)). Then

α(S0) = S0 and α(S1) = Mp−1S1.

Proof. It was proved in [JKS12] that Up commutes with S0 for all odd p,so α(S0) = S0. Since Up is unitary, we have U∗

pUp = UpU∗p = I. We prove that

Mp−1S1Up = UpS1, which is thus equivalent to the statement of the lemma.

UpS1eγ = Upe4γ+1

= e4pγ+p, and

Mp−1S1Upeγ = Mp−1S1epγ

= Mp−1e4pγ+1 by extension of S1 to N


= e4pγ+p

Therefore,

α(S1) = UpS1U∗p = Mp−1S1UpU

∗p = Mp−1S1.

�

We now discover a connection between the scaled spectrum pΓ and what wecall an additive spectrum E(4Γ) ∪E(4Γ + p). It will turn out that this connectiontells us more about the additive spectra than the scaled spectra.

Theorem 4.3. Given any odd natural number p, if E(pΓ) is an ONB thenE(4Γ) ∪E(4Γ + p) is also an ONB.

Proof. Since E(pΓ) is an ONB, we have that the operator Up from Equation(4.1) is a unitary operator. We define the map on B(L2(μ 1

4))

(4.2) α(X) = UpXU∗p .

Since Up is unitary, it is straightforward to verify that α is a ∗-automorphism onB(L2(μ 1

4)).

If we apply α to our operators S0 and S1, we have by Lemma 4.2,

α(S0) = S0 and α(S1) = UpS1U∗p = Mp−1S1.

Define the operator

W := α(S0)S∗0 + α(S1)S

∗1 = S0S

∗0 +Mp−1S1S

∗1 .

Then W is unitary by Lemma 4.1.

We see that if γ ∈ 4Γ; i.e., γ = 4γ′ for some γ′ ∈ Γ, that Weγ = WS0eγ′ = eγsince S∗

0S0 = I and S∗1S0 = 0 by the Cuntz relations. Similarly, if γ ∈ 4Γ+1, hence

γ = 4γ′ + 1 for some γ′ ∈ Γ, then Weγ = WS1eγ′ = Mp−1S1eγ′ = e4γ′+p. In fact,

W maps E(4Γ + 1) bijectively onto E(4Γ + p). Therefore, since W is unitary, wecan conclude that E(4Γ) ∪ E(4Γ + p) is an ONB for L2(μ 1

4). �

We now address the spectral functions—recall Equation (2.3)—for our additivesets. We can use the splitting Γ = (4Γ) ∪ (4Γ + 1) to divide the spectral functionfor Γ into the corresponding terms

cΓ(t) =∑γ∈Γ

|μ 14(t− 4γ)|2 +

∑γ∈Γ

|μ 14(t− 4γ − 1)|2.

Denote the sums on the right-hand side of the equation above by c0(t) and c1(t)respectively. More generally, denote

(4.3) cm(t) =∑γ∈Γ

|μ 14(t− 4γ −m)|2.

Proposition 4.4. The function c1 is 2-periodic.

Proof. By Theorem 4.3 the sets (4Γ)∪ (4Γ+ 5) and (4Γ)∪ (4Γ+ 7) are bothspectra for μ 1

4— this follows because it is known (see, for example, [�LW02]) that

the scaled sets 5Γ and 7Γ are spectra. We therefore have

1 = c0(t) +∑γ∈Γ

|μ 14(t− 4γ − 5)|2 = c0(t) +

∑γ∈Γ

|μ 14(t− 4γ − 7)|2.


Using the fact that the set Γ itself is also a spectrum, we have

1 = c0(t) + c1(t) = c0(t) + c5(t) = c0(t) + c7(t)

for all t ∈ R. Hence,

c1(t) = c5(t) = c7(t) ∀t ∈ R.

But we also observe that c5(t) = c1(t− 4) and c7(t) = c1(t− 6), so the function c1is both 4-periodic and 6-periodic, hence is 2-periodic. �

−4 −2 2 4

0.2

0.4

0.6

0.8

1.0

Figure 1. Numerical estimate of c1, with 16 factors in the productand 128 terms in the sum.

Corollary 4.5. The function c0 is 2-periodic.

We next observe that Theorem 4.3 is a stepping stone to the following result.

Theorem 4.6. Given any odd integer p, the set E[(4Γ)∪ (4Γ+ p)] is an ONBfor L2(μ 1

4).

Proof. This is a direct result of Proposition 4.4. The spectral function forE[(4Γ)∪(4Γ + p)] can be written in the two parts∑

γ∈Γ

|μ 14(t− 4γ)|2 +

∑γ∈Γ

|μ 14(t− (4γ + p))|2.

When p = 1, we have the canonical ONB in the 14 case. Otherwise, using the

2-periodicity of c0, we have

c0(t) + cp(t) =∑γ∈Γ

|μ 14(t− 4γ)|2 +

∑γ∈Γ

|μ 14(t− (4γ + p))|2

=∑γ∈Γ

|μ 14(t− (p− 1)− 4γ)|2 +

∑γ∈Γ

|μ 14(t− (4γ + 1)− (p− 1))|2

= c0(t− p+ 1) + c1(t− p+ 1) ≡ 1.

Since the spectral function is identically 1, the set E[(4Γ)∪ (4Γ+p)] is an ONBfor L2(μ 1

4). �


Acknowledgements

We mention here that the existence of the spectra that we call the additivespectra for μ 1

4is not new. They are among the examples described, from a different

perspective, in Section 5 of [DHS09].The authors would like to thank Allan Donsig for helpful conversations while

writing an earlier version of this work.Some of the work on this paper was done at the special session on the Harmonic

Analysis of Frames, Wavelets, and Tilings at the AMS Spring Western SectionalMeeting in April, 2013. We wish to thank the organizers of that session and theAmerican Mathematical Society for making the conference possible.

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Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419

E-mail address: [email protected]: http://www.math.uiowa.edu/~jorgen/

Department of Mathematics, The University of Oklahoma, Norman, Oklahoma

73019

E-mail address: [email protected]: http://www.math.ou.edu/~kkornelson/

Department of Mathematics and Statistics, Grinnell College, Grinnell, Iowa 50112-

1690

E-mail address: [email protected]: http://www.math.grinnell.edu/~shumank/


Necessary density conditions for sampling and interpolationin de Branges spaces

Sa’ud al-Sa’di and Eric Weber

Abstract. We consider the problems of sampling and interpolation in deBranges spaces. The class of de Branges spaces considered are those whoseweight function has a phase function whose derivative is bounded below. For

this class, we prove that the Homogeneous Approximation Property holds forthe reproducing kernel. As a consequence, necessary conditions for samplingand interpolating sequences are shown, which generalize some well-known sam-pling and interpolation results in the Paley–Wiener space. We also prove suffi-cient conditions for which a sequence satisfies the Plancherel–Polya Inequality.

1. Introduction

The theory of Hilbert spaces of entire functions was first introduced by L.de Branges in the series of papers [4–7]. These spaces, which are now called deBranges spaces, generalize the classical Paley–Wiener space which consists of theentire functions of exponential type π which are square integrable on the real line.

An entire function E(z) is said to be of Hermite−Biehler class, denoted by HB,if it satisfies the condition

(1.1) |E(z)| < |E(z)|,for all z ∈ C+ = {z ∈ C : Im z > 0}. An analytic function f on C+ is said to be ofbounded type in C+ if it can be represented as a quotient of two bounded analyticfunctions in C+. The mean type of f in C+ is defined by

(1.2) mt+(f) := lim supy→+∞

log |f(iy)|y

Given a function E ∈ HB, the de Branges space H(E) consists of all entirefunctions f(z) such that both f(z)/E(z) and f∗(z)/E(z) are of bounded type andnonpositive mean type in the upper half-plane, and

(1.3) ||f ||2E:=

∫R

∣∣∣∣ f(t)E(t)

∣∣∣∣2dt < ∞.

Here f∗(z) = f(z). H(E) is a Hilbert space with inner product defined by

〈f, g〉E =

∫R

f(t)g(t)

|E(t)|2 dt,

2010 Mathematics Subject Classification. Primary 94A20, 42C30.


129

130 SA’UD AL-SA’DI AND ERIC WEBER

for all f, g ∈ H(E). Moreover, H(E) is a reproducing kernel Hilbert space [8], withkernel given by:

(1.4) K(w, z) =E(w)E(z)− E(w)E∗(z)

2πi(w − z).

Therefore, for every f ∈ H(E) and every w ∈ C.

(1.5) f(w) = 〈f(t),K(w, t)〉E.Every de Branges space H(E) possesses a phase function defined by the gen-

erating function E. Indeed, for E ∈ HB, there exists a continuous and strictlyincreasing function ϕ : R → R such that E(x)eiϕ(x) ∈ R for all x ∈ R, so that E(x)can be written as

(1.6) E(x) = |E(x)|e−iϕ(x), x ∈ R.

The phase function need not be unique; however, if ϕ(x) is any such function, thenusing (1.4) and (1.6), an easy computation yields

(1.7) 〈K(x, t),K(x, t)〉E = K(x, x) =1

πϕ′(x)|E(x)|2 x ∈ R.

It follows that ϕ′ is uniquely defined, and any two phase functions for a given Ediffer by an additive constant.

Remarkably, any de Branges space has an orthogonal basis consisting of repro-ducing kernels corresponding to real points [8].

Theorem 1.1. Let H(E) be a de Branges space with phase function ϕ(x), andlet α ∈ R. If Γ = {γn}n∈Z is a sequence of real numbers, such that ϕ(γn) = α+πn,n ∈ Z, then the functions {K(γn, z)}n∈Z form an orthogonal set in H(E).

If, in addition, eiαE(z) − e−iαE∗(z) /∈ H(E), then{ K(γn,z)‖K(γn,.)‖

}n∈Z

is an or-

thonormal basis for H(E). Moreover, for every f(z) ∈ H(E),

(1.8) ‖f‖2 =∑n∈Z

∣∣∣∣ f(γn)E(γn)

∣∣∣∣2 π

ϕ′(γn)=∑n∈Z

|f(γn)|2‖K(γn, .)‖2E

,

and

(1.9) f(z) =∑n∈Z

f(γn)K(γn, z)

‖K(γn, .)‖2E.

There is at most one real number α modulo π such that the function eiαE(z)−e−iαE∗(z) belongs to H(E) [4]. It follows that for Γ which correspond to an αfor which {K(γn, z)} is complete, then Equation (1.8) is a sampling formula andEquation (1.9) is a reconstruction formula. The above theorem is the generalizationof the Shannon-Whitaker-Kotelnikov Sampling Theorem in the Paley–Wiener spaceto all de Branges spaces. See [2] for a detailed history of the Shannon-Whitaker-Kotelnikov Sampling Theorem.

A sequence Λ := {λn} is said to be a sampling sequence for H(E) if there existpositive constants A and B such that for all f ∈ H(E)

A‖f‖2E ≤∑n∈Z

|f(λn)|2‖K(λn, .)‖2E

≤ B‖f‖2E .

The sequence Λ is said to be an interpolating sequence for H(E) if for every sequence

of scalars {cn} that satisfies the condition∑

n∈Z

|cn|2‖K(λn,.)‖2

E< ∞, there exist f ∈

SAMPLING IN DE BRANGES SPACES 131

H(E) such that f(λn) = cn for all n ∈ Z. If for every such sequence {cn} thefunction f is unique then the sequence Λ is said to be a complete interpolatingsequence for H(E).

By the reproducing kernel property these definitions can be seen from the frametheory viewpoint, and the standard problems of sampling and interpolation in H(E)can be rephrased as follows: a sequence Λ = {λn}n∈Z is a sampling sequence inH(E) if and only if the corresponding sequence of normalized reproducing ker-

nels{ K(λn,.)

‖K(λn,.)‖}n∈Z

is a frame for H(E). Then, any function f ∈ H(E) can be

reconstructed from its samples on the sequence Λ by the (sampling) formula

f(z) =∑n∈Z

f(λn) kn(z)

where {kn}n∈Z is a dual frame of{ K(λn,.)‖K(λn,.)‖

}. Likewise, a sequence Λ is an inter-

polating sequence in H(E) if and only if the corresponding normalized reproducingkernels is a Riesz sequence in H(E). A complete interpolating sequence is a se-quence which is both interpolating and sampling.

A sequence Λ is said to be separated (or ρ-uniformly separated) if there existsρ > 0, such that inf

n�=m|λn − λm| ≥ ρ. The constant ρ is called the separation

constant of Λ. Without loss of generality, we will assume that a separated sequenceis monotone increasing.

The notion of Beurling density is one of the main ingredients in sampling theory.For R > 0, define n+(R), n−(R) by:

(1.10) n+(R) = supx∈R

�(Λ ∩ [x−R, x+R)),

and

(1.11) n−(R) = infx∈R

�(Λ ∩ [x−R, x+R)),

where �A denotes the cardinality of the set A. The upper and lower Beurling densityof Λ is defined, respectively, by

(1.12) D+(Λ) := lim supR→∞

n+(R)

2R, D−(Λ) := lim inf

R→∞

n−(R)

2R

If D+(Λ) = D−(Λ) = D(Λ), then the sequence Λ is said to have uniform Beurlingdensity D(Λ).

The canonical example of a de Branges space is the Paley–Wiener space PWa,a > 0. In this case we could write PWa = H(E), where E(z) = exp(−iaz),where the two spaces are equal as sets, and have equivalent norms. Landau provednecessary density conditions for sampling and interpolating sequences in the Paley–Wiener space PWπ ([13], see also [12] and [19]). Landau’s results were reprovenby Grochenig and Razafinjatovo [9] using an argument based on the HomogeneousApproximation Property. A complete characterization of which sequences are sam-pling in PWπ was obtained by Ortega-Cerda and Seip [16].

Characterizing sampling sequences in de Branges spaces (other than the Paley–Wiener spaces) is unresolved. Lyubarskii and Seip [14] extend Landau’s necessarydensity criteria to de Branges spaces which satisfy the condition that α ≤ ϕ′(x) ≤ β.Marzo, Nitzan, and Olsen [15] extend Landau’s results to de Branges spaces whichhave the property that the measure ϕ′(x)dx is a “doubling measure”. Our mainresult is an extension of Landau’s results for de Branges spaces with the property


that α ≤ ϕ′(x) only. While this is clearly weaker than the assumptions made byLyubarskii and Seip, it is unclear whether it is weaker than the doubling measurecondition. We note that if α ≤ ϕ′(x) ≤ β, then ϕ′(x)dx is a doubling measure.Also, Marzo et.al. give an example of a ϕ whose derivative is not bounded belowand ϕ′(x)dx satisfies a “locally doubling” condition, but it does not satisfy thedoubling condition. We are not aware of an example of a phase function whichsatisfies our condition but not the doubling measure condition.

Our proof of Landau’s necessary density conditions uses the Homogeneous Ap-proximation Property introduced by Grochenig and Razafinjatovo. We show theHAP holds for our class of de Branges spaces in Section 2. We prove our mainresult in Section 3. In an attempt to begin finding sufficient conditions for a se-quence to be a sampling sequence in a de Branges space, we prove in Section 4 thatfinite Beurling density is sufficient for a sequence to satisfy the Plancherel–PolyaInequality, which is the upper inequality in the sampling sequence criterion. Wenote that, generally, this upper inequality is much easier to establish.

2. Homogeneous Approximation Property

The Homogeneous Approximation Property and the Comparison Theorem wereintroduced by Ramanathan and Steger [18] in the context of Gabor frames. Aversion of the HAP for frames of translates of band-limited functions was provedby Grochenig and Razafinjatovo [9] where they used it to derive density conditionsfor sampling and interpolating sequences in the Paley–Wiener space. Furthermore,the HAP is extended to weighted wavelet frames by C. Heil and G. Kutyniok[10,11]. We will prove that the Homogeneous Approximation Property holds forthe reproducing kernel in H(E) from which will follow a comparison theorem forsampling and interpolating sequences in H(E).

Let Λ = {λn}n∈Z ⊂ R be a sequence such that the corresponding normalized re-

producing kernels kλn= K(λn,.)

‖K(λn,.)‖ form a Riesz basis inH(E), and M = {μn}n∈Z ⊆R be such that the corresponding normalized reproducing kernels kμn

= K(μn,.)‖K(μn,.)‖

form a frame in H(E). Let r > 0, and y ∈ R, and define the index sets

Jr(y) = {n ∈ Z : |λn − y| ≤ r}, Ir(y) = {n ∈ Z : |μn − y| ≤ r}Define the subspaces

Vr(y) := span{kλn: n ∈ Jr(y)}

andWr(y) := span{kμn

: n ∈ Ir(y)},where {kμn

} is the canonical dual frame of {kμn}. Denote the corresponding or-

thogonal projections by

Py,r : H(E) −→ Vr(y), Qy,r : H(E) −→ Wr(y)

lemma 2.1. For all f ∈ H(E)

(2.1) ‖f −Qy,rf‖ = infcn

‖f −∑

n∈Ir(y)

cnkμn‖ ≤

∥∥∥∥ ∑n/∈Ir(y)

f(μn)

‖K(μn, .)‖kμn

(z)

∥∥∥∥.Proof. The equality follows by definition of the orthogonal projection Qy,r.

The inequality follows by expanding f in terms of the dual frames {kμn}, {kμn

},

and setting cn = 〈f, kμn〉 = f(μn)

‖K(μn, .)‖. �


The Homogeneous Approximation Property:

Theorem 2.2. Let H(E) be a de Branges space such that the phase functionof E(z) satisfies 0 < δ ≤ ϕ′(x) for all x ∈ R. Let {μn}n∈Z ⊂ R be a separatedsequence such that {kμn

(z)}n∈Z is a frame in H(E). Then given ε > 0 there existsR = R(ε) > 0 such that for all y ∈ R and all r > 0

(2.2) sup|x−y|≤r

∥∥kx(.)−Qy,r+Rkx(.)∥∥ < ε,

where kx(z) =K(x,z)‖K(x,.)‖ , and the supremum is taken over x ∈ R.

Proof. First we will show that (2.2) holds when the function kx(z) is replace

by the function K(x,z)

E(x). Since the function K(x,z)

E(x)∈ H(E) for all x ∈ R, it can be

expanded in terms of the frame:

K(x, z)

E(x)=∑n∈Z

K(μn, x)

E(x)‖K(μn, .)‖kμn

(z)

Fix R > 0; we shall alter it later to suit our purposes. By Lemma 2.1,∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥ ≤∥∥∥∥ ∑

n/∈Ir+R(y)

K(μn, x)

E(x)‖K(μn, .)‖kμn

(z)

∥∥∥∥Since { K(μn,z)

‖K(μn,.)‖} is a frame, there exists a constant C > 0 such that

(2.3)

∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥2 ≤ C

|E(x)|2∑

n/∈Ir+R(y)

∣∣∣∣ K(μn, x)

‖K(μn, .)‖

∣∣∣∣2Using the assumption that 0 < δ ≤ ϕ′(x) for all x ∈ R together with (1.7), Inequal-ity (2.3) becomes∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥2 ≤ C

|E(x)|2

(π

δ

∑n/∈Ir+R(y)

∣∣∣∣K(μn, x)

E(μn)

∣∣∣∣2)(2.4)

Note that since μn, x ∈ R, we have by (1.4),∣∣∣∣ K(μn, x)

E(x)E(μn)

∣∣∣∣ = ∣∣∣∣ Im(E(x)E(μn))

E(x)E(μn) · π(μn − x)

∣∣∣∣;and therefore, Inequality (2.4) becomes

∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥2 ≤ C

πδ

∑n/∈Ir+R(y)

1

(x− μn)2≤ C

πδ

∑n/∈Lx(R)

1

(x− μn)2

(2.5)

where LR(x) := {n ∈ Z : |x − μn| ≤ R}. The assumption |x− y| ≤ r implies thatLR(x) ⊆ Ir+R(y).

Since the sequence {μn}n∈Z is separated, then for each x ∈ R and for eachfinite R > 0, the index set LR(x) is finite. Hence, we may assume that LR(x) ={n1, n2, . . . , nL} with n1 < n2 < · · · < nL. Note that |μnL+1 − x| ≥ R and fork > 1, |μnL+k −μnL+1| ≥ (k− 1)ρ (where ρ is the separation constant). Therefore,|μnL+k − x| ≥ R + (k − 1)ρ for all k ≥ 1. Likewise, |μn1−k − x| ≥ R + (k − 1)ρ forall k ≥ 1.


We have that∑n/∈LR(x)

1

(x− μn)2=

∑n<n1

1

(x− μn)2+

∑n>nL

1

(x− μn)2

=

∞∑k=1

1

(x− μn1−k)2+

∞∑k=1

1

(x− μnL+k)2

≤∞∑k=1

1

(R+ (k − 1)ρ)2+

∞∑k=1

1

(R+ (k − 1)ρ)2

= 2∞∑k=0

1

(R+ kρ)2.

Inequalities (2.5) and the one above imply that∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥2 <2C

πδ

∞∑k=0

1

(R+ kρ)2

for all x, y ∈ R with |x− y| ≤ r. Therefore,

sup|x−y|≤r

∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥2 <2C

πδ

∞∑k=0

1

(R+ kρ)2.

Since the latter sum is finite then we can choose R = R(ε) > 0 sufficiently largeso that the latter sum is less than επδ/2C. That is, the homogeneous approximationproperty holds for the function K(x, z)/E(x), for all x ∈ R.

Given ε > 0, choose R = R(ε) > 0, as above, such that

sup|x−y|≤r

∥∥∥∥K(x, .)

E(x)−Qy,r+R

K(x, .)

E(x)

∥∥∥∥ < ε

√δ

π

Using (1.7) and the assumption that δ ≤ ϕ′(x), we obtain for all r > 0 and ally ∈ R

sup|x−y|≤r

∥∥∥∥ K(x, .)

‖K(x, .)‖ −Qy,r+RK(x, .)

‖K(x, .)‖

∥∥∥∥ < ε.

�As a consequence of The Homogeneous Approximation Property, the Compar-

ison Theorem demonstrates that the Beurling density of a frame must be greaterthan the Beurling density of any orthonormal basis or Riesz basis in H(E). Thisis consistent with the fact that frames provide redundant non-orthogonal expan-sions in Hilbert space–accordingly, they should be “denser” than orthonormal bases(Riesz bases).

Theorem 2.3. Let H(E) be a de Branges space, and the corresponding phasefunction of E satisfies 0 < δ ≤ ϕ′(x) for all x ∈ R. Suppose that M = {μn},Γ ={γn} ⊆ R are two separated sequences, such that {kμn

(z)}n∈Z is a frame in H(E),and {kγn

(z)}n∈Z is a Riesz basis for a closed subspace of H(E). Then for everyε > 0, there exists R = R(ε) > 0, such that for all r > 0 and y ∈ R, we have

(1− ε) �(Γ ∩ [y − r, y + r)

)≤ �

(M∩ [y − r −R, y + r +R)

).

Therefore,D−(Γ) ≤ D−(M), and D+(Γ) ≤ D+(M)


Proof. Let kγndenote the biorthogonal basis of kγn

; then there exists Co > 0

such that ‖kγn‖ ≤ Co, for all n ∈ Z.

Given ε > 0, choose R = R(ε) > 0 such that the homogeneous approximationproperty holds for the functions kx(z), x ∈ R, with ε/Co, i.e., for all r > 0 andy ∈ R

sup|x−y|≤r

∥∥kx(.)−Qy,r+Rkx(.)∥∥ < ε/Co.

Given r > 0 and y ∈ R we define the operators Ty,r : Vr(y) → Vr(y) by

(2.6) Ty,r = Py,rQy,r+R.

By definition, the sequence {kγn}n∈Jr(y) is a basis for Vr(y) with the dual basis

{Py,rkγn}n∈Jr(y). Hence, by the biorthogonality of the sequences {kγn

}n∈Jr(y) and

{Py,rkγn}n∈Jr(y), the trace of Ty,r can be written as

tr(Ty,r) =∑

n∈Jr(y)

〈Ty,rkγn, kγn

〉

Since Py,r is an orthogonal projection, it is self adjoint, and Py,rkγn= kγn

, for alln ∈ Jr(y), hence we have

〈Ty,rkγn, kγn

〉 = 〈Py,rQy,r+Rkγn, kγn

〉= 〈Qy,r+Rkγn

, Py,rkγn〉

= 〈Py,rkγn, kγn

〉+ 〈Qy,r+Rkγn− kγn

, Py,rkγn〉

= 1 + 〈Qy,r+Rkγn− kγn

, Py,rkγn〉

So we have

(2.7) 〈Ty,rkγn, kγn

〉 − 1 = 〈Qy,r+Rkγn− kγn

, Py,rkγn〉

Applying the Cauchy – Schwarz inequality to the right hand side of the previousequation, and using the fact that ‖Py,rkγn

‖ ≤ Co, we obtain

|〈Qy,r+Rkγn− kγn

, Py,rkγn〉| ≤ ‖kγn

(.)−Qy,r+Rkγn(.)‖ ‖Py,rkγn

‖≤ sup

|x−y|≤r

∥∥kx(.)−Qy,r+Rkx(.)∥∥ ‖Py,r‖‖kγn

‖

< (ε/Co).Co

= ε

whenever |γn − y| ≤ r. Therefore, by (2.7) we get

(2.8) |〈Ty,rkγn, kγn

〉 − 1| = |〈Qy,r+Rkγn− kγn

, Py,rkγn〉| ≤ ε

Now, note that∣∣ ∑n∈Jr(y)

1−∑

n∈Jr(y)

〈Ty,rkγn, kγn

〉∣∣ ≤

∑n∈Jr(y)

∣∣1− 〈Ty,rkγn, kγn

〉∣∣

Hence, by the definition of the trace of Ty,r and (2.8) we have⎛⎝ ∑n∈Jr(y)

1

⎞⎠ − tr(Ty,r) ≤∣∣ ∑n∈Jr(y)

1−∑

n∈Jr(y)

〈Ty,rkγn, kγn

〉∣∣ ≤ ∑

n∈Jr(y)

ε


Therefore, we can estimate a lower bound to the trace of Ty,r by

tr(Ty,r) ≥∑

n∈Jr(y)

(1− ε) = (1− ε)�(Γ ∩ [y − r, y + r])

On the other hand, since the operator norm of Ty,r satisfies

‖Ty,r‖ = ‖Py,rQy,r+R‖ ≤ ‖Py,r‖‖Qy,r+R‖ = 1,

all the eigenvalues of Ty,r have modulus less than or equal to 1, this in turn providesus with an upper bound for the trace of Ty,r. Indeed,

tr(Ty,r) =∑

(non-zero eigenvalues of Ty,r) ≤ rank(Ty,r)

Also, since

rank(Ty,r) = dim(range(Ty,r)) = dim(range(Py,rQy,r+R)) ≤ dim(Wr+R),

then

tr(Ty,r) ≤ dim(Wr+R) ≤ �{μn : |μn − y| ≤ r +R}= �(M∩ [y − r −R, y + r +R])

Therefore, combining these two estimates of the trace of Ty,r we get

(1− ε)�(Γ ∩ [y − r, y + r]) ≤ �(M∩ [y − r − R, y + r +R])

for all r > 0 and all y ∈ R. Moreover,

(1− ε)�(Γ ∩ [y − r, y + r])

2r≤ (2r + 2R)

2r

�(M∩ [y − r −R, y + r +R])

(2r + 2R),

so taking the infimum over all y ∈ R for both sides yields

(1− ε) infy∈R

�(Γ ∩ [y − r, y + r])

2r≤ (2r + 2R)

2rinfy∈R

�(M∩ [y − r −R, y + r +R])

(2r + 2R)

and by taking liminf as r → ∞ yields the estimates

(1− ε)D−(Γ) ≤ D−(M)

Since ε is arbitrary, we conclude that

D−(Γ) ≤ D−(M)

A similar calculation shows that

D+(Γ) ≤ D+(M)

�

3. Necessary Conditions for Sampling and Interpolating

We are now in a position to prove our main result concerning necessary densityconditions for sequences to be sampling or interpolating in H(E).

Theorem 3.1. Let E ∈ HB, with phase function satisfying 0 < δ ≤ ϕ′(x), forall x ∈ R. If M = {μn}n∈Z is a uniformly separated sampling sequence in H(E),then D−(M) ≥ δ

π .


Proof. Let Λ = {λn}n∈Z ⊂ R be such that ϕ(λn) = α + nπ, for all n ∈Z, for some α ∈ [0, π). Thus, the corresponding normalized reproducing kernels{kλn

(z)}n∈Z forms an orthonormal set for H(E), and D−(ϕ(Λ)) = 1π . On the other

hand, since 0 < δ ≤ ϕ′(x) for all x ∈ R, a simple calculation gives D−(ϕ(Λ)) ≤1δ D

−(Λ) hence, D−(Λ) ≥ δπ . If M = {μn}n∈Z is a sampling sequence in H(E),

then by the Comparison Theorem we have

D−(M) ≥ D−(Λ) ≥ δ

π,

as desired. �

Theorem 3.2. Let E ∈ HB, with phase function satisfying 0 < δ ≤ ϕ′(x) ≤M < ∞, for all x ∈ R. If Γ = {γn}n∈Z is a uniformly separated interpolatingsequence in H(E), then D+(Γ) ≤ M

π .

Proof. Let Γ = {γn}n∈Z be an interpolating sequence in H(E), then thecorresponding normalized reproducing kernels {kγn

(z)}n∈Z is a Riesz basis for somesubspace of H(E). Let Λ = {λn}n∈Z ⊂ R be such that ϕ(λn) = α + nπ, for alln ∈ Z, where α ∈ [0, π) is chosen so that the corresponding normalized reproducingkernels of λn’s forms an orthonormal basis for H(E). Note that D+(ϕ(Λ)) = 1

π , and

since ϕ′(x) ≤ M for all x ∈ R, a simple calculation gives 1M D+(Λ) ≤ D+(ϕ(Λ))

hence D+(Λ) ≤ Mπ . The Comparison Theorem implies that

D+(Γ) ≤ D+(Λ) ≤ M

π.

�

Corollary 3.3. Let E ∈ HB, with phase function satisfying 0 < δ ≤ ϕ′(x) ≤M , for all x ∈ R. If Γ = {γn}n∈Z is a uniformly separated complete interpolatingsequence in H(E), then δ

π ≤ D−(Γ) ≤ D+(Γ) ≤ Mπ .

Example 3.4. The Paley–Wiener space PWπ is the de Branges space associ-ated to E(z) = e−iπz, which has phase function ϕ(x) = πx. Therefore, ϕ′(x) = π.Applying Theorems 3.1 and 3.2, we obtain that a sampling sequence for PWπ musthave density at least 1, and an interpolating sequence must have density at most1. Thus, we reproduce Landau’s necessary conditions for the Paley–Wiener space[13].

4. The Plancherel–Polya Inequality

In the previous section, we proved necessary conditions for a sequence to be asampling sequence or an interpolation sequence in a de Branges space. Naturally, itis desirable to also have sufficient conditions for a sequence to be sampling or inter-polating. We do not have results along these lines; however, we do have sufficientconditions for a sequence to be a Plancherel–Polya sequence.

We say that a sequence {μn}n∈Z of real numbers is a Plancherel–Polya sequencein H(E) if there exists a positive constant B, independent of f , such that∑

n∈Z

|f(λn)|2‖K(λn, .)‖2H

≤ B‖f‖2H

for all f ∈ H(E), where K(w, z) is the reproducing kernel of H(E). If the inequalityholds, we say that the sequence satisfies the Plancherel–Polya Inequality.


Recall that in the case of the Paley–Wiener space PWπ, a sufficient (and neces-sary) condition for a sequence to be a Plancherel–Polya sequence is that the upperBeurling density is finite. We will prove in Theorem 4.10 that the same result holdsin H(E) provided E possesses a bounded logarithmic derivative, so that H(E) pos-sesses a certain Berstein type inequality. Our proof involves a several step process:we first prove that sequences which interlace an orthonormal set is a Plancherel–Polya sequence; we then prove that perturbations of Plancherel–Polya sequencesare again Plancherel–Polya analogous to the Paley–Wiener perturbation theorem.We then prove the full result.

We require a few technical lemmas to begin.

lemma 4.1. Let E be a de Branges function with E′

E ∈ L∞(R), and ϕ be aphase function of E, then ϕ′ is bounded on R, moreover, ‖ϕ′‖∞ ≤ ‖E′/E‖∞.

Proof. By the definition of the phase function in (1.6), E(x) = |E(x)|e−iϕ(x),so for all x ∈ R the logarithmic derivative of E

E′(x)

E(x)= −iϕ′(x) +

|E(x)|′|E(x)| ,

therefore, ϕ′(x) = −Im(E′(x)

E(x) ). Hence,∣∣∣∣E′(x)

E(x)

∣∣∣∣2 = |ϕ′(x)|2 +∣∣∣∣ |E(x)|′|E(x)|

∣∣∣∣2,for all x ∈ R, and ‖ϕ′‖∞ ≤ ‖E′/E‖∞. �

lemma 4.2. Let H(E) be a de Brange space, and ϕ(x) be the correspondingphase function of E(z). Let α ∈ [0, π), and {λn}n∈Z be a sequence of real numberssuch that ϕ(λn) = α+ nπ, n ∈ Z. If 0 < δ ≤ ϕ′(x) ≤ M , for all x ∈ R, then

(4.1)π

M≤ λn+1 − λn ≤ π

δ

for all n ∈ Z.

Proof. Let n ∈ Z, then by the Mean Value Theorem there exist a point νnbetween λn and λn+1 such that

ϕ(λn+1)− ϕ(λn)

λn+1 − λn= ϕ′(νn),

hence,

λn+1 − λn =ϕ(λn+1)− ϕ(λn)

ϕ′(νn).

Since ϕ(λn) = α + nπ, then ϕ(λn+1) − ϕ(λn) = π. Also, since δ ≤ ϕ′(x) ≤ M forall x ∈ R, then

π

M≤ ϕ(λn+1)− ϕ(λn)

ϕ′(νn)≤ π

δ.

Thus, πM ≤ λn+1 − λn ≤ π

δ , as desired. �

lemma 4.3. Let H(E) be a de Brange space, and ϕ(x) be the correspondingphase function of E(z), and K(w, z) be the corresponding reproducing kernel. Letα ∈ [0, π), and {λn}n∈Z be a sequence of real numbers such that ϕ(λn) = α + nπ,


n ∈ Z. Let fn(z) =K(λn,z)

E(λn), n ∈ Z, z ∈ C. If 0 < δ ≤ ϕ′(x) ≤ M , for all x ∈ R,

thenδ

π‖f‖2E ≤

∑n

|〈f, fn〉|2 ≤ M

π‖f‖2E ,

for all f ∈ H(E), i.e., the sequence {fn}n∈Z is a frame for H(E).

Proof. Let f ∈ H(E). Since ϕ(λn) = α+nπ, n ∈ Z, then by Theorem 1.1 the

corresponding normalized reproducing kernels { K(λn,.)‖K(λn,.)‖} is an orthonormal basis

in H(E), thus we have∑n

|f(λn)|2K(λn, λn)

=∑n

∣∣〈f, K(λn, .)

‖K(λn, .)‖〉∣∣2 = ‖f‖2E .

Using the fact that K(x, x) = 1πϕ

′(x)|E(x)|2 for all x ∈ R, and that ϕ′(x) ≤ M ,we obtain ∑

n

|〈f, fn〉|2 =∑n

∣∣∣∣〈f(t), K(λn, t)

E(λn)〉∣∣∣∣2

=∑n

|f(λn)|2|E(λn)|2

=∑n

π|f(λn)|2ϕ′(λn)|E(λn)|2

ϕ′(λn)

π

=∑n


ϕ′(λn)

π

≤ M

π

∑n


(4.2)

=M

π‖f‖2E .

Similarly, since 0 < δ ≤ ϕ′(x), we also get∑n

|〈f, fn〉|2 =∑n


ϕ′(λn)

π

≥ δ

π

∑n


=δ

π‖f‖2E .

Since f is arbitrary, then the sequence {fn}n∈Z is a frame for H(E), completingthe proof. �

lemma 4.4. Let ϕ : R → R be a phase function of E(z) satisfying 0 < δ ≤ϕ′(x) ≤ M , for all x ∈ R. Let M = {μn}n∈Z ⊆ R. Then

(4.3)1

MD−(M) ≤ D−(ϕ(M)) ≤ 1

δD−(M)

and

(4.4)1

MD+(M) ≤ D+(ϕ(M)) ≤ 1

δD+(M)


Proof. Let r > 0, first we will show that

(4.5) [ϕ(x− r), ϕ(x+ r)] ⊆ [ϕ(x)−Mr,ϕ(x) +Mr]

for all x ∈ R, or equivalently, ϕ(x)−Mr ≤ ϕ(x− r) ≤ ϕ(x+ r) ≤ ϕ(x) +Mr forall x ∈ R. To begin with, let x ∈ R, then since ϕ is continuously differentiable onR we have

ϕ(x+ r) = ϕ(x) +

∫ x+r

x

ϕ′(x)dx

≤ ϕ(x) +

∫ x+r

x

M dx

= ϕ(x) +Mr

hence, ϕ(x+ r) ≤ ϕ(x) +Mr. On the other hand, we have

ϕ(x) = ϕ(x− r + r)

= ϕ((x− r) + r)

≤ ϕ(x− r) +Mr, (by applying the result above for x− r),

and we get ϕ(x)−Mr ≤ ϕ(x− r). Since ϕ is a nondecreasing function and r > 0,then ϕ(x−r) < ϕ(x+r). Therefore, ϕ(x)−Mr ≤ ϕ(x−r) ≤ ϕ(x+r) ≤ ϕ(x)+Mrfor all x ∈ R.

Using the fact that ϕ is bijective and relation (4.5), we get

�(M∩ [x− r, x+ r]) = �(ϕ(M) ∩ [ϕ(x− r), ϕ(x+ r)])

≤ �(ϕ(M) ∩ [ϕ(x)−Mr,ϕ(x) +Mr])

for all x ∈ R. Hence,

infx∈R

�(M∩ [x− r, x+ r])

Mr≤ inf

x∈R

�(ϕ(M) ∩ [ϕ(x)−Mr,ϕ(x) +Mr])

Mr

= infy∈R

�(ϕ(M) ∩ [y −Mr, y +Mr])

Mr

for all r > 0. Taking liminf as r → ∞ yields

1

MD−(M) = lim inf

r→∞infx∈R

�(M∩ [x− r, x+ r])

Mr≤ lim inf

r→∞infy∈R

�(ϕ(M) ∩ [y −Mr, y +Mr])

Mr

= D−(ϕ(M)).

Again, let r > 0, we will show that

(4.6) [ϕ(x)− δr, ϕ(x) + δr] ⊆ [ϕ(x− r), ϕ(x+ r)]

for all x ∈ R, or equivalently, ϕ(x− r) ≤ ϕ(x)− δr ≤ ϕ(x) + δr ≤ ϕ(x+ r) for allx ∈ R. Let x ∈ R, then

ϕ(x+ r) = ϕ(x) +

∫ x+r

x

ϕ′(x)dx

≥ ϕ(x) +

∫ x+r

x

δ dx

= ϕ(x) + δr


hence, ϕ(x+ r) ≥ ϕ(x) + δr. On the other hand, we have

ϕ(x) = ϕ(x− r + r)

= ϕ((x− r) + r)

≥ ϕ(x− r) + δr,

and we get ϕ(x)− δr ≥ ϕ(x− r). Therefore, ϕ(x− r) ≤ ϕ(x)− δr ≤ ϕ(x) + δr ≤ϕ(x+ r) for all x ∈ R.

Again, using the fact that ϕ is bijective, and relation (4.6), we get

�(M∩ [x− r, x+ r]) = �(ϕ(M) ∩ [ϕ(x− r), ϕ(x+ r)])

≥ �(ϕ(M) ∩ [ϕ(x)− δr, ϕ(x) + δr])

for all x ∈ R. Hence,

infx∈R

�(M∩ [x− r, x+ r])

δr≥ inf

x∈R

�(ϕ(M) ∩ [ϕ(x)− δr, ϕ(x) + δr])

δr

≡ infy∈R

�(ϕ(M) ∩ [y − δr, y + δr])

δr

for all r > 0. Taking liminf as r → ∞ yields

1

δD−(M) = lim inf

r→∞infx∈R

�(M∩ [x− r, x+ r])

δr≥ lim inf

r→∞infy∈R

�(ϕ(M) ∩ [y − δr, y + δr])

δr

= D−(ϕ(M))

Similar computations show that

1

MD+(M) ≤ D+(ϕ(M)) ≤ 1

δD+(M)

�

The following is well known (see Lemma 7.1.3 of [3]).

lemma 4.5. Let Λ = {λn}n∈I be a sequence of real numbers. Then the followingare equivalent:

(a) D+(Λ) < ∞.(b) Λ is relatively separated.(c) For every R > 0, there exists an integer NR > 0 such that

supn∈Z

�(Λ ∩

[(n− 1)R, (n+ 1)R

))= NR < ∞

(d) For some R > 0, there exists an integer NR > 0 such that

supn∈Z

�(Λ ∩

[(n− 1)R, (n+ 1)R

))= NR < ∞

We will require the Bernstein inequality in our de Branges spaces. The neces-sary assumption is that the weight function E has bounded logarithmic derivative,i.e. E′/E ∈ L∞(R). The following result appears in [1].

Theorem 4.6 (The Bernstein Inequality). Let E(z) ∈ HB. If E′

E ∈ L∞(R),then

(4.7)

∥∥∥∥f ′

E

∥∥∥∥2

≤ C

∥∥∥∥E′

E

∥∥∥∥∞||f ||E

for all f ∈ H(E), with C ≤ 4 +√6.


The following lemma is a direct application of the Bernstein inequality.

lemma 4.7. Let H(E) be a de Branges space. If E′/E ∈ L∞(R), then (f/E)′ ∈L2(R) for all f ∈ H(E).

Proof. Let f(z) ∈ H(E). Using the identity |a + b|2 ≤ 2(|a|2 + |b|2) for anya, b ∈ R we get

∫ ∞

−∞

∣∣∣∣( f(t)

E(t)

)′∣∣∣∣2 dt =

∫ ∞

−∞

∣∣∣∣f ′(t)

E(t)− E′(t)

E(t)

f(t)

E(t)

∣∣∣∣2 dt

≤ 2

(∫ ∞

−∞

∣∣∣∣f ′(t)

E(t)

∣∣∣∣2 dt+ ∥∥∥∥E′

E

∥∥∥∥2∞

∫ ∞

−∞

∣∣∣∣ f(t)E(t)

∣∣∣∣2 dt)= 2

(∥∥∥∥f ′

E

∥∥∥∥22

+

∥∥∥∥E′

E

∥∥∥∥2∞‖f‖2E

)≤ 2

(C2

∥∥∥∥E′

E

∥∥∥∥2∞

‖f‖2E +

∥∥∥∥E′

E

∥∥∥∥2∞‖f‖2E

)= 2(C2 + 1)‖E′/E‖2∞ ‖f‖2E

where we used the Bernstein inequality with constant C ≤ 4+√6. The right-hand

side of the last inequality is finite by the assumptions. �

Our first result concerning Plancherel–Polya sequences is the following theorem,which states that if a sequence of points interlaces an orthogonal basis of kernels,then the sequence satisfies the Plancherel–Polya Inequality.

Theorem 4.8. Let H(E) be a de Branges space where E has no real zeros,E′

E ∈ L∞(R), and ϕ′(x) is bounded away from zero. Let {λn}n∈Z, {μn}n∈Z be twosequences of real numbers, such that ϕ(λn) = α + nπ for all n ∈ Z. If λn ≤ μn ≤λn+1, for all n ∈ Z, then {μn}n∈Z is a Plancherel–Polya sequence in H(E).

Proof. Since ϕ′ is bounded away from zero, then there exist δ > 0 such that

ϕ′(x) ≥ δ, for all x ∈ R. Also, since E′

E ∈ L∞(R) then, by Lemma 4.1, ϕ′(x) ≤ Mfor all x ∈ R, for some M > 0. Let λn ≤ μn ≤ λn+1, and ϕ(λn) = α + nπfor all n ∈ Z. Then by Lemma 4.2 we have λn+1 − λn ≤ π

δ . Consequently,maxn |μn − λn| ≤ ρ ≤ π

δ .

Set fn(z) =K(λn,z)

E(λn), and gn(z) =

K(μn,z)

E(μn), n ∈ Z. We need to show that there

exist a constant Bμ > 0, such that

∑n

∣∣∣∣〈f, K(μn, .)

‖K(μn, .)‖〉∣∣∣∣2 ≤ Bμ‖f‖2

for all f ∈ H(E).


To begin with, note that given f ∈ H(E), the function f(t)/E(t) is continuousand differentiable for all t ∈ R. Hence, using Holder’s inequality we get

|〈f, gn − fn〉|2 =

∣∣∣∣⟨f, K(μn, .)

E(μn)− K(λn, .)

E(λn)

⟩∣∣∣∣2=

∣∣∣∣⟨f, K(μn, .)

E(μn)

⟩−⟨f,

K(λn, .)

E(λn)

⟩∣∣∣∣2=

∣∣∣∣ f(μn)

E(μn)− f(λn)

E(λn)

∣∣∣∣2=

∣∣∣∣ ∫ μn

λn

(f(t)

E(t)

)′dt

∣∣∣∣2≤

∫ μn

λn

∣∣∣∣( f(t)

E(t)

)′∣∣∣∣2 dt .

∫ μn

λn

1 dt

≤ maxn

|μn − λn|∫ μn

λn

∣∣∣∣( f(t)

E(t)

)′∣∣∣∣2 dt

≤ ρ

∫ λn+1

λn

∣∣∣∣( f(t)

E(t)

)′∣∣∣∣2 dt

Hence, by Lemma 4.7 we get

∑n

|〈f, gn − fn〉|2 ≤ ρ∑n

∫ λn+1

λn

∣∣∣∣( f(t)

E(t)

)′∣∣∣∣2 dt

≤ ρ

∫ ∞

−∞

∣∣∣∣( f(t)

E(t)

)′∣∣∣∣2 dt

≤ 2ρ

(C2

∥∥∥∥E′

E

∥∥∥∥2∞

‖f‖2E +

∥∥∥∥E′

E

∥∥∥∥2∞‖f‖2E

)Therefore,

(4.8)∑n

|〈f, gn − fn〉|2 ≤ R ‖f‖2E

where ρ ≤ π/δ, and R = 2ρ(C2 +1)‖E′/E‖2∞. Since ϕ(λn) = α+ nπ for all n ∈ Z,then by Lemma 4.3, the sequence {fn}n∈Z is a frame with frame bounds δ/π andM/π. Therefore, the Minkowski inequality implies that

(∑n

|〈f, gn〉|2) 1

2

≤(∑

n

|〈f, gn − fn〉|2) 1

2

+

(∑n

|〈f, fn〉|2) 1

2

≤√R ||f ||+

√M

π||f ||

=

(√R+

√M

π

)||f ||


for every f ∈ H(E). It follows that∑n

∣∣∣∣〈f, K(μn, .)

‖K(μn, .)‖〉∣∣∣∣2 =

∑n

|f(μn)|2K(μn, μn)

=∑n

π|f(μn)|2ϕ′(μn)|E(μn)|2

≤ π

δ

∑n

|f(μn)|2|E(μn)|2

=π

δ

∑n

|〈f, gn〉|2(4.9)

≤ π

δ

(√R+

√M

π

)2‖f‖2Efor all f ∈ H(E). That is, the sequence {μn}n∈Z is a Plancherel–Polya sequence in

H(E), with bound (at most) Bμ = πδ

(√R+

√Mπ

)2, completing the proof.

�

The following result is a variation of the well-known Paley–Wiener perturbationidea [17].

Theorem 4.9. Let H(E) be a de Branges space where E has no real zeros,E′

E ∈ L∞(R), and 0 < δ ≤ ϕ′(x) for all x ∈ R. Let N = {νn}n∈Z ⊂ R be aδo-uniformly separated sequence. Let

M := {νn + εn : εn ∈ [−η, η], n ∈ Z},

where 0 < η < δo/2. If N is a Plancherel–Polya sequence in H(E) with bound Bν ,then M is also a Plancherel–Polya sequence in H(E) with bound Bμ = Bμ(Bν , η).

Proof. Let M = {μn}n∈Z, then |νn − μn| = |εn| ≤ η, for all n ∈ Z. LetM1 = {μn ∈ M : εn ≥ 0} and M2 = {μn ∈ M : εn < 0}, then M = M1 ∪ M2.Since the union of Plancherel–Polya sequences is again such sequence, it is enoughto show that M1 is Plancherel–Polya sequences in H(E) (the same proof will applyfor M2). Therefore, without loss of generality we may assume that εn ≥ 0 for alln.

First note that since E′

E ∈ L∞(R), then ϕ′(x) ≤ M for all x ∈ R by Lemma

4.1. Set fn(z) =K(νn,z)

E(νn), and gn(z) =

K(μn,z)

E(μn), n ∈ Z. Let f ∈ H(E). Following

the same computations in the proof of Theorem 4.8 we get∑n

|〈f, gn − fn〉|2 ≤ 2max |νn − μn|(C2

∥∥∥∥E′

E

∥∥∥∥2∞

‖f‖2E +

∥∥∥∥E′

E

∥∥∥∥2∞‖f‖2E

)≤ R‖f‖2E

where R = 2η(C2 + 1)‖E′

E ‖2∞.Since N is a Plancherel–Polya sequence in H(E) with bound Bν , then∑

n

|f(νn)|2K(νn, νn)

=∑n

|〈f, K(νn, .)

‖K(νn, .)‖〉|2 ≤ Bν‖f‖2.


Hence, by inequality (4.2) we have∑n

|〈f, fn〉|2 ≤ M

π

∑n

|f(νn)|2K(νn, νn)

≤ M

πBν‖f‖2.

Therefore,(∑n

|〈f, gn〉|2) 1

2

≤(∑

n

|〈f, gn − fn〉|2) 1

2

+

(∑n

|〈f, fn〉|2) 1

2

≤√R ‖f‖+

√M

πBν ‖f‖

=

(√R+

√M

πBν

)‖f‖

Let B =√R+

√Mπ Bν . Inequality (4.9) implies that∑

n

∣∣∣∣〈f, K(μn, .)

‖K(μn, .)‖〉∣∣∣∣2 ≤ π

δ

∑n

|〈f, gn〉|2

≤ π

δB2‖f‖2E .

Since f ∈ H(E) is arbitrary, this implies that the sequence {μn}n∈Z is a Plancherel–Polya sequence in H(E), with bound (at most) Bμ = π

δB2, completing the proof.

�

Theorem 4.10. Given a de Brange space H(E) with E has no real zeros,E′

E ∈ L∞(R), and the derivative of the corresponding phase function of E is boundedaway from zero. Let M = {μn}n∈Z be a sequence of real numbers. If D+(M) < ∞,then M is a Plancherel–Polya sequence in H(E).

Proof. Assume that E′

E ∈ L∞(R), then by Lemma 4.1 there is a constantM > 0 such that ϕ′(x) ≤ M < ∞, for all x ∈ R. Also, since ϕ′ is bounded awayfrom zero on R, then there exist δ > 0 such that δ ≤ ϕ′(x), for all x ∈ R. ByTheorem 1.1 we can find a sequence {λn}n∈Z ⊂ R, such that ϕ(λn) = α+nπ for alln ∈ Z, for some α ∈ [0, π) and the corresponding normalized reproducing kernels

is an orthonormal basis in H(E). Set kλn(z) = K(λn,z)

||K(λn,.)|| , and kμn(z) = K(μn,z)

||K(μn,.)|| ,

n ∈ Z. We need to show that there is some constant Bμ > 0, such that∑n∈Z

|〈f, kμn〉|2 ≤ Bμ||f ||2, for every f ∈ H(E).

Since D+(M) < ∞, then D+(ϕ(M)) < ∞ by Lemma 4.4. Lemma 4.5 impliesthat the number of points of the sequence {ϕ(M)} in any interval of a given finitelength is bounded, that is, given R > 0 there exist an integer NR > 0 such that

supy∈R

�({ϕ(M)} ∩ [y, y +R)) ≤ supy∈R

�({ϕ(M)} ∩ [y −R, y +R)) ≤ NR < ∞.

In particular, for R = π, then there exist Nπ ∈ N, such that

�({ϕ(M)} ∩

[α+ kπ, α+ (k + 1)π

) )≤ Nπ, for all k ∈ Z

or equivalently,

�({ϕ(M)} ∩

[ϕ(λk), ϕ(λk+1)

) )≤ Nπ, for all k ∈ Z


Since the function ϕ is bijective we can trace the points ϕ(μn) back to get

�(M∩

[λk, λk+1

) )≤ Nπ, for all k ∈ Z.

This means that we can partition the sequence M = {μn}n∈Z into a finitenumber of (disjoint) subsequences Mj in a way such that for each 1 ≤ j ≤ Nπ

there is at most one point of the sequence Mj in [λk, λk+1], for all k ∈ Z:

M =

Nπ⋃j=1

Mj , Mj := {μ(j)i }i∈Z, j = 1, 2, . . . , Nπ

For j ∈ {1, 2, . . . , Nπ}, define the index set Ij := {ki ∈ Z : �(Mj∩[λki, λki+1)) =

1}. Then the sequences Mj and Λj := {λki}ki∈Ij are interlaced. Since ϕ(λki

) =α+ kiπ for all ki ∈ Ij , then∑

ki∈Ij

|〈f, kλki〉|2 ≤

∑n∈Z

|〈f, kλn〉|2 = ‖f‖2.

Therefore, applying Theorem 4.8 for the sequencesMj and Λj , j = 1, 2, . . . , Nπ,implies that the sequence Mj is a Plancherel–Polya sequence for H(E), with

bound at most Bμ := πδ

(√2πδ

√C2 + 1

∥∥E′

E

∥∥∞ +

√Mπ

)2, for every j = 1, 2, . . . , Nπ,

where C is the Bernstein inequality constant. Hence, the sequence {μn}n∈Z is aPlancherel–Polya sequence for H(E), with bound at most NπBμ. �

References

[1] A. D. Baranov, The Bernstein inequality in the de Branges spaces and embedding theo-rems, Proceedings of the St. Petersburg Mathematical Society, Vol. IX, Amer. Math. Soc.Transl. Ser. 2, vol. 209, Amer. Math. Soc., Providence, RI, 2003, pp. 21–49. MR2018371(2004m:30043)

[2] Modern sampling theory, Applied and Numerical Harmonic Analysis, Birkhauser Boston,Inc., Boston, MA, 2001. Mathematics and applications; Edited by John J. Benedetto andPaulo J. S. G. Ferreira. MR1865678 (2003a:94003)


[4] Louis de Branges, Some Hilbert spaces of entire functions, Proc. Amer. Math. Soc. 10 (1959),840–846. MR0114002 (22 #4833)

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(1961), 73–115. MR0133457 (24 #A3289c)[8] Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs,

N.J., 1968. MR0229011 (37 #4590)[9] K. Grochenig and H. Razafinjatovo, On Landau’s necessary density conditions for sampling

and interpolation of band-limited functions, J. London Math. Soc. (2) 54 (1996), no. 3, 557–565, DOI 10.1112/jlms/54.3.557. MR1413898 (98m:42029)

[10] Christopher Heil and Gitta Kutyniok, Density of weighted wavelet frames, J. Geom. Anal.13 (2003), no. 3, 479–493, DOI 10.1007/BF02922055. MR1984851 (2004d:42065)

[11] Christopher Heil and Gitta Kutyniok, The homogeneous approximation property for waveletframes, J. Approx. Theory 147 (2007), no. 1, 28–46, DOI 10.1016/j.jat.2006.12.011.MR2346801 (2008g:42031)

[12] S. Jaffard, A density criterion for frames of complex exponentials, Michigan Math. J. 38(1991), no. 3, 339–348, DOI 10.1307/mmj/1029004386. MR1116493 (92i:42001)

[13] H. J. Landau, Necessary density conditions for sampling and interpolation of certain entirefunctions, Acta Math. 117 (1967), 37–52. MR0222554 (36 #5604)


[14] Yurii I. Lyubarskii and Kristian Seip, Weighted Paley-Wiener spaces, J. Amer. Math. Soc.15 (2002), no. 4, 979–1006 (electronic), DOI 10.1090/S0894-0347-02-00397-1. MR1915824(2003m:46039)

[15] Jordi Marzo, Shahaf Nitzan, and Jan-Fredrik Olsen, Sampling and interpolation in de Brangesspaces with doubling phase, J. Anal. Math. 117 (2012), 365–395, DOI 10.1007/s11854-012-0026-2. MR2944102

[16] Joaquim Ortega-Cerda and Kristian Seip, Fourier frames, Ann. of Math. (2) 155 (2002),

no. 3, 789–806, DOI 10.2307/3062132. MR1923965 (2003k:42055)[17] Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain,

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[18] Jayakumar Ramanathan and Tim Steger, Incompleteness of sparse coherent states, Appl.Comput. Harmon. Anal. 2 (1995), no. 2, 148–153, DOI 10.1006/acha.1995.1010. MR1325536(96b:81049)

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Department of Mathematics, Hashemite University, Zarqa, Jordan


Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, Iowa

50011



Dynamical sampling in hybrid shift invariant spaces

Roza Aceska and Sui Tang

Abstract. In the modeling of smooth spatio - temporal fields (eg. in wirelesssensor networks) it is desirable to have locally adapted smoothness of the field.Hybrid shift invariant spaces (HSIS), studied in this paper, are a good fit forthis purpose. We explore the properties of these spaces and formulate the dy-namical sampling problem in this new setting. We generalize the techniques ofdynamical sampling in shift-invariant spaces to HSIS. We solve the dynamicalsampling problem in hybrid shift invariant spaces under special assumptionsand the related results in shift invariant spaces become corollaries.

1. Introduction

1.1. What is the dynamical sampling problem? Let a function f , definedon a domain D, be the initial state of a physical system that evolves in time underthe action of a family of operators At (indexed by t ≥ 0). It is known that f can berecovered if we take samples on D, i.e, f is uniquely determined by all values on D.An interesting question to ask is if we can under-sample the function f = f0 and stillrecover it by using its subsampled states Atif = fti , i = 1, . . . , N , at appropriatesampling sets Xi ⊂ D. For example, if f = f0 is an initial temperature distributionand At is the heat diffusion operator, then ft is the temperature distribution attime t. Samples are taken at varying times, and the reconstruction of f = f0 usessamples of these various states, f0, ft1 , . . . , ftN . The dynamical sampling problemis solved when conditions on the sampling sets and N are found, so that recovery ofthe signal is possible. This new problem is related to sensing networks [11,12] andthe work in [9,10]. In [9] Lu and Vetterli study the problem of dynamical samplingfor the specific case of bandlimited functions, with a heat kernel as an evolutionoperator. Several related mathematical models addressing special cases have beenstudied in [1,4,5] and we briefly reflect on them in the next subsection.

1.2. Dynamical sampling in special cases. When f ∈ �2(Z), defined onthe domainD = Z, the evolution operator family is given in [5] by powers of discreteconvolution, denoted by An(f) = a ∗ a ∗ a.... ∗ a ∗ f , which in short is labeled asan ∗f . The samples of f are taken on a uniform grid X = mZ ⊂ D for some m > 1and the related sampling operator is labeled as Sm(f) = f |mZ. The dynamical

2010 Mathematics Subject Classification. Primary 94A20, 94A12, 42C15, 15A29.Key words and phrases. Dynamical sampling, hybrid shift invariant spaces.The second author was supported in part by the NSF grant DMS- 1322099.


149

150 ROZA ACESKA AND SUI TANG

sampling problem under these assumptions can be stated as follows: Under whatconditions on a and N can a function f ∈ �2(Z) be recovered from the samples

(1.1) {f(X), a ∗ f(X), · · · , (aN−1 ∗ f)(X)}, for X ⊂ Z?

WhenX = mZ, we call the sampling procedure for obtaining the data (1.1) am,N-dynamical sampling scheme. If in addition, N = m, then we call it a m-dynamicalsampling scheme.

In [4, 5] the authors take the Fourier transform (labeled as or F) of therepeated samples and use Poisson’s summation formula to derive a matrix repre-sentation for the sampling process. In this way, all the sampling information isintergrated into a matrix and the recovery problem can be studied using matrixproperties.

In shift invariant spaces (SIS1) the crucial ingredient towards a stable recon-struction of a sampled function, influenced by a convolution operator a, is describedby [1]:

Lemma 1.1. Let a SIS V (ϕ) be defined as in (2.1) and take f =∑

k ckϕ(.−k) ∈V (ϕ). Let

(1.2) ϕj = aj ∗ ϕ, fj = aj ∗ f, hj = fj |Z and Φj = ϕj |Zfor j = 0, 1, ...,m− 1. Then

(1.3) F(Smhj)(ξ) =1

m

m−1∑l=0

c(ξ + l

m

)Φj

(ξ + l

m

).

In short notation, it holds

(1.4) y(ξ) = Am(ξ)cm(ξ),

with Am(ξ) defined as in (1.5). Therefore, it holds

Theorem 1.2. Let ϕ ∈ W0(L1) and a ∈ W (L1), then Φj ∈ C(T) for j =

1, . . . ,m. Moreover, let

(1.5) Am(ξ) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Φ0(ξm ) Φ0(

ξ+1m ) ... Φ0(

ξ+m−1m )

Φ1(ξm ) Φ1(

ξ+1m ) ... Φ1(

ξ+m−1m )

. . . .

. . . .

. . . .

Φm−1(ξm ) Φm−1(

ξ+1m ) ... Φm−1(

ξ+m−1m )

⎞⎟⎟⎟⎟⎟⎟⎟⎠,

ξ ∈ T. Then a vector f ∈ V (ϕ) can be recovered in a stable way (i.e. the inverse isbounded) from the measurements yn, for n = 0, . . . ,m−1 if and only if detAm(ξ) =0 for every ξ ∈ [0, 1].

By Theorem 1.2, whenever Am(ξ) is singular, a stable recovery is impossible.Overcoming this problem for finitely many singularities is possible by taking extrasamples. Let Tc be a operator that shifts a vector in �2(Z) to the right by c unitsso that Tcz(k) = z(k− c). Let SmnTc represent shifting by c and then sampling bymn for some positive integer n. It holds

1Subsection 2.1 contains a short review on SIS

DYNAMICAL SAMPLING IN HYBRID SHIFT INVARIANT SPACES 151

Theorem 1.3. Suppose Am(ξ) is singular only when ξ ∈ {ξi}i∈I with |I| <∞. Let n be a positive integer such that |ξi − ξj | = k

n for any i, j ∈ I and k ∈{1, . . . , n−1}. Then the additional sampling given by {SmnTc}c∈{1,...,m−1} provides

enough additional information to stably recover any f ∈ V (ϕ).

1.3. Dynamical sampling in hybrid shift invariant spaces. For mod-eling smooth spatio - temporal fields in wireless sensor networks, locality such assmoothness is an important factor [11]. On behalf of locality, we introduce a hy-brid shift invariant space (HSIS), which is in fact a generalization of a conventionalSIS and can adapt to the local smoothness properties of the field i.e. allows theamount of smoothness of the field to be adapted more locally. We are motivatedby the possibility of a generalization of the developed dynamical sampling schemefor SIS [1] in this new setting. Specifically, we define a hybrid shift invariant spaceV (ϕ−, ψ+) as a patchwork (see (2.6)) of two semi-shift invariant spaces, under somemild conditions on the building blocks ϕ, ψ. This construction can be generalizedto a patchwork of finitely many subspaces, which is a part of our future work (seesubsection 5.3).

We assume that any signal of interest h ∈ V (ϕ−, ψ+), when sampled on asampling set S, can be fully recovered; if we undersample h on a sampling set X ⊂S, the data is insufficient for reconstruction. We assume h is evolving in time underthe influence of an evolution operator family, denoted by An(h), n = 1, 2, ..., N − 1.We undersample h and its evolved states A1h, A2h, ..., AN−1h on a sampling setX ⊂ S and label the sampling operator as SX . In applications this means we onlyneed to use |X| sensor nodes repeatedly working N times, and do not require afull sensor network with |S| sensor nodes, which is useful as it saves on equipment.The dynamical sampling problem in a hybrid shift invariant space under theseassumptions is:

What are the conditions imposed on A1, ..., AN−1, X and N , so that a functionh ∈ V (ϕ−, ψ+) can be recovered from its repeated subsamples

(1.6) {SXh, SXA1h, SXA2h, ..., SXAN−1h}?

We propose two notions here that we find are of importance in our setting, butalso in the cases discussed in [1,4,5]:

(1) Invertibility sampling condition. Operators A1, ..., AN−1, the sam-pling set X and the number of repeated samplings N satisfy this conditionin a fixed HSIS, if any function h in that HSIS has a related sample dataset (1.6) that uniquely determines h. In other words, this condition en-sures the uniqueness of recovery.

(2) Stability sampling condition. Operators A1, ..., AN−1, the samplingset X and the number of repeated samplings N satisfy this condition ina fixed HSIS, if for any two functions h, h1 in that HSIS it holds

‖h− h1‖22 ∼N−1∑i=0

‖SXAi(h− h1)‖2�2 .

Within this paper we consider our sampling set X to be mZ, a sampling grid withfixed width, as our basic tool (3.6) is not adapted to a non-uniform grid.


1.4. Organization and contribution. The organization of this paper is asfollows: First, we define a hybrid shift invariant space and study its properties inSection 2. In Section 3 we formulate the dynamical sampling problem in HSISand state several useful results related to dynamical sampling in HSIS. We givethe invertibility sampling condition and stability sampling condition for the specialcase when X = mZ and {A0, A1, ..., AN−1} is a family of convolution operators inSection 4. We further explore some singularity problems related to stability andprove that it is possible to add extra samples to recover the signal in a stable way.Last, in Section 5 we discuss several related problems, such as the sensor motionproblem and the sampling rate varying problem. We have placed several propertiesand examples of interest in the Appendix.

2. Constructing a hybrid shift invariant space

We first review the basic properties of shift invariant spaces and then introducethe notion of hybrid shift invariant spaces.

2.1. Shift invariant spaces. A subspace V of L2(R) is a shift invariant space(SIS) if it is invariant under integer translations, i.e. f ∈ V iff f(· − k) ∈ V for allk ∈ Z. A typical shift invariant space considered in sampling theory is generatedby a single building block ϕ ∈ L2(R). Specifically, V is of type

(2.1) V (ϕ) = {∑k∈Z

ckϕ(· − k)|(ck)k∈Z ∈ �2(Z)},

with series convergence of its elements ensured by the conditions in Theorem 2.1(see below). Recall that {fk}∞k=1 is a Riesz basis for a separable Hilbert space Hif there exist constants A,B >0 such that

(2.2) A ‖f‖2 ≤∞∑k=1

|〈f, fk〉|2 ≤ B ‖f‖2 for all f ∈ H.

Theorem 2.1. If there exist some positive constants M,m > 0 such that

m ≤∑k∈Z

|ϕ(ξ + k)|2 ≤ M a.e. for ξ ∈ R,

then V (ϕ) as defined in (2.1) is a well defined closed linear subspace of L2(R) withRiesz basis {ϕ(· − k)}k∈Z.

If ϕ satisfies the assumptions of Theorem 2.1, then V (ϕ) is equivalently defined

as V (ϕ) = span{ϕ(· − k)} with coefficients in �2(Z).The local behavior and global decay of ϕ can be described in terms of Wiener

amalgam spaces [3,7]. A measurable function f belongs to the Wiener amalgamspace W (Lp), 1 ≤ p < ∞, if it satisfies

(2.3) ‖f‖pW (Lp) :=∑k∈Z

ess sup{|f(x+ k)|p;x ∈ [0, 1]} < ∞.

Because ideal sampling makes sense only for continuous functions, we work in theamalgam spaces

(2.4) W0(Lp) := W (Lp(R)) ∩ C(R).


If ϕ ∈ W0(L1) and satisfies the conditions of Theorem 2.1, then V (ϕ) is a subspace

of W0(L2). Under these conditions on ϕ, any function f ∈ V (ϕ) is continuous and

can be sampled at any x ∈ R. Moreover, there exists C > 0 such that∑k∈Z

|f(k)|2 ≤ C‖f‖22 ∀ f ∈ V (ϕ).

Lemma 2.2. If ϕ ∈ W0(L1) satisfies the conditions of Theorem 2.1 and for

almost all ξ ∈ R it holds ∑j∈Z

ϕ(ξ + j) = 0,

then there exists an interpolating atom ϕI ∈ V (ϕ) that vanishes on intergers exceptϕI(0) = 1 such that {ϕI(· − k)}k∈Z is Riesz basis of V (ϕ), i.e.V (ϕ) = V (ϕI).

Any function f ∈ V (ϕI) can be uniquely written as f =∑

k∈Z ckϕI(·−k) withck = f(k), k ∈ Z. In other words, f can be recovered from its samples on Z; i.e.,from f(Z).

For more on sampling and reconstruction in SIS we refer the reader to [2,3,6,8,13–15,17].

2.2. Hybrid shift invariant spaces. Let ϕ, ψ ∈ W0(L1) generate two SISs

V (ϕ) and V (ψ). It is trivial to show that the sets {ϕ(·−k)|k < 0} and {ψ(·−l)|l ≥ 0}are then Riesz bases for the restricted, semi shift invariant spaces

V (ϕ−) := {∑k<0

ckϕ(· − k)|(ck)k<0 ∈ �2(Z−)}

and

V (ψ+) := {∑l≥0

dlψ(· − l)|(dl)l≥0 ∈ �2(Z+0 )}.

Given elements f =∑

k<0 ckϕ(· − k) ∈ V (ϕ−) and g =∑

l≥0 dlψ(· − l) ∈ V (ψ+),we consider

(2.5) h = f + g =∑k<0

ckϕ(· − k) +∑l≥0

dlψ(· − l)

to be an element of a hybrid shift invariant space

(2.6) V (ϕ−, ψ+) := V (ϕ−) + V (ψ+),

where the joint coefficients sequence (ck)k<0∪ (dl)l≥0 is naturally in �2(Z). We callthe shifts of ψ right-oriented ; respectively, the shifts of ϕ are left-oriented.

Under some mild conditions (see Proposition 2.3), the union of the Riesz basesfor V (ϕ−) and V (ψ+) is a Riesz basis for V (ϕ−, ψ+). In fact, a sum of two Rieszbases-generated spaces V1 and V2 will often be a Riesz basis-generated space V1+V2.

Proposition 2.3. Given two Riesz bases R1 = {ϕk : k < 0} and R2 = {ψk : k ≥ 0},where ϕk and ψk are not necessarily generated as shifts of single functions, let V1

and V2 be the closed subspaces of some Hilbert space H, spanned by R1 and R2

respectively. Then{ϕk}k<0 ∪ {ψk}k≥0 is a Riesz basis for V1 + V2 if and only if there exists some

M ∈ (0, 1) such that for all f ∈ V1 and g ∈ V2, it holds:

(2.7) |〈f, g〉| ≤ M ‖f‖ ‖g‖ .


Remark 2.4. For Propostion 2.3 to hold true, it is necessary to have V1∩V2 ={0}. In finite dimensions, V1 ∩ V2 = {0} is equivalent to (2.7). The equivalencyno longer holds true in infinite dimension spaces. So (2.7) is an essential conditionthat cannot be weakened. In Example 6.1 (see Appendix), we show that there existtwo infinite-dimensional spaces V1 and V2 with V1 ∩ V2 = {0}, but (2.7) fails.

Remark 2.5. In special cases, for instance when the building blocks are com-pactly supported, the uniform M -inequality always holds true (see Example 6.2 inthe Appendix). In this case, it is possible to verify (2.7) via some prior informationon the building blocks.

We now give the proof of Proposition 2.3:

Proof. Let f =∑

k<0 ckϕk ∈ V1, g =∑

l≥0 dlψl ∈ V2. As {ϕk}k<0 and

{ψk}l≥0 are Riesz bases for V1 and V2, we know,

(2.8)√

‖f‖22 + ‖g‖22 ∼√∑

k<0

|ck|2 +∑l≥0

|dl|2.

Notice that, {ϕk}k<0 ∪ {ψl}l≥0 is a Riesz basis for V1 + V2 if and only if for f andg defined as above, it holds

(2.9) ‖f + g‖2 ∼√∑

k<0

|ck|2 +∑l≥0

|dl|2.

From the above, we know that (2.8) holds if and only if

(2.10) ‖f + g‖2 ∼√

‖f‖22 + ‖g‖22.

Now the problem reduced to show there exists some 0 < M < 1 such that (2.7)holds iff (2.10) holds for all f ∈ V1, g ∈ V2.

“ ⇐ ”Suppose we have M ∈ (0, 1) is such that (2.7) holds true. Then

‖f + g‖2 = ‖f‖2 + ‖g‖2 + 2Re 〈f, g〉 ≥ ‖f‖2 + ‖g‖2 − 2M‖f‖‖g‖≥ (1−M)(‖f‖2 + ‖g‖2) +M(‖f‖ − ‖g‖)2 ≥ (1−M)(‖f‖2 + ‖g‖2).

As ‖f + g‖2 ≤ 2(‖f‖2 + ‖g‖2), we conclude that (2.10) holds true.

“ ⇒ ” Since ‖f + g‖2 ∼√

‖f‖22 + ‖g‖22, there exists M > 0, such that

‖f + g‖22 ≥ M(‖f‖22 + ‖g‖22).This implies 2Re 〈f, g〉 ≥ (M−1)(‖f‖22+‖g‖22). The left-hand side of the inequalityis not always positive, thus M − 1 cannot be positive, since ‖f‖22 + ‖g‖22 ≥ 0. i.e.M ≤ 1.

Case 1: If M = 1, then Re 〈f, g〉 ≥ 0. Thus 〈f, g〉 = 0 for all f, g. Otherwise,one of Re 〈f, g〉 and Re 〈f,−g〉 could be negative. Therefore (2.7) follows directly.

Case 2: Let 0 < M < 1. Then |2Re 〈f, g〉 | ≤ (1 − M)(‖f‖22 + ‖g‖22). Let usverify this is true:

Assume |2Re 〈f, g〉 | > (1−M)(‖f‖22+‖g‖22) for some f and g. If Re 〈f, g〉 ≥ 0,then

2Re 〈f,−g〉 < −(1−M)(‖f‖22 + ‖g‖22) = (M − 1)(‖f‖22 + ‖ − g‖22),which is a contradiction.


If Re 〈f, g〉 < 0,then −2Re 〈f, g〉 > (1 − M)(‖f‖22 + ‖g‖22). This implies2Re 〈f, g〉 < (M − 1)(‖f‖22 + ‖g‖22), which is a contradiction.

For f and g, there exists some θ, such that | 〈f, g〉 | = Re⟨f, eiθg

⟩. Hence,

|2 〈f, g〉 | = |2Re⟨f, eiθg

⟩| ≤ (1−M)(‖f‖22 + ‖eiθg‖22 = (1−M)(‖f‖22 + ‖g‖22).

Thus | 〈f, g〉 | ≤ 1−M2 (‖f‖22 + ‖g‖22).

Hence, for all ‖f‖22 = 1, ‖g‖22 = 1, we have | 〈f, g〉 | ≤ 1 − M . For nonzero

f, g, f‖f‖ ,

g‖g‖ are of unit norm. So we derive | 〈f, g〉 | ≤ (1 − M)‖f‖‖g‖, where

0 < 1−M < 1. �

3. Dynamical sampling in hybrid shift invariant spaces

Let ϕ, ψ ∈ W0(L1) generate two SIS V (ϕ) and V (ψ) such that the union of

the respective Riesz bases for the related V (ϕ−) and V (ψ+) is a Riesz basis for aHSIS V (ϕ−, ψ+).

3.1. Reasons to work with interpolating atoms in the HSIS case. Mostof the results presented in this paper hold true for any reasonable choice of atomsϕ and ψ, if we assume that sampling on Z gives us enough data for reconstruction.However, sampling a function h on Z, if h is a linear combination of shifts ofinterpolating atoms, surely gives us sufficient information for full function recovery- namely, the samples are the function expansion coefficients (similar to the SISresult in Lemma 2.2, see Property 6.3 in the Appendix). Taking samples on Zmay not be enough for function recovery in a HSIS when working with atoms thatare not interpolating (see Example 6.4 in the Appendix). Therefore, working withatoms that are not interpolating will require the assumption that sampling thehybrid functions on Z would be sufficient for recovery. In addition, working withinterpolating atoms simplifies some of the proofs in Section 5.

This is why we work with interpolating atoms ϕ and ψ i.e. they satisfy ϕ(0) =1 = ψ(0) and ϕ(k) = 0 = ψ(k) for all k /∈ Z \ {0}. However, we alert the readerthat our results hold true in any HSIS, under the assumption that Z is a sufficientsampling set for full function reconstruction in that HSIS. For this reason, weformally keep the functions Φ0 and Ψ0 introduced in (3.4) in all the listed results,even though for the special case when using interpolating atoms it holds Φ0 = Ψ0 =1.

3.2. Stating the dynamical sampling problem. In a smooth spatio - tem-poral field, the initial state of a signal h ∈ V (ϕ−, ψ+) is evolving under the influenceof an evolution operators family. We work with an evolution operators family, givenby powers of convolution2, denoted by An(f) = a∗a∗a....∗a∗f , in short an ∗f . Weundersample N evolved states of h at a uniform grid X = mZ for some m > 1 andlabel the sampling operator as Sm(h) = h|mZ. The dynamical sampling problem ina hybrid shift invariant space V (ϕ−, ψ+) under these assumptions can be stated asfollows:

What conditions imposed on a, m and N satisfy the invertibility samplingcondition and the stability sampling condition for a m,N - dynamical sam-pling scheme in HSIS?

2Note that, given f ∈ V (ϕ−, ψ+), even under severe constraints on the convolutor a, theevolved element a∗f may not be an element of V (ϕ−, ψ+) anymore. However, this does not affectthe results presented here.


3.3. Useful results. The Fourier transform of a function h of form (2.5) is

(3.1) h(ξ) =∑k<0

cke−2πikξϕ(ξ) +

∑l≤0

dle−2πilξψ(ξ).

Wework with expanded coefficient sequences c := (ck)k∈Z and d := (dl)l∈Z, meaningthat ck = 0 for k ≥ 0 and dl = 0 for l < 0. Then c(ξ) =

∑k<0 cke

−2πikξ,

d(ξ) =∑

l≥0 dle−2πilξ and we have

(3.2) h(ξ) = c(ξ)ϕ(ξ) + d(ξ)ψ(ξ).

Given an evolution operator A described by Ah := a ∗h (for a ∈ W 3) and labelingaj := a ∗ a ∗ a · · · ∗ a︸︷︷︸

j− terms

, we have

Lemma 3.1. Let yj = Ajh|mZ = aj ∗ h|mZ. Then,

(3.3) yj(ξ) = F(aj ∗ h|mZ)(ξ) =1

m

m−1∑l=0

F(aj ∗ h|Z)(ξ + l

m).

For j = 0, 1, ..., N − 1, we define

(3.4) ϕj = aj ∗ ϕ, Φj = F(ϕj |Z), ψj = aj ∗ ψ, and Ψj = F(ϕj |Z).

Lemma 3.2. Let V (ϕ−, ψ+) be a HSIS and h ∈ V (ϕ−, ψ+) be of form (2.5).The functions

(3.5) Φj(ξ) =∑k∈Z

aj(ξ + k)ϕ(ξ + k) and Ψj(ξ) =∑k∈Z

aj(ξ + k)ψ(ξ + k)

for integers j = 0, 1, ..., N − 1 are 1-periodic and it holds

(3.6) yj(ξ) =1

m

m−1∑l=0

(c(ξ + l

m)Φj(

ξ + l

m) + d(

ξ + l

m)Ψj(

ξ + l

m)

).

Proof. By Poisson’s Summation Formula, it holds

aj ∗ h|Z(ξ) =∑k∈Z

aj ∗ h(ξ + k) =∑k∈Z

aj(ξ + k)h(ξ + k)

= c(ξ)∑k∈Z

aj(ξ + k)ϕ(ξ + k) + d(ξ)∑k∈Z

aj(ξ + k)ψ(ξ + k)

= c(ξ)Φj(ξ) + d(ξ)Ψj(ξ).(3.7)

We compare formula (3.7) to Lemma 3.1 and conclude that (3.6) holds true. �

4. Main results

After taking N -times repeated subsamples

(4.1) yj = Sm

(aj ∗ h

), j = 0, 1, ..., N − 1,

we apply the Fourier transform to the obtained samples and organize them in a col-

umn vector yN (ξ) := (y0(ξ) y1(ξ) ... yN−1(ξ))T . We introduce the vector columns

(4.2) cm(ξ) =

(c(

ξ

m) c(

ξ + 1

m) ... c(

ξ +m− 1

m)

)T

and

3see (2.4) for the definition of W


(4.3) dm(ξ) =

(d(

ξ

m) d(

ξ + 1

m) ... d(

ξ +m− 1

m)

)T

.

For fixed values of m, let

A(ξ) = Am,N (ξ) :=1

m

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Φ0(ξm ) Φ0(

ξ+1m ) ... Φ0(

ξ+m−1m )

Φ1(ξm ) Φ1(

ξ+1m ) ... Φ1(

ξ+m−1m )

. . . .

. . . .

. . . .

ΦN−1(ξm) ΦN−1(

ξ+1m ) ... ΦN−1(

ξ+m−1m )

⎞⎟⎟⎟⎟⎟⎟⎟⎠and

B(ξ) = Bm,N (ξ) :=1

m

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Ψ0(ξm ) Ψ0(

ξ+1m ) ... Ψ0(

ξ+m−1m )

Ψ1(ξm ) Ψ1(

ξ+1m ) ... Ψ1(

ξ+m−1m )

. . . .

. . . .

. . . .

ΨN−1(ξm ) ΨN−1(

ξ+1m ) ... ΨN−1(

ξ+m−1m )

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

Note that, if a ∈ W0(L1), then all the entries in Am,N (ξ),Bm,N (ξ) are continuous.

Proposition 4.1 (Matrix Representation form). The dynamical samples (4.1)can be represented in matrix form (in short notation) on the Fourier side as

(4.4) yN (ξ) = A(ξ)cm(ξ) + B(ξ)dm(ξ) = [A(ξ)B(ξ)] [cm(ξ)dm(ξ)]T .

Proposition 4.1 provides a matrix representation of our sampling process. Be-fore we discuss the minimum sampling requirement on the time side (that is, theminimal value of N) to make sure this sampling process is injective, we first discussthe conditions under which any h ∈ V (ϕ−, ψ+) (that is, its coefficients sequence(ck)k<0 ∪ (dl)l≥0) can be recovered in N -times repeated sampling steps.

Note that G : L2(T) → (L2(T))m, defined by

(Gz)(ξ) =1√m

(z(

ξ

m), z(

ξ + 1

m), · · · , z(ξ +m− 1

m)

),

is an isometric isomorphism from L2(T ) to (L2(T ))m. We define two spaces H l andHr, called left and right Hardy spaces by

H l = {∑k<0

cke−2πkξ|(ck)k<0 ∈ l2(Z−)}, Hr = {

∑l≥0

dle−2πlξ|(dl)l≥0 ∈ l2(Z+

0 )}.

It is easy to see that L2(T ) = H l ⊕ Hr. Let H l, Hr denote the image of H l, Hr

under the isometric isomorphism G.Clearly, yN is in (L2(T ))N and we can observe that, for fixed values of m

and N , the matrices Am,N (ξ) and Bm,N (ξ) define linear operators A and B from(L2(T ))m to (L2(T ))N . Namely, since G is an isometric isomorphism, every ele-

ment z ∈ (L2(T ))m is of form 1√m

(z( ξ

m ), z( ξ+1m ), · · · , z( ξ+m−1

m )). Then (Az)(ξ) =

Am,N (ξ)z(ξ) and (Bz)(ξ) = Bm,N (ξ)z(ξ). Operators A ◦G and B ◦G jointly mapL2(T ) into (L2(T ))N and it is the qualities of operators A and B that determine ifsignal recovery is possible or not.


Theorem 4.2 (Invertibility sampling condition). Recovery of the sampled sig-nal h ∈ V (ϕ−, ψ+) from its samples (4.1) is feasible if and only if the restricted

operators A|Hl and B|Hr are injective and AH l ∩ BHr = {0}.

Proof. If we assume that c(ξ) and d(ξ) (related to the sampled signal h =c ∗ϕ+d∗ψ) form a solution to (4.4), then we can recover the coefficients (ck)k<0∪(dl)l≥0 if and only if that solution is unique. This means that our sampling operatormust be injective, that is, different signals have different samples.

“ ⇐ ” Suppose h1 has the same dynamical samples as h. Let c1(ξ) and d1(ξ)be the vector columns as in (4.2) and (4.3) respectively, related to h1. Then c1(ξ)and d1(ξ) form a solution to the equation in Proposition 4.1 too, so we have

(4.5) A(c(ξ)− c1(ξ)) = B(d1(ξ)− d(ξ)).

It follows that c− c1 ∈ H l and d1 − d ∈ Hr. Since AH l ∩ BHr = {0}, we have

(4.6) A(c(ξ)− c1(ξ)) = B(d1(ξ)− d(ξ)) = 0.

By the injectivity of A|Hl and B|Hr we conclude that c = c1 and d1 = d, henceh1 = h.

“ ⇒ ” Let us assume that our sampling process is injective, which meansdifferent signals have different dynamical samples. Since V (ϕ−) ⊂ V (ϕ−, ψ+), if

h ∈ V (ϕ−), then d(ξ) = 0 and respectively d(ξ)) = 0, so (4.4) becomes

(4.7) yN (ξ) = Ac(ξ).

To preserve the injectivity, A|Hl must be injective. In a similar manner, we concludethat B|Hr must be injective too.

If we assume that AH l∩BHr = {0}, then there exist nonzero c1(ξ) and nonzerod1(ξ) satisfying

(4.8) Ac1(ξ) = −B(ξ)d1(ξ).

These c1 and d1 determine a unique function h1 ∈ V (ϕ−, ψ+). But the zerosequence also provides a solution to (4.8), which recovers the zero function h ≡ 0.By assumption h1 = h, which is a contradiction to having injective A|Hl and B|Hr .

Therefore, it must hold AH l ∩ BHr = {0}. �

Corollary 4.3. Undersampling at X = mZ requires at least N = m timesrepeated subsamples i.e. sampling m times is the minimum requirement to recoverthe signal.

When N = m, we label the involved matrices as Am,m = Am and Bm,m = Bm

in synchrony with the labels used in the SIS case. Observe that if our atoms coincide(ϕ ≡ ψ), then V (ϕ−, ψ+) is reduced to a SIS V (ψ). In this case, Am(ξ) ≡ Bm(ξ).

Corollary 4.4. If ϕ = ψ, then V (ϕ−, ψ+) ≡ V (ϕ). We can recover allh ∈ V (ϕ) via a N = m dynamical sampling scheme if and only if the related matrixAm(ξ) is invertible for all ξ ∈ [0, 1].

Theorem 4.5 (Stability sampling condition). The dynamical sampling processin HSIS is stable if and only if both A|Hl ,B|Hr are injective mappings with closed

range, and for all f ∈ AH l, g ∈ BHr, there exists some M ∈ (0, 1) such that

(4.9) |〈f, g〉| ≤ M ‖f‖ ‖g‖ .


Proof. We use the labels A = Am,N and B = Bm,N . Similar to the reasoningin the proof of Proposition 2.3, if h is the sampled signal, stability means that‖h‖ ∼ ‖yN‖. That is, there exist constants c1, c2 > 0 such that c1‖h‖ ≤ ‖yN‖ ≤c2‖h‖. We have

(4.10) yN (ξ) = Ac(ξ) + Bd(ξ),

which means ‖y‖2 ∼ ‖Ac+ Bd‖2.Let f(ξ) = Ac(ξ), g(ξ) = Bd(ξ). By our assumption, ‖f‖ ∼ ‖c(ξ)‖ ∼ ‖c‖,

‖g‖ ∼ ‖d(ξ)‖ ∼ ‖d‖. Then ‖c(ξ)+d(ξ)‖2 ∼ ‖c+ d‖2 = ‖c‖2+‖d‖2 ∼ ‖h‖2. Hence,‖h‖2 ∼ ‖f‖2 + ‖g‖2.

Therefore, the stability of the recovered signal is equivalent to

(4.11) ‖f + g‖ ∼√

‖f‖2 + ‖g‖2.

�

In case Theorem 4.5 does not hold true, there will be multiple solutions to thedynamical sampling problem. One solution candidate is the min norm solution (seeLemma 6.6 in the Appendix).

It is interesting that, if we sample N = 2m times, then we will produce a2m×2m square matrix in the adapted matrix equation from Proposition 4.1. Thatequation now becomes

(4.12) y2m(ξ) =

[Am(ξ) Bm(ξ)

Am(ξ) Bm(ξ)

]·[

cm(ξ)dm(ξ)

],

where

Am(ξ) :=1

m

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Φm( ξm ) Φm( ξ+1

m ) ... Φm( ξ+m−1m )

Φm+1(ξm ) Φm+1(

ξ+1m ) ... Φm+1(

ξ+m−1m )

. . . .

. . . .

. . . .

Φ2m−1(ξm ) Φ2m−1(

ξ+1m ) ... Φ2m−1(

ξ+m−1m )

⎞⎟⎟⎟⎟⎟⎟⎟⎠and

Bm(ξ) :=1

m

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Ψm( ξm ) Ψm( ξ+1

m ) ... Ψm( ξ+m−1m )

Ψm+1(ξm ) Ψm+1(

ξ+1m ) ... Ψm+1(

ξ+m−1m )

. . . .

. . . .

. . . .

Ψ2m−1(ξm ) Ψ2m−1(

ξ+1m ) ... Ψ2m−1(

ξ+m−1m )

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

Corollary 4.6. If the square matrix

(4.13) M2m(ξ) =

[Am(ξ) Bm(ξ)

Am(ξ) Bm(ξ)

]is invertible for all ξ ∈ [0, 1], then we can stably recover the sampled signal viaequation (4.12).


4.1. Adding extra samples. In SIS, when a,Φ are symmetric, the problemsrelated to singularities are resolved via extra samples (Theorem 1.3). Singularitiesmay occur in the HSIS case as well, say when the left inverse of the matrix involvedin (4.4), Mm,N = [Am,N (ξ)Bm,N (ξ)], does not exist. We explore the option ofadding extra samples in an attempt to avoid the singularities.

As a direct result of Theorem 1.3, it is possible to fix finitely many singularitiesof Am(ξ) and Bm(ξ) and achieve stability on whole shift invariant space V (ϕ), V (ψ)respectively. It follows naturally that this is feasible with the semi-shift invariantspaces V (ϕ−), V (ψ+).

We explore the case when M2m(ξ) has finitely many singularities. It is knownthat if M2m(ξ) is invertible for ξ ∈ [0, 1], then we can recover the sampled signalin a stable way. Now assume M2m(ξ) has finitely many sigularities {ξi}i∈I , |I| <+∞. We prove that we can achieve stable recovery in a HSIS with interpolatingatoms through adding extra samples. In synchrony with Theorem 1.3, we have thefollowing result

Theorem 4.7. Let the samples Sm(aj∗h), j = 0, 1, ...,m−1 for h ∈ V (ϕ−, ψ+)be given, where ϕ = ψ are interpolating atoms. Let {ξi}i∈I be the finite set ofsingularities of M2m(ξ) and allow a choice for n such that |ξi − ξj | = k

n , k ∈{1, 2, ..., n−1} for any i = j ∈ I.Then along with the additional samples SmnTch forshifts c = 1, 2, ...,m, there is enough information to stably recover h ∈ V (ϕ−, ψ+)via a 2m-dynamical sampling scheme.

Proof. For interpolating atoms, it holds Φ0 = Ψ0 = 1, so the additionalsamples produce the following equation

F(SmnTch)(nξ) =1

mn

mn−1∑s=0

e−2πic

m (ξ+s/n)

(c(ξ + s/n

m) + d(

ξ + s/n

m)

).

We organize all the samples in a vector

(4.14) Ym(ξ) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

mne2πi1ξ/mF(SmnT1h)(nξ)...

mne2πimξ/mF(SmnTmh)(nξ)y2m(ξ)

y2m(ξ + 1n )

...y2m(ξ + n−1

n )

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.

In addition, we define vector functions

Uc(l) := e−2πicl/(mn)[1 e−2πic/m ... e−2πic(m−1)/m

], l = 0, 1, ..., n−1, c = 1, 2, ...,m

and introduce Uc(l) = [Uc(l) Uc(l)]. We employ the following matrices

(4.15) M2m(ξ) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

U1(0) . . . . . . U1(n− 1)Um(0) . . . . . . Um(n− 1)M2m(ξ) 0 . . . 0

0 M2m(ξ + 1n ) . . . 0

......

......

......

......

0 0 . . . M2m(ξ + n−1n )

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


and

X(ξ) =

⎛⎜⎜⎜⎜⎝cm(ξ)dm(ξ). . .

cm(ξ + n−1n )

dm(ξ + n−1n )

⎞⎟⎟⎟⎟⎠in a matrix equation

(4.16) Ym(ξ) = M2m(ξ)X(ξ).

For all ξ ∈ [0, 1n ), the giant matrix M2m(ξ) has a bounded inverse. The key

ingredient here is that the matrix

U(l) =

⎛⎝ U1(l). . .

Um(l)

⎞⎠ ,

involved in M2m(ξ), is invertible for l ∈ {0, 1, 2, ..., n−1} (as its rows are rotated andre-ordered roots of unity,which causes orthogonality among the rows). In addition,for any fixed l, at most one of the block matrices M2m(ξ + l

n ) is singular (because

for a fixed value of ξ, at most one ξ + ln ∈ {ξi}i∈I). For more details, we refer to

the proof of the analogous result for �2(Z) [5]. �

5. Conclusions and future work

We have defined a hybrid shift invariant space and discussed the dynamicalsampling problem in that setting. We establish the invertibility and stability ofthe sampling scheme. Then we studied the possibility of adding extra samples toovercome singularities. In addition, we present some initial encouraging resultstowards a locally adapted sampling rate in hybrid spaces. But first, we reflect onthe option of sensor motion when using the techniques of dynamical sampling. Wehave solid reasons to believe that sensor motion is not a strategy that will improvethe reconstruction process in terms of stability or the size of the necessary data forfull function recovery (see subsection 5.2).

Given that a hybrid SIS means there is a locally varying quality of the functionswe sample (h = f + g ∈ V (ϕ−, ψ+) with f ∈ V (ϕ−) and g ∈ V (ψ+)), the naturalquestion to pose here is: can we recover the function h by a dynamical samplingscheme that has different (locally adapted) rates of sampling in V (ϕ−) and V (ψ+)respectively? Varying the sampling rate in the dynamical sampling scheme mayeventually provide more locality but we have not proved a general result yet. How-ever, we have an encouraging example toward a generalized sampling rate variancepresented in subsection 5.1.

5.1. Varying the sampling rate - first results. Let h = f+g ∈ V (ϕ−, ψ+).For interpolating atoms ϕ and ψ, the samples of h on Z− give us (f(k))k<0, due toProperty 6.3 (see bellow) and we can fully compute

(5.1) f =∑k<0

f(k)ϕ(· − k) and aj ∗ f for j = 1, ...,m− 1.


If we sample h on mZ+0 m-times at time instances t = 0, 1, ...,m−1, the samples

at t = 0 are (g(ml))ml∈mZ+0, while the samples at t = j are

(5.2)(aj ∗ h(ml)

)ml∈mZ

+0

for j = 1, ...,m− 1.

We can use the data from (5.1) to clean up the data in (5.2) and obtain sufficientsubsamples for g, a ∗ g upto am−1 ∗ g, namely

yj(mZ+0 ) = aj ∗ h(mZ+

0 )− aj ∗ f(mZ+0 ), j = 1, ...,m− 1.

Then we can reconstruct g via the standard matrix equation y(ξ) = Bm(ξ)d(ξ),provided Bm(·) is invertible and recover h as f + g. This means that, if we allowsensor motion, we must be willing to take a lot more samples than in the case of aimmobile sensor network.

5.2. Sensor motion. The dynamical sampling problem studied so far in-volved repeated subsampling via a network of sensors with fixed locations (samplingon mZ). Here we briefly discuss the difficulty we would encounter if we allow forchanges of the sensors locations in a SIS setting:

Let m = 2. We sample f =∑

k∈Z ckϕ(· − k) ∈ V 2(ϕ) on 2Z at time instancej = 0. If the atom ϕ is interpolating, then the samples are

(5.3) f(2k) = c2k, k ∈ Z.

If we take samples at 2Z at time instance j = 1 of a ∗ f , we have sufficient data torecover via the standard formula (ck)k∈Z = F−1 (A2 · y).

Let’s assume the sampling grid has changed from 2Z (at j = 0) to 2Z + 1 atj = 1. We can compute f2 =

∑k∈Z c2kϕ(· − 2k) from (5.3) and thus compute

a ∗ f2|2Z+1 to clear up what

a ∗ f1|2Z+1 = a ∗ f |2Z+1 − a ∗ f2|2Z+1

is for f1 =∑

k∈Z c2k+1ϕ(· − 2k − 1). Note that a ∗ ϕ is not an interpolating atomand we now need either repeated subsampling or some anti-convolution techniquesto recover f1 from the samples a ∗ f1|2Z+1 and then to produce f = f1 + f2. Thismeans two consecutive subsamples’ harvests are not sufficient as in the stationarysensors case.

5.3. When three or more atoms are involved to build a hybrid space.Let us consider a hybrid space V = V (γ, ϕ, ψ), composed of three different partsof SISs V (γ), V (ϕ) and V (ψ) with γ, ϕ, ψ ∈ W0 and influenced by the standardevolution operator, defined by a convolution a ∈ W . We consider a typical elementof V to be of form

h = b ∗ γ + c ∗ ϕ+ d ∗ ψ,where b = (bk)k∈Z ∈ �2(Z) has non-zero entries for at most k < K < 0, d =(dk)k∈Z ∈ �2(Z) has non-zero entries for at most k > L > 0 and c = (ck)k∈Z ∈ �2(Z)has non-zero entries for at most K ≤ k ≤ L. We let yj = Ajh|mZ = aj ∗ h|mZ andΓj := aj ∗ γ|Z for j = 0, 1, ..., N − 1. Formula (3.6) then takes form(5.4)

myj(ξ) =m−1∑l=0

(b(ξ + l

m)Γj(

ξ + l

m) + c(

ξ + l

m)Φj(

ξ + l

m) + d(

ξ + l

m)Ψj(

ξ + l

m)

)


and the related matrix equation (4.4) becomes

(5.5) yN (ξ) = [GN (ξ)AN(ξ)BN(ξ)] [bN (ξ)cN(ξ)dN(ξ)]T .

Whenever the left inverse of Mn(ξ) = [GN (ξ)AN(ξ)BN (ξ)] exists, we can recoverthe coefficients sequence b ∪ c ∪ d via (5.5). Clearly, the conditions imposed onMn(ξ) to ensure its left inverse existence will be more complex than in the case westudied in this paper (a HSIS generated with only two atoms).

6. Appendix

6.1. In Proposition 2.3, we have stated the uniform M -inequality as crucialfor a Riesz basis-generated patched space. But a uniform M -inequality does notalways hold true, even if the patched spaces have a zero intersection.

Example 6.1. Let B = {ϕn | n ≥ 0} be an orthonormal basis for some spaceV . Let the subspaces V1 and V2 be generated by the bases

B1 = {ϕ2k+1 | k ≥ 0} , B2 =

{2k + 1√

(2k + 1)2 + 1ϕ2k+1 +

1√(2k + 1)2 + 1

ϕ2k | k ≥ 0

}.

There is no uniform M ∈ (0, 1) such that for all f ∈ V1 = spanB1 and g ∈ V2 =spanB2 it holds | 〈f, g〉 | < M‖f‖‖g‖.

Let us clarify why in Example 6.1 the uniform M -inequality does not hold true:Given two elements f ∈ V1 and g ∈ V2, there exist coefficients ck, k ≥ 0 and

dk, k ≥ 0 such that

f =∑k

dkϕ2k+1, g =∑k

ck

(√(2k + 1)2 + 1ϕ2k+1 +

1√(2k + 1)2 + 1

ϕ2k

).

Then

(6.1) f + g =∑k

(dk +

2k + 1√(2k + 1)2 + 1

ck

)ϕ2k+1 +

ck√(2k + 1)2 + 1

ϕ2k.

By (6.1), f +g has a V -space representation and by the orthonormality of the basisB it holds

‖f + g‖22 =∑k≥0

(dk +

2k + 1√(2k + 1)2 + 1

ck

)2

+c2k

(2k + 1)2 + 1

=∑k≥0

(d2k + c2k + 2ckdk

2k + 1√(2k + 1)2 + 1

)(6.2)

If fk = ϕ2k+1, gk = 2k+1√(2k+1)2+1

ϕ2k+1 +1√

(2k+1)2+1ϕ2k, then ‖fk‖2 = ‖gk‖2 = 1,

but 〈fk, gk〉 = 2k+1√(2k+1)2+1

approaches 1 as k → ∞. Also, note that if wk = gk −fk,

then ‖wk‖2 → 0, but the �2-norm of the related coefficients of wk is√2.


6.2. The uniform M -inequality introduced in Proposition 2.3 easily holds truefor spaces V1, V2 generated by Riesz bases of compactly supported elements.

Example 6.2. If ϕ, ψ are compactly supported atoms with minimal overlapsuch that their oriented shifts sets B1 = {ϕ(· − k)|k < 0}, B2 = {ψ(· − l)|l ≥ 0}are Riesz bases for two spaces V1, V2 and only suppϕ(·+1)∩ suppψ = ∅, then theuniform M -inequality from Proposition 2.3 holds true.

Let’s assume that ϕ, ψ are compactly supported, normalized atoms and let|〈ϕ(·+ 1), ψ〉| = Mo > 0. Then for any f ∈ V1 = V (ϕ−), g ∈ V2 = V (ψ+

0 ), it holds

(6.3) |〈f, g〉| =

∣∣∣∣∣∣∑k<0

∑l≥0

ckdl 〈ϕ(· − k), ψ(· − l)〉

∣∣∣∣∣∣ = |c−1| |do| |〈ϕ(·+ 1), ψ〉| .

Thus, for M = Mo

A1A2and A1, A2 the lower Riesz constants in (2.2), under the

assumption Mo < A1A2, it holds

M < 1 and |〈f, g〉| ≤ ‖f‖‖g‖A1A2

Mo = M‖f‖‖g‖.

In (6.3) there will be only finitely many nonzero inner products, if we assumethat more than one pair of atom shifts overlap. Therefore it is possible to expandthe result from Example 6.2 on a HSIS with finitely many overlaps of its compactlysupported, shift-oriented atoms.

6.3. In this paper, we chose to work with interpolating atoms in the HSIS caseas sampling a function of type (2.5) on Z surely gives us sufficient information forfull function recovery (the samples are the function expansion coefficients).

Property 6.3. Let ϕ and ψ be two interpolating atoms. Then any h ∈V (ϕ−, ψ+) can be uniquely represented as

(6.4) h =∑k<0

h(k)ϕ(· − k) +∑l≥0

h(l)ψ(· − l).

Proof. For any h ∈ V (ϕ−, ψ+), given that {ϕ(x − k)}k<0 ∪ {ψ(x − l)}l≥0 isRiesz basis, there exists a unique sequence {ck}k<0 ∪ {dl}l≥0 such that

h =∑k<0

ckϕ(· − k) +∑l≤0

dlψ(· − l).

For a fixed j ∈ Z, we have

h(j) =∑k<0

ckϕ(j − k) +∑l≤0

dlψ(j − l) =∑k<0

ckδjk +∑l≤0

dlδlj ,

since ϕ and ψ are interpolating. That is to say if j ≥ 0, then h(j) = dj ; otherwiseh(j) = cj . Hence (6.4) holds true. �

However, a hybrid space generated by two atoms ϕ and ψ is not identical tothe hybrid space generated by the two respective interpolating atoms ϕI and ψI ,even though V (ϕ) = V (ϕI) and V (ψ) = V (ψI) hold true.

Example 6.4. Let ψI be the interpolating atom respective to atom ψ suchthat V (ψ) = V (ψI). But V (ψ+) = V (ψ+

I )!


To clarify, we consider g ∈ V (ψ+) ⊂ V (ψ) with representation

g =∑l≥0

dlψ(· − l)

such that g(−1) = 0. If we assume that g ∈ V (ψ) = V (ψI), then there existcoefficients bs, s ∈ Z such that g =

∑s∈Z bsψI(· − l). But then b−1 = g(−1) = 0,

which means that g /∈ V (ψ+I ).

6.4. The row rank of a matrix A is the maximum number of linearly indepen-dent row vectors of A. Equivalently, the row rank of A is the dimension of the rowspace of A.

Proposition 6.5. Let A be a full row-rank N ×M -matrix such that N < M .Then AAT is an invertible matrix and the matrix equation Ax = y has infinitelymany solutions. In particular, the minimal norm solution is AT (AAT )−1y.

Due to Proposition 6.5, one solution to the matrix equation (4.4) (that involvesM(ξ) = Mm,N (ξ) = [Am,N (ξ)Bm,N (ξ)]) in Proposition 4.1 is

(6.5) MT (ξ)(M(ξ)MT (ξ))−1yN (ξ).

Lemma 6.6. The min norm solution to the dynamical sampling problem inHSIS is h = c ∗ϕ+ d ∗ψ, where the coefficients sequences c∪ d = (ck)k<0 ∪ (dl)l≥0

are extracted from (6.5).

It is still unclear if the min norm solution is (ever) the desired solution to thedynamical sampling problem. Notice that it must hold

(6.6) cm(ξ) = ATm,N ((Am,NBm,N )(Am,NBm,N )T )−1yN (ξ),

(6.7) dm(ξ) = BTm,N ((Am,NBm,N )(Am,NBm,N )T )−1yN (ξ).

Equations (6.6) and (6.7) imply that proper injective behavior of the respectiveoperators is required, restricted on the half-Hardy spaces.

Acknowledgment

The authors would like to thank Akram Aldroubi for his inspiring insights andsuggestions in several stages of preparing this manuscript.

References

[1] R. Aceska, A. Aldroubi, J. Davis and A. Petrosyan. Dynamical sampling in shift-invariantspaces, AMS Contemporary Mathematics (CONM) book series, 2013

[2] Akram Aldroubi and Michael Unser, Sampling procedures in function spaces and asymptoticequivalence with Shannon’s sampling theory, Numer. Funct. Anal. Optim. 15 (1994), no. 1-2,1–21, DOI 10.1080/01630569408816545. MR1261594 (95a:94002)

[3] Akram Aldroubi and Karlheinz Grochenig, Nonuniform sampling and reconstructionin shift-invariant spaces, SIAM Rev. 43 (2001), no. 4, 585–620 (electronic), DOI10.1137/S0036144501386986. MR1882684 (2003e:94040)

[4] A. Aldroubi, J. Davis, and I. Krishtal. Dynamical sampling: time space trade-off, Appl. Com-put. Harmon. Anal., Vol.34(3), 2013, pp. 495-503

[5] A. Aldroubi, J. Davis, and I. Krishtal. Exact reconstruction of spatially undersampled signalsin evolutionary systems, preprint, arXiv:1312.3203.

[6] Nikolaos D. Atreas, Perturbed sampling formulas and local reconstruction in shift invariantspaces, J. Math. Anal. Appl. 377 (2011), no. 2, 841–852, DOI 10.1016/j.jmaa.2010.12.011.MR2769180 (2012e:42070)


[7] Hans G. Feichtinger, Wiener amalgams over Euclidean spaces and some of their applications,Function spaces (Edwardsville, IL, 1990), Lecture Notes in Pure and Appl. Math., vol. 136,Dekker, New York, 1992, pp. 123–137. MR1152343 (93c:46038b)

[8] Antonio G. Garcıa and Gerardo Perez-Villalon, Multivariate generalized sampling in shift-invariant spaces and its approximation properties, J. Math. Anal. Appl. 355 (2009), no. 1,397–413, DOI 10.1016/j.jmaa.2009.01.057. MR2514475 (2010d:94052)

[9] Y. Lu and M. Vetterli. Spatial super-resolution of a diffusion field by temporal oversampling

in sensor networks, in Acoustics, Speech and Signal Processing, 2009. ICASSP 2009. IEEEInt. Conf. on, April 2009, pp. 2249 –2252.

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[11] Gunter Reise, Gerald Matz, and Karlheinz Grochenig, Distributed field reconstruction inwireless sensor networks based on hybrid shift-invariant spaces, IEEE Trans. Signal Process.60 (2012), no. 10, 5426–5439, DOI 10.1109/TSP.2012.2205918. MR2979004

[12] G. Reise and G. Matz. Distributed sampling and reconstruction of non-bandlimited fieldsin sensor networks based on shift-invariant spaces,Proc. ICASSP, Taipeh, Taiwan, (2009),2061-2064.

[13] Hrvoje Sikic and Edward N. Wilson, Lattice invariant subspaces and sampling, Appl. Com-put. Harmon. Anal. 31 (2011), no. 1, 26–43, DOI 10.1016/j.acha.2010.09.006. MR2795873(2012i:42042)

[14] Wenchang Sun, Sampling theorems for multivariate shift invariant subspaces, Sampl. TheorySignal Image Process. 4 (2005), no. 1, 73–98. MR2115475 (2005k:94013)

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Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville,

Tennessee 37240


Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville,

Tennessee 37240



Dynamical sampling in infinite dimensions with and withouta forcing term

Jacqueline Davis

Abstract. Dynamical sampling is the study of sampling in dynamical sys-tems. In [1, 3, 4], it is shown that temporal oversampling of a signal in anevolutionary system can offset spatial undersampling, allowing for exact re-construction of the signal.

In this paper, the dynamical sampling problem in infinite dimensions isexplained and results from [4] are discussed. The results of [4] are extendedhere as a new stable sampling set is given that requires fewer spatial samplesthan required by earlier results. Additionally, the problem of dynamical sam-pling with a forcing term is introduced. In this problem, an unknown sourceterm enters the system during the time period when samples are taken.

1. Introduction

Traditional sampling theory asks the question: when can a signal be recon-structed from partial knowlege of the signal? Perhaps the most famous result insampling theory is the Shannon-Whitaker Sampling Theorem [9], which states thata bandlimited signal can be reconstructed exactly from discrete evenly spaced sam-ples of the signal. Specifically, if f(x) is T -bandlimited, i.e. the Fourier transformof f has support contained in [−T, T ], then

f(x) =∞∑

n=−∞f( n

2T

) sin π(2Tx− n)

π(2Tx− n).

Dynamical sampling is a new type of sampling problem that results from sam-pling an evolving signal at various times and asks the question: when do coarsesamplings taken at varying times contain the same information as a finer samplingtaken at the earliest time? In other words, under what conditions on an evolvingsystem, can time samples be traded for spatial samples? This question has beenstudied in [1, 3,4]. Because dynamical sampling uses samples from varying timelevels for a single reconstruction, it departs from classical sampling theory in whicha single signal is sampled and then reconstructed.

Some intuition about dynamical sampling can come from considering a diffu-sive process. The value at each location depends on earlier values at surroundinglocations. Thus, it is logical to think that information from different times could

2010 Mathematics Subject Classification. Primary 94A12, 94A20; Secondary 42A99.Key words and phrases. Sampling, reconstruction.This work was partially supported by the NSF grant DMS-1322099.

c©2014 American Mathematical Society167

168 JACQUELINE DAVIS

allow us to conclude something about the values at locations where measurementswere not taken. For instance, if the value at one location is increasing, we mightassume that there is a high concentration somewhere nearby that is spreading tothis location.

The dynamical sampling problem is formulated mathematically as follows. Letx ∈ �2(Ω) be an initial state of a signal in a dynamical system with evolution rulegiven by the operator A : �2(Ω) → �2(Ω) so that the signal at time t = n is givenby Anx. The sampling scheme is defined by the sets {Ωn}Nn=0, where each Ωn ⊂ Ω.The signal is measured at location vi ∈ Ω at time t = n if and only if vi ∈ Ωn. IfS(Ωn) represents the subsampling operator at time t = n, then yn = S(Ωn)A

nx isthe measured signal at time t = n. Under what conditions on A and {Ωn}n=N

n=0 canthe initial signal x can be recovered from the measurements yn for n = 0, . . . N?

Dynamical sampling is certainly not the first setting in which a signal x is tobe recovered from samples of related signals, rather than from samples of x itself.In wavelet theory, a high pass filter H and a low pass filter L are applied to thesignal x. The goal is to design filters H and L so that reconstruction of x fromsamples of Hx and Lx is feasible. In dynamical sampling there is only one filter A,and it is applied iteratively to the signal x. Furthermore, the filter A may be highpass, low pass, or neither and is given in the problem formulation, not designed.

Similarly, in inverse problems an operator A, which often represents a physicalprocess, is given. The goal is to recover x from knowledge of Ax. When theoperator A does not have a bounded inverse, the problem is considered an ill-posedinverse problem. In dynamical sampling, it is also possible that the operator Adoes not have a bounded inverse. However, partial knowledge of x is known, as thesubsampled signals are x,Ax,A2x, . . . , ANx. Because each of these signals is onlypartially known, dynamical sampling differs from the classical inverse problem.

A natural setting for dynamical sampling is wireless sensor networks (WSN). InWSN large amounts of physical sensors are distributed to gather information aboutthe field to be monitered, such as temperature, pressure, or pollution. WSN are usedin many industries, including the health, military, and environmental industries[2]. The authors of [5,6] seek to minimize the cost of reconstructing temperaturefields in wireless sensor networks by using fewer sensors activated more frequently.Other algorithms for reconstructing signals in WSN are proposed in [7, 8], butthese approaches take a more traditional sampling theory perspective in that theevolutionary nature of the system is mostly ignored.

1.1. Organization and contribution. In section 2, the dynamical samplingproblem in infinite dimensions is formulated and examined. We give an improve-ment of a result in [4] by showing that for special cases fewer spatial samples areneeded than required by earlier results. In section 3, dynamical sampling with aforcing term is introduced. In this problem, we remove an assumption made indynamical sampling by allowing for an unknown source term to enter the systemduring the sampling period. Under strong assumptions, results similar to those insection 2 are given.

2. Dynamical sampling in infinite dimensions

In this section, we formulate the dynamical sampling problem in infinite di-mensions when the subsampling is regular and the evolutionary rule is given byconvolution. The formulation is given in more detail in [4]. The last result in this

DYNAMICAL SAMPLING IN �2(Z) WITH AND WITHOUT A FORCING TERM 169

section is a new result, which allows for reconstruction in special cases with fewerspatial samples than required by the results in [4].

As in the previous papers, we assume A is a convolution operator so thatAx = a ∗ x and the signal at time t = j is given by

aj ∗ x = (a ∗ . . . ∗ a︸︷︷︸j

) ∗ x.

We also restrict ourselves to regular subsampling, so that S = Sm is the operationof subsampling by a factor of m, i.e., (Smz)(k) = z(mk). The sampled signal yjat time t = j is given by yj = Sm(aj ∗ x). The dynamical sampling procedure iswritten as

(2.1) y = Ax

where A is the dynamical sampling operator from (�2(Z)) to (�2(Z))N and

y = (y0, y1, . . . , yN−1) = (Smx, Sm(a ∗ x), . . . , Sm(aN−1 ∗ x)).

The goal of this paper is to study when A has a bounded left inverse, giving abounded reconstruction operator. The boundedness is necessary for stability whenadditive noise is present in the samples. The expected discrepancy, x− x, betweenthe recovered signal x and the original signal x is controlled by the norm of thereconstruction operator.

If A does not have a bounded left inverse, it may still be the case that A isinjective. In this case, we are often able to define a new sampling operator A thatdoes have a bounded left inverse by expanding the dynamical sampling operator Ato include an additional sampling set. This is illustrated in theorems 2.3 and 2.4.

The method of proof for the results below and those in [1,3,4] is as follows.Fourier techniques are used to reduce the study of the dynamical sampling operatorA to the study of an operator defined by pointwise matrix multiplication on thetorus. Then a left inverse is defined in the Fourier domain by pointwise multiplica-tion by the left inverse of such matrices, when such left inverses exist. For z ∈ �1(Z)the Fourier transform is defined on the torus T � [0, 1) by

z(ξ) =∑n∈Z

z(n)e−2πinξ, ξ ∈ T.

Here we state and prove a result from [4], which gives necessary and sufficientconditions for A to have a bounded left inverse.

Theorem 2.1. Assume that a ∈ L∞(T) and fix m ∈ Z+. Define

(2.2) A(ξ) =

⎛⎜⎜⎜⎝1 1 . . . 1

a( ξm ) a( ξ+1

m ) . . . a( ξ+m−1m )

......

......

a(m−1)( ξm ) a(m−1)( ξ+1

m ) . . . a(m−1)( ξ+m−1m )

⎞⎟⎟⎟⎠ ,

ξ ∈ T. Then A in (2.1) has a bounded left inverse for some N ≥ m if and onlyif there exists α > 0 such that the set {ξ : | detA(ξ)| < α} has zero measure.Consequently, A in (2.1) has a bounded left inverse for some N ≥ m if and only ifA has a bounded left inverse for all N ≥ m.


Proof. The proof uses the identity (a ∗ z)∧(ξ) = a(ξ)z(ξ), the Poisson sum-mation formula

(2.3) (Smz)∧(ξ) =1

m

m−1∑l=0

z(ξ + l

m)

and the following lemma, which is proved in [4].

Lemma 2.2. Suppose an operator B : (L2(T))m → (L2(T))n is defined by(Bx)(ξ) = B(ξ)x(ξ) where the map ξ → B(ξ) from T to the space of n×m matricesMnm is measurable. Then ‖B‖op = ess supT ‖B(ξ)‖op.

Using equation (2.3), the subsampled signal yj at time t = j can be written as

yj(ξ) = (Sm(aj ∗ x))∧(ξ) = 1

m

m−1∑l=0

aj(ξ + l

m

)x

(ξ + l

m

)(2.4)

Expressing this in matrix form, we have the equation

(2.5) m

⎛⎜⎜⎜⎝y0(ξ)y1(ξ)...

yN−1(ξ)

⎞⎟⎟⎟⎠ = AN (ξ)

⎛⎜⎜⎜⎝x( ξ

m )

x( ξ+1m )...

x( ξ+m−1m )

⎞⎟⎟⎟⎠ , where

(2.6) AN (ξ) =

⎛⎜⎜⎜⎝1 . . . 1

a( ξm ) . . . a( ξ+m−1

m )...

......

aN−1( ξm ) . . . aN−1( ξ+m−1

m )

⎞⎟⎟⎟⎠ .

Define H : L2(T) → (L2(T))m to be the isometry given by

(2.7) (Hz)(ξ) =1√m

(z

(ξ

m

), z

(ξ + 1

m

), . . . , z

(ξ +m− 1

m

))T

,

and define x =√mHx. Define FN : (�2(Z))N → (L2(T))N to be the one-

dimensional Fourier transform applied to each component of the product space(�2(Z))N , and define y = FNy. Then we can write (2.5) in more compact notationas

(2.8) y(ξ) =1

mAN (ξ)x(ξ),

Define the operator AN : (L2(T))m → (L2(T))N by (AN x)(ξ) = AN (ξ)x(ξ).Then the dynamical sampling operator can be written as a product of operators asfollows

(2.9) A =1√m

F−1N ANHF .

Since F ,FN and H are isometries, the operator A has a bounded left inverseif and only if the operator AN has a bounded left inverse. By lemma 2.2, it sufficesto study the left invertibility of AN (ξ) for each ξ ∈ T.

The submatrix A(ξ) of AN (ξ) is a Vandermonde matrix, which is invertiblewhen no two columns coincide. When two columns of A(ξ) coincide, the corre-sponding columns of AN (ξ) also coincide. Thus, AN (ξ) has a left inverse if andonly if | detA(ξ)| = 0. The conclusion follows from lemma 2.2. �


Figure 1. An example of a stable sampling scheme in Theorem2.4 with m = 5, n = 7, and c = 1. The sampling locations aremarked by crosses and the extra samples are marked as crossesinside squares.

If the hypotheses of theorem 2.1 are not satisfied, it may still be possible torecover the signal x by taking some additional samples. The additional samples aretaken as follows. Let Tc : �

2(Z) → �2(Z) be the operator that shifts a vector to theright by c units so that Tcz(k) = z(k − c). For n ∈ N, the operator SmnTc is theoperator of shifting by c and then subsampling by mn.

Here we give an example of an additional sample scheme that resolves thedynamical sampling problem when the matrix in (2.2) is singular for finitely manyξ ∈ T. This result is a special case of a result in [4], and the proof is omitted here.

Theorem 2.3. Let m ∈ Z+ be fixed. Suppose that a is continuous and thatA(ξ) is singular only when ξ ∈ {ξi}i∈I with |I| < ∞. Let n be a positive integersuch that |ξi − ξj | = k

n for any i, j ∈ I and k ∈ {1, . . . , n − 1}. Then the extrasamples given by {(SmnTc)x}c∈{1,...,m−1} provide enough additional information to

stably recover any x ∈ �2(Z), i.e. the reconstruction operator is bounded.

In many physical applications, the convolution operator is such that a is real,symmetric, continuous, and decreasing on [0, 1

2 ]. In this case, it is shown in [4] that

A(ξ) is singular if and only if ξ ∈ {0, 12} and that the number of shifts required by

theorem 2.3 can be reduced to m−12 shifts.

The additional sampling schemes given in [1, 3, 4] use samples taken only atthe initial time. Here we present a new additional sampling scheme that includessamples taken at varying times. When a has the properties discussed above, thenumber of required shifts is reduced from m−1

2 to just 1, thus, reducing the numberof spatial samples required. An example of such a sampling set is illustrated infigure 1.

Theorem 2.4. Let m ∈ Z+ be odd. Suppose a is real, symmetric, continuous,and decreasing on [0, 1

2 ]. In this case, the additional sampling given by {(SmnTc)(aj∗

x)}j∈{0,...,m−1} for n odd and any fixed c = 1, . . . ,m−1 provides enough additional

information to stably recover any x ∈ �2(Z), i.e. the reconstruction operator isbounded.

Proof. The dynamical sampling procedure with the additional samples iswritten as y = Ax where A is the dynamical sampling operator from (�2(Z))to (�2(Z))N+m and

y =(y0, y1, . . . , yN−1, SmnTcx, SmnTc(a ∗ x), . . . , SmnTc(a

m−1 ∗ x))T

.

It is shown below that A has a left inverse. The techniques are similar to, but morecomplicated than, those in the proof of theorem 2.1.


In the proof of 2.1, we used equation (2.3) to relate the Fourier transform ofthe subsampled signal yj to the Fourier transform of the original signal x by writing

each yj(ξ) as a linear combination of the m-unknowns x( ξm), x( ξ+1

m ), . . . , x( ξ+m−1m ).

Similarly, the additional samples are taken by subsampling by a factor of mn andso equation (2.3) expresses the Fourier transform of the additional samples as alinear combination of mn-unknowns. The goal is to write the linear combinationsfrom both of these systems in such a way that they can be combined to create asystem of mn-unknowns and (n+ 1)m-equations.

In order to choose equations with the same unknowns, we consider the formulabelow for (SmnTcz)

∧(nξ) and note that right hand side contains the same variablesgiven by equation (2.4) for y(ξ+ k

n ) for k = 0, . . . , n−1. Using (2.3) and the identity

(Tcz)∧(ξ) = e−i2πcξ z(ξ), we obtain the following formula.

(SmnTcz)∧(nξ) =

1

mne

−i2πcξm

mn−1∑l=0

e−i2πcl

mn z(ξ

m+

l

mn)(2.10)

=1

mne

−i2πcξm

n−1∑k=0

e−i2πck

mn

m−1∑j=0

e−i2πcj

m z(ξ + k

n + j

m).

The original and additional samples are related to the original signal in the Fourierdomain by matrix multiplication:

(2.11) m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

nei2πcξ

m (SmnTcy0)∧(nξ)

...nei2πcξ(SmnTcym−1)

∧(nξ)y(ξ)

y(ξ + 1n )

...y(ξ + n−1

n )

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠= AN (ξ)

⎛⎜⎜⎜⎝x(ξ)

x(ξ + 1n )

...x(ξ + n−1

n )

⎞⎟⎟⎟⎠ ,

where y(ξ) and x(ξ) are given in (2.8) and AN (ξ) is the block matrix

(2.12) AN (ξ) =

⎛⎜⎜⎜⎜⎜⎝Ac,0(ξ) Ac,1(ξ + 1

n ) . . . Ac,n−1(ξ + n−1n )

AN (ξ) 0 . . . 00 AN (ξ + 1

n ) . . . 0...

.... . .

...0 0 . . . AN (ξ + n−1

n )

⎞⎟⎟⎟⎟⎟⎠ , where

Ac,k(ξ) = e−i2πck

mn

⎛⎜⎜⎜⎜⎝1 e

−i2πcm . . . e

−i2πc(m−1)m

a( ξm ) e

−i2πcm a( ξ+1

m ) . . . e−i2πc(m−1)

m a( ξ+m−1m )

......

......

a(m−1)( ξm ) e

−i2πcm a(m−1)( ξ+1

m ) . . . e−i2πc(m−1)

m a(m−1)( ξ+m−1m )

⎞⎟⎟⎟⎟⎠ ,

Similar to the technique of the proof of theorem 2.1, we want to reduce thestudy of the dynamical sampling operator A to the study of the matrices AN (ξ).


To accomplish this, we define the invertible map J : (L2(T))m → (L2(T/n))mn by

(J z)(ξ) =

⎛⎜⎜⎜⎝z (ξ)

z(ξ + 1

n

)...

z(ξ + n−1

n

)⎞⎟⎟⎟⎠ .

Using J , the Fourier transform, and H from (2.7), the dynamical sampling operator

A can be expressed as a product of operators so that finding a left inverse of Areduces to finding a left inverse of the matrix AN (ξ) in (2.11) for each ξ ∈ T/n.The remainder of the proof is showing that AN (ξ) has a left inverse.

In block form,

(2.13) AN (ξ) =

(D(ξ) F (ξ)0 G(ξ)

)

where D(ξ) =

(Ac,0(ξ)AN (ξ)

), F (ξ) =

(Ac,1(ξ + 1

n ) . . . Ac,n−1(ξ + n−1n )

0 . . . 0

), and

G(ξ) is the block diagonal matrix with AN (ξ+ 1n ), . . . ,AN (ξ+ n−1

n ) on the diagonal.

IfD�(ξ) and G�(ξ) are left inverses ofD(ξ) and G(ξ), respectively, then a left inverse

of AN (ξ) is given by

(2.14) A�N (ξ) =

(D�(ξ) −D�(ξ)F (ξ)G�(ξ)0 G�(ξ)

).

It remains to show that G(ξ) and D(ξ) have left inverses. Because n is oddand AN (ξ) is singular only when ξ ∈ {0, 1

2}, for any fixed ξ, AN (ξ + kn ) is singular

for at most one k = 0, . . . , n− 1. It is shown in [4] that without loss of generality,we can assume AN (ξ + 1

n ), . . . ,AN (ξ + n−1n ) have left inverses. Thus, G(ξ) has a

left inverse.The proof that D(ξ) has a left inverse is more complicated and relies on the

structure of kerAN (0) and kerAN ( 12 ). In [4], it is shown that

kerA(0) = span

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎛⎜⎜⎜⎜⎜⎜⎜⎝

010...0−1

⎞⎟⎟⎟⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎜⎜⎜⎝

001...

−10

⎞⎟⎟⎟⎟⎟⎟⎟⎠, . . . ,

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...1−1...0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭= span{vj}

m−12

j=1 and

kerA(1

2) = span

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

10...0...0−1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

01...0...

−10

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, . . . ,

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...10−1...0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭= span{wj}

m−32

j=0 .


Notice that

(2.15) Ac,k(ξ)

⎛⎜⎜⎜⎝z0z1...

zm−1

⎞⎟⎟⎟⎠ = e−i2πck

mn A(ξ)

⎛⎜⎜⎜⎝z0

e−i2πc

m z1...

e−i2πc(m−1)

m zm−1

⎞⎟⎟⎟⎠ ,

Define the map Mc : kerA(ξ) → kerAc,k(ξ) by Mcz =

⎛⎜⎜⎜⎝z0

ei2πcm z1...

ei2πc(m−1)

m zm−1

⎞⎟⎟⎟⎠ .

We can now study kerAc,k(ξ) by looking at kerA(ξ). This is summarized in thefollowing lemma.

Lemma 2.5. The matrix D(ξ) has a left inverse if and only if

ker(AN (ξ)) ∩Mc(kerA(ξ)) = {0}.

Since kerAN (ξ) = {0} for ξ /∈ {0, 12}, we need only check the cases of ξ = 0

and ξ = 12 .

Let v1, . . . , vm−12

be the basis for kerA(0) given in (2.15). Using the equality

for the dimension of subspaces

dim(U ∩W ) = dim(U) + dim(W )− dim(U ∪W ),

we see ker(AN (0)) ∩Mc(kerA(0)) = {0} if and only if

dim(span{v1, . . . , vm−1

2,Mcv1, . . . ,Mcvm−1

2})= m− 1.

Note that vj has zeros everywhere except the j-th and (m − j)-th positions.Thus, there is linear dependence among the set of vectors

{v1, . . . , vm−12

,Mcv1, . . . ,Mcvm−12

}

if and only if Mcvj = αvj for some α ∈ C and j = 1, . . . m−12 . This condition is

satisfied if and only if

(2.16) ei2πcj

m = −ei2πc(m−j)

m = −e−i2πcj

m ,

which requires that

(2.17)4πjc

m= (2s+ 1)π for some s ∈ Z.

Since m is odd and j and c are integers, equation (2.17) is never satisfied. Thus,Mc(kerA(0)) kerAN (0) = {0}. A similar argument shows that

Mc(kerA(1

2)) ∩ kerAN (

1

2) = {0}.

�


3. Dynamical Sampling with a Forcing Term

In this section, we explore the dynamical sampling problem when an unknownsource term enters the system during the time period when samples are taken.For example, if the signal being measured is a pollutant, the dynamical samplingproblem in the previous section and in [1, 3, 4] assumes that the pollution wasreleased at time t=0 and that no additional pollution enters the system whilethe samples are taken. Here we explore what happens when this assumption isremoved and allow for pollution to enter the system during the sampling period.It is assumed that the source is independent of time, that is the forcing term isconstant in the time variable.

If A is the evolution rule, x0 is the unknown initial signal, and x is the unknownforcing term, then the signal yn at time t = n is modeled by

y0 = x0

y1 = Ay0 + x = Ax0 + x

y2 = Ay1 + x = A2x0 +Ax+ x

...

yN = AyN−1 + x = ANx0 +AN−1x+AN−2x+ . . .+Ax+ x

Note that the notation in this section differs from that in section 2, in which yjrepresented the subsampled signal at time t = j. Here yj denotes the signal beforeit is subsampled. If for any j ∈ N, both yj and yj+1 are completely known, then xcan be easily recovered by the relationship x = yj+1−Ayj . It’s quite simple, and ofcourse, not the question we are interested in. Let S be some subsampling operatorso that the measured signal at time j is given by Syj . In general, S and A do notcommute, so that

(3.1) Syj+1 = S(Ayj + x) = ASyj + Sx

This means that knowledge of Syj and Syj+1 does not allow for reconstruction ofSx by taking differences as above.

In this section, we keep the assumptions of section 2 that A is a convolutionoperator so that Ax = a ∗ x where a ∈ L∞(T) and that the subsampling is regularand constant in time. The measured signal at time t = j is Smyj . The dynamicalsampling procedure in the presence of a forcing term can be written as

(3.2) y = AFx

where AF is an operator from (�2(Z))2 to (�2(Z))N , x = (x0, x), and

y = (Smy0, Smy1, . . . , SmyN−1).

In general, the operator AF does not have a bounded left inverse. In fact,when nothing is known a priori about x and x0, the operator AF is not even

injective. In particular, if ‖A‖ < 1, x0(mk) = 0 for all k ∈ Z, and x0 =∞∑

n=0Anx,

then (x0, x) ∈ kerAF. It is not surprising that dynamical sampling fails in thiscase because the system is static, that is yj = yj+1 = x0. The spirit of dynamicalsampling is to use the dynamics of a system to compensate for spatial undersampling


by temporal oversampling. In a static system, temporal sampling does not providethe necessary additional information to offset spatial undersampling.

Because the operatorAF is not injective, the additional sampling schemes givenin theorems 2.3 and 2.4 are not enough to allow for the recovery of x. The dynamicalsampling operator in these theorems is injective and the additional samples areneeded only for stable reconstruction. In contrast, in order to even define a leftinverse ofAF, one must have some a priori knowledge of x0 and x that appropriatelyrestricts the domain of AF. One such example is discussed below.

3.1. The initial signal is the forcing term. Perhaps the simplest case ofdynamical sampling with a forcing term is when it is assumed that the initial signalis the forcing term, i.e. x0 = x. In the wireless sensor network setting, this maymean that sensors are in place and actively sampling a null field when a physicalphenomena occurs, producing the forcing term. The phenomena is captured fromthe beginning. The results in this special case parallel those in section 2.

Theorem 3.1. Let m ∈ Z+ be fixed. Suppose it is known that x0 = x. Assumethat a ∈ L∞(T) and define

C(ξ) =

⎛⎜⎜⎜⎝1 . . . 1

a( ξm ) + 1 . . . a( ξ+m−1

m ) + 1...

......

am−1( ξm ) + . . .+ a( ξ

m ) + 1 . . . am−1( ξ+m−1m ) + . . .+ a( ξ+m−1

m ) + 1

⎞⎟⎟⎟⎠ ,

ξ ∈ T. Then AF in (3.2) has a bounded left inverse for some N ≥ m if andonly if there exists α > 0 such that the set {ξ : | detC(ξ)| < α} has zero measure.Consequently, AF has a bounded left inverse for some N ≥ m if and only if AF

has a bounded left inverse for all N ≥ m.

Remark 3.2. The matrix C(ξ) can be written as the matrix product

(3.3) C(ξ) =

⎛⎜⎜⎜⎜⎝1 0 . . . 0

1 1. . . 0

......

. . . 01 1 . . . 1

⎞⎟⎟⎟⎟⎠A(ξ),

where A(ξ) is defined in (2.2). Since the first matrix in the product of (3.3) isobviously invertible, C(ξ) has an inverse if and only if A(ξ) has an inverse. AlsoA(ξ) and C(ξ) have the same kernel. Thus, the additional sampling schemes definedin theorems 2.3 and 2.4 to stabilize a left inverse of A apply here and stabilize aleft inverse of AF.

Proof. The assumption that x0 = x reduces the domain of AF to �2(Z), andthe signal at time t = j is given by yj = aj ∗ x+ aj−1 ∗ x+ . . . + a ∗ x + x, whereaj ∗ x = (a ∗ . . . ∗ a︸︷︷︸

j

) ∗ x. The remainder of the proof is identical to the proof of

theorem 2.1. �

References

[1] R. Aceska, A. Aldroubi, J. Davis, and A. Petrosyan. Dynamical sampling in shift invariantspaces,Contemp. Math., Amer. Math. Soc. (2013), to appear.


[2] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. Wireless sensor networks: Asurvey, Computer Networks, 38, (2002) 393-422.

[3] A. Aldroubi, J. Davis, and I. Krishtal. Dynamical Sampling: Time Space Trade-off, Appl.Comput. Harmon. Anal., (2012), http://dx.doi.org/10.1016/j.bbr.2011.03.031.

[4] A. Aldroubi, J. Davis, and I. Krishtal. Exact Reconstruction of Spatially Undersampled Signalsin Evolutionary Systems, preprint.

[5] Y. Lu and M. Vetterli. Spatial super-resolution of a diffusion field by temporal oversampling

in sensor networks, in Acoustics, Speech and Signal Processing, 2009. ICASSP 2009. IEEEInternational Conference on, april 2009, 2249–2252.

[6] J. Ranieri, A. Chebira, Y. M. Lu, and M. Vetterli. Sampling and reconstructing diffusionfields with localized sources, in Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEEInternational Conference on, May 2011, 4016 –4019.

[7] Gunter Reise, Gerald Matz, and Karlheinz Grochenig, Distributed field reconstruction in wire-less sensor networks based on hybrid shift-invariant spaces, IEEE Trans. Signal Process. 60(2012), no. 10, 5426–5439, DOI 10.1109/TSP.2012.2205918. MR2979004

[8] G. Reise and G. Matz, Distributed sampling and reconstruction of non-bandlimited fields insensor networks based on shift-invariant spaces,Proc. ICASSP, Taipeh, Taiwan, (2009), 2061-2064.

[9] C.E. Shannon, Classic paper: Communication in the presence of noise, Proc. IEEE, 86, (1998)447-457.

Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240


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626 Veronika Furst, Keri A. Kornelson, and Eric S. Weber, Editors, OperatorMethods in Wavelets, Tilings, and Frames, 2014

614 James W. Cogdell, Freydoon Shahidi, and David Soudry, Editors, AutomorphicForms and Related Geometry, 2014

613 Stephan Stolz, Editor, Topology and Field Theories, 2014

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609 Dinh Van Huynh, S. K. Jain, Sergio R. Lopez-Permouth, S. Tariq Rizvi,

and Cosmin S. Roman, Editors, Ring Theory and Its Applications, 2014

608 Robert S. Doran, Greg Friedman, and Scott Nollet, Editors, Hodge Theory,Complex Geometry, and Representation Theory, 2014

607 Kiyoshi Igusa, Alex Martsinkovsky, and Gordana Todorov, Editors, ExpositoryLectures on Representation Theory, 2014

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603 Azita Mayeli, Alex Iosevich, Palle E. T. Jorgensen, and Gestur Olafsson,Editors, Commutative and Noncommutative Harmonic Analysis and Applications, 2013

602 Vyjayanthi Chari, Jacob Greenstein, Kailash C. Misra, K. N. Raghavan,and Sankaran Viswanath, Editors, Recent Developments in Algebraic andCombinatorial Aspects of Representation Theory, 2013

601 David Carfı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel vanFrankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure andApplied Mathematics II, 2013

600 David Carfı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel vanFrankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure andApplied Mathematics I, 2013

599 Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, FernandoSchwartz, and Gilbert Weinstein, Editors, Geometric Analysis, MathematicalRelativity, and Nonlinear Partial Differential Equations, 2013

598 Eric Todd Quinto, Fulton Gonzalez, and Jens Gerlach Christensen, Editors,Geometric Analysis and Integral Geometry, 2013

597 Craig D. Hodgson, William H. Jaco, Martin G. Scharlemann, and StephanTillmann, Editors, Geometry and Topology Down Under, 2013

596 Khodr Shamseddine, Editor, Advances in Ultrametric Analysis, 2013

595 James B. Serrin, Enzo L. Mitidieri, and Vicentiu D. Radulescu, Editors, Recent

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594 James B. Serrin, Enzo L. Mitidieri, and Vicentiu D. Radulescu, Editors, RecentTrends in Nonlinear Partial Differential Equations I, 2013

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume contains the proceedings of the AMS Special Session on Harmonic Analysisof Frames, Wavelets, and Tilings, held April 13–14, 2013, in Boulder, Colorado.

Frames were first introduced by Duffin and Schaeffer in 1952 in the context of nonhar-monic Fourier series but have enjoyed widespread interest in recent years, particularly asa unifying concept. Indeed, mathematicians with backgrounds as diverse as classical andmodern harmonic analysis, Banach space theory, operator algebras, and complex analysishave recently worked in frame theory. Frame theory appears in the context of wavelets,spectra and tilings, sampling theory, and more.

The papers in this volume touch on a wide variety of topics, including: convex geometry,direct integral decompositions, Beurling density, operator-valued measures, and splines.These varied topics arise naturally in the study of frames in finite and infinite dimensions.In nearly all of the papers, techniques from operator theory serve as crucial tools to solvingproblems in frame theory.

This volume will be of interest not only to researchers in frame theory but also to thosein approximation theory, representation theory, functional analysis, and harmonic analysis.

ISBN978-1-4704-1040-7

9 781470 410407

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