Applied and Numerical Harmonic Analysis

Martha Abell, Emil IacobAlex Stokolos, Sharon TaylorSergey Tikhonov, Jiehua ZhuEditors

Topics in Classical and Modern AnalysisIn Memory of Yingkang Hu

Applied and Numerical Harmonic Analysis

Series EditorJohn J. BenedettoUniversity of MarylandCollege Park, MD, USA

Advisory Editors

Akram Aldroubi Gitta KutyniokVanderbilt University Technical University of BerlinNashville, TN, USA Berlin, Germany

Douglas Cochran Mauro MaggioniArizona State University Johns Hopkins UniversityPhoenix, AZ, USA Baltimore, MD, USA

Hans G. Feichtinger Zuowei ShenUniversity of Vienna National University of SingaporeVienna, Austria Singapore, Singapore

Christopher Heil Thomas StrohmerGeorgia Institute of Technology University of CaliforniaAtlanta, GA, USA Davis, CA, USA

Stéphane Jaffard Yang WangUniversity of Paris XII Hong Kong University ofParis, France Science & Technology

Kowloon, Hong KongJelena KovacevicCarnegie Mellon UniversityPittsburgh, PA, USA

More information about this series at http://www.springer.com/series/4968

Martha Abell • Emil Iacob • Alex Stokolos •Sharon Taylor • Sergey Tikhonov • Jiehua ZhuEditors

Topics in Classicaland Modern AnalysisIn Memory of Yingkang Hu

EditorsMartha AbellGeorgia Southern UniversityStatesboroGA, USA

Emil IacobGeorgia Southern UniversityStatesboroGA, USA

Alex StokolosGeorgia Southern UniversityStatesboroGA, USA

Sharon TaylorGeorgia Southern UniversityStatesboroGA, USA

Sergey TikhonovICREA, Passeig de Lluís CompanysBarcelona, Spain

Centre de Recerca MatemàticaBarcelona, Spain

Jiehua ZhuGeorgia Southern UniversityStatesboroGA, USA

ISSN 2296-5009 ISSN 2296-5017 (electronic)Applied and Numerical Harmonic AnalysisISBN 978-3-030-12276-8 ISBN 978-3-030-12277-5 (eBook)https://doi.org/10.1007/978-3-030-12277-5

Mathematics Subject Classification (2010): 41A, 42A, 30E, 65D

© Springer Nature Switzerland AG 2019This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

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ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims toprovide the engineering, mathematical, and scientific communities with significantdevelopments in harmonic analysis, ranging from abstract harmonic analysis tobasic applications. The title of the series reflects the importance of applicationsand numerical implementation, but richness and relevance of applications andimplementation depend fundamentally on the structure and depth of theoreticalunderpinnings. Thus, from our point of view, the interleaving of theory andapplications and their creative symbiotic evolution is axiomatic.

Harmonic analysis is a wellspring of ideas and applicability that has flourished,developed, and deepened over time within many disciplines and by means ofcreative cross-fertilization with diverse areas. The intricate and fundamentalrelationship between harmonic analysis and fields such as signal processing, partialdifferential equations (PDEs), and image processing is reflected in our state-of-the-art ANHA series.

Our vision of modern harmonic analysis includes mathematical areas such aswavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis,and fractal geometry, as well as the diverse topics that impinge on them.

For example, wavelet theory can be considered an appropriate tool to deal withsome basic problems in digital signal processing, speech and image processing,geophysics, pattern recognition, biomedical engineering, and turbulence. Theseareas implement the latest technology from sampling methods on surfaces to fastalgorithms and computer vision methods. The underlying mathematics of wavelettheory depends not only on classical Fourier analysis, but also on ideas from abstractharmonic analysis, including von Neumann algebras and the affine group. This leadsto a study of the Heisenberg group and its relationship to Gabor systems, and of themetaplectic group for a meaningful interaction of signal decomposition methods.The unifying influence of wavelet theory in the aforementioned topics illustrates thejustification for providing a means for centralizing and disseminating informationfrom the broader, but still focused, area of harmonic analysis. This will be a key roleof ANHA. We intend to publish with the scope and interaction that such a host ofissues demands.

v

vi ANHA Series Preface

Along with our commitment to publish mathematically significant works at thefrontiers of harmonic analysis, we have a comparably strong commitment to publishmajor advances in the following applicable topics in which harmonic analysis playsa substantial role:

Antenna theory Prediction theory

Biomedical signal processing Radar applications

Digital signal processing Sampling theory

Fast algorithms Spectral estimation

Gabor theory and applications Speech processing

Image processing Time-frequency and

Numerical partial differential equations time-scale analysis

Wavelet theory

The above point of view for the ANHA book series is inspired by the history ofFourier analysis itself, whose tentacles reach into so many fields.

In the last two centuries Fourier analysis has had a major impact on thedevelopment of mathematics, on the understanding of many engineering andscientific phenomena, and on the solution of some of the most important problemsin mathematics and the sciences. Historically, Fourier series were developed inthe analysis of some of the classical PDEs of mathematical physics; these serieswere used to solve such equations. In order to understand Fourier series and thekinds of solutions they could represent, some of the most basic notions of analysiswere defined, e.g., the concept of “function.” Since the coefficients of Fourierseries are integrals, it is no surprise that Riemann integrals were conceived to dealwith uniqueness properties of trigonometric series. Cantor’s set theory was alsodeveloped because of such uniqueness questions.

A basic problem in Fourier analysis is to show how complicated phenomena,such as sound waves, can be described in terms of elementary harmonics. There aretwo aspects of this problem: first, to find, or even define properly, the harmonics orspectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second,to determine which phenomena can be constructed from given classes of harmonics,as done, for example, by the mechanical synthesizers in tidal analysis.

Fourier analysis is also the natural setting for many other problems in engineer-ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem inFourier analysis not only characterizes the behavior of the prime numbers, but alsoprovides the proper notion of spectrum for phenomena such as white light; thislatter process leads to the Fourier analysis associated with correlation functions infiltering and prediction problems, and these problems, in turn, deal naturally withHardy spaces in the theory of complex variables.

Nowadays, some of the theory of PDEs has given way to the study of Fourierintegral operators. Problems in antenna theory are studied in terms of unimodulartrigonometric polynomials. Applications of Fourier analysis abound in signalprocessing, whether with the fast Fourier transform (FFT), or filter design, or the

ANHA Series Preface vii

adaptive modeling inherent in time-frequency-scale methods such as wavelet theory.The coherent states of mathematical physics are translated and modulated Fouriertransforms, and these are used, in conjunction with the uncertainty principle, fordealing with signal reconstruction in communications theory. We are back to theraison d’être of the ANHA series!

University of Maryland John J. BenedettoCollege Park Series Editor

Preface

On May 9, 2017, we celebrated the life of Yingkang Hu with an Afternoon in hishonor as a special part of the International Conference on Approximation Theory.1

While this was an exceptional mathematical event, a gathering of about 40 analysts,students, and junior and senior researchers, from all over the USA, Austria, Canada,India, Israel, Russia, Spain, and Ukraine, we also paused to remember Yingkang’spersonality, his sharp mind and gentle sense of humor, as well as his contributionsto mathematics and to the life of the university and academic community.

Many speakers talked about how their research was inspired and supported byYingkang’s knowledge, curiosity, and intuition. Dany Leviatan (Tel Aviv Univer-sity), in a special lecture “Characterizing smoothness of functions via the degree ofapproximation and some shape preservation in Lp and C,” shared with the audiencehis memory of Yingkang and explained Yingkang’s contribution to the field ofapproximation theory.

Martha Abell (Georgia Southern University), in a special lecture about Yingkangand his contribution to the department as a teacher, scholar, and servant, highlightedthe manner in which Yingkang positively impacted all around him through hissupportive and collaborative nature. His remarkable ability to take on new researchareas, service roles, and teaching duties and become an expert set him apart frommost and allowed him to become a leader in scholarly endeavors, in mentoring newfaculty, and in encouraging students to strive for high levels of accomplishment intheir studies.

Yingkang, who grew up in Beijing, China, earned a B.S. in Mathematics fromBeijing University of Chemical Technology in 1982 and a Ph.D. in Mathematicsfrom the University of South Carolina in 1989 under the supervision of RonaldDeVore. He passed away in his home in the early hours of March 11, 2016. At thetime of his death, he was survived by his wife, Xia Liu (Sherry) of Statesboro, GA,

1The Conference was sponsored by the National Science Foundation (Grant #1700153) and by theCollege of Science and Mathematics of Georgia Southern University.

ix

x Preface

his son and daughter-in-law, Harry Hu and Jess Oakley of Durham, NC, and threebrothers and their families.

As the sad news of Yingkang’s passing spread throughout the ApproximationTheory community in March 2016, many individuals reached out to the departmentwith their condolences. These heartfelt thoughts helped lift up the departmentthrough the difficult challenge of dealing with a loss of this magnitude. Facultyand staff members in the department joined students, colleagues across campus, andfriends in the community in the Russell Union on the Georgia Southern Universitycampus on March 22, 2016, to share memories and give support for Yingkang’sfamily.

In this volume, friends and colleagues have contributed research papers, surveys,and/or short remembrances about Yingkang. The remembrances were sometimesweaved into the article submitted (either at the beginning or at the end), and wehave respected the format each author chose.

Many of the authors gave talks at the International Conference in ApproximationTheory held in Savannah in May 2017, while others were unable to attend theseevents but took this opportunity to contribute to the volume.

The mathematical contributions naturally align with Yingkang’s mathematicalinterests: approximation theory, harmonic and complex analysis, splines, andclassical Fourier theory.

The volume contains articles about Yingkang, his mathematics, remembrancesby colleagues and friends, and survey and research articles on approximation theory,harmonic and complex analysis, splines, signal processing, and dynamical systems.

Contents of Volume In Part I, a collection of photos of Yingkang Hu in varioushypostases is presented, courtesy of Yingkang’s family. Yingkang was a pivotalmember of the Department of Mathematical Sciences at Georgia Southern Univer-sity with deep interest in mathematics. In his free time, as many remember him,he was an avid photographer and an active member of the community. Part I thencontinues with a collection of reminiscences by Yingkang’s colleagues and friends.It concludes with an article in the spirit of Yingkang Hu’s work. In this article, KirillA. Kopotun, Dany Leviatan, and Igor A. Shevchuk discuss some properties of therecently introduced moduli of smoothness with Jacobi weights. These results areapplied to characterize the behavior of the polynomials of best approximation of afunction in a Jacobi-weighted Lp space, 0 < p ≤ ∞. Finally, sharp Marchaud-and Jackson-type inequalities proved by Ditzian, Dai, and Tikhonov, in the case1 < p < ∞, are discussed.

Part II contains survey and research articles submitted by an array of mathemati-cians representing one or several of the mathematical themes close to Yingkang’sinterests. We have organized the rest of the volume according to these themes.

All the articles were invited and peer reviewed; we thank the authors andanonymous referees for their help in making these volumes possible.

In chapter “Special Difference Operators and the Constants in the ClassicalJackson-Type Theorems”, Alexander Babenko and Yuriy Kryakin outline theproofs of certain sharp (with respect to the order of the modulus of continuity)estimates of the Jackson–Stechkin constants for classical trigonometric and alge-

Preface xi

braic approximation. The authors show how one can use some special case ofgeneralized difference operators first considered by H. Shapiro in 1967 to indicatethe sharp order of the Jackson–Stechkin constants in the main theorems of classicalapproximation theory.

In chapter “Comparison Theorems for Completely and Multiply MonotoneFunctions and Their Applications”, Yiulia Babenko, Valdislav Babenko, and OlegKovalenko present a survey of existing results, analyze and compare used methods,propose new methods, and present new sharp inequalities that estimate L2- or L∞-norm of an “intermediate” derivative of a univariate or multivariate function withthe help of L2-norms of the function itself and its “higher” derivatives. There existnumerous well-known sharp inequalities of this kind: Hardy–Littlewood–Polyainequality, Taikov inequality, etc. Some of these inequalities have been generalizedto the case of functions of unbounded operators in a Hilbert space, which led tointeresting applications. However, many questions in this direction remain open.

In chapter “Concerning Exponential Bases on Multi-Rectangles of Rd”, LauraDe Carli produces exponential bases with explicit frame constants on finite union ofdisjoint rectangles in R

d with rational vertices. It is well known that finite unionof disjoint rectangles in R

d (or: multi-rectangles) has exponential bases. Multi-rectangles QN made of N unit cubes with rational vertices and bases in the form

of BN = B( �δ1, . . . , �δN

)= ∪N

j=1

{e2πi〈�n+ �δj 〉

}�n∈Zd

for some �δ1, . . . , �δN ∈ Rd , are

of special interest. This chapter provides a complete characterization of the sets BN

that form an exponential basis on a given multi-rectangle QN and precise estimateof the frame constants of BN . Necessary and sufficient conditions that ensure thatBN is an exponential basis on QN can be deduced from various theorems in theliterature, so this result is not new; however, the frame constants of BN are notexplicit in the literature and the proof in this chapter may have an interest on its ownbecause it uses the semigroup properties and precise norm estimates of a remarkablegroup of isometries on l2(Zd ). New corollaries and examples complete the chapter.

In chapter “Hankel Transforms of General Monotone Functions”, AlbertoDebernardi studies the boundedness and uniform convergence of the Hankeltransform Hαf of order α ≥ − 1

2 . Relying on an Abel–Olivier test for real-valued functions, the author shows that the Hankel transform of a general monotonefunction converges uniformly if and only if the limit function is bounded. Moreover,upper estimates for |Hαf | in terms of α and f are obtained, and correspondingresults for the cosine transform (which is H−1/2) and cosine series (with generalmonotone coefficients, analogue to general monotone functions) are derived.

In chapter “Univalence of a Certain Quartic Function”, Jimmy Dillies provesthe univalence of the polynomial f4(z), from a general family of polynomialsindexed by the natural numbers, fN(z),N ∈ N. The univalence question appears,for instance, in studying the relation between the stability of equilibrium in discretedynamical systems and a problem of optimal covering of the interval (−μ,μ) by theinverse of the polynomial image of the unit disk. Even though the author essentiallyrestricts himself to studying this specific example, his elegant method has a widerrange of applications and definitely deserves a bit of attention.

xii Preface

In chapter “Finding, Stabilizing, and Verifying Cycles of Nonlinear DynamicalSystems”, Dmitriy Dmitrishin, Ionut E. Iacob, Ivan Skrinnik, and Alex Stokolosaddress the problem of cycle detection, which is one of the most fundamentalin mathematics. The authors present a new solution for finding, verifying, andstabilizing cycles in nonlinear dynamical systems. The solution relies on a newcontrol method based on mixing previous states of the system (or the functions ofthese states). This approach allows local stabilization and finding a priori unknowncycles of a given length. Several numerical examples are considered.

In chapter “Finding Orbits of Functions Using Suffridge Polynomials”, DmitriyDmitrishin, Paul Hagelstein, and Alex Stokolos indicate how Suffridge polynomialsmay be used to find orbits of functions. In particular, they describe a controlmechanism that, given a function f : R → R and a positive integer T , yieldsa dynamical system G : R

Tn → RTn which under quantifiable conditions has

(x, . . . , x) as an attractor, provided x lies on a T -cycle of f . An explicit example ofthis control mechanism is provided using a logistic function.

In chapter “The Sharp Remez-Type Inequality for Even Trigonometric Poly-nomials on the Period”, Tamás Erdélyi discusses the Remez-type inequality fortrigonometric polynomials with complex coefficients. Remez-type inequalities forvarious classes of functions have been studied by several authors, and they haveturned out to be applicable and connected to various problems in approximationtheory. While the sharp Remez-type inequality for trigonometric polynomialsremains open, a proof is provided in this chapter for the sharp Remez-type inequalityfor even trigonometric polynomials.

In chapter “The Lebesgue Constraints of Fourier Partial Sums”, MichaelGanzburg and Elijah Liflyand present a first attempt to discuss monotonicityproperties of the multidimensional Lebesgue constants of partial sums in a generalsetting. It is hardly believable that in the general case such norms may possessmonotonicity properties in full. The main result of the chapter states that they,however, are subject to partial monotonicity. It is of great interest to find certainapplications of this result. For completeness, the chapter is supplied with a surveyof the known results on the behavior of the Lebesgue constants of partial sumsgenerated by various sets in the Euclidean space. The latter may be useful for thosewho work on this topic and wish to have the knowledge of the state of affairs invarious directions of this subject.

In chapter “Liouville-Weyl Derivatives of Double Trigonometric Series”, AinurJumabayeva and Boris Simonov study estimates of norms and the angle best approx-imations of the generalized Liouville–Weyl derivatives by the angle approximationof functions themselves in the two-dimensional case. The main goal of this workis to extend previous results in the following respects. First, the authors considerthe generalized Liouville–Weyl derivatives in place of the classical mixed Weylderivatives. Second, similarly to one-dimensional inequalities, the authors obtainestimates of the angle approximations of these derivatives by angle approximation offunctions themselves in the two-dimensional case. The concept of general monotonesequences plays a key role in our study.

Preface xiii

In chapter “Inequalities in Approximation Theory Involving Fractional Smooth-ness in Lp , 0 < p < 11”, Yurii Kolomoitsev and Tetiana Lomako study inequalitiesfor the best trigonometric approximations and fractional moduli of smoothnessinvolving the Weyl and Liouville–Grünwald fractional derivatives in Lp , 0 < p <

1. The authors extend a series of known inequalities in approximation theory tothe whole range of parameters of smoothness as well as establish a new type ofestimates. The following results are proved: estimates from above and from belowfor the errors of the best approximation and moduli of smoothness of a function andits fractional derivatives; the direct and inverse approximation theorems involvingfractional derivatives and moduli of smoothness; and complete description of theclass of functions with the optimal rate of decay of the fractional modulus ofsmoothness in Lp for 0 < p < 1.

In chapter “On de Boor-Fix Type Functionals for Minimal Splines”,Egor K. Kulikov and Anton Makarov consider minimal coordinate splines(polynomial or nonpolynomial). These splines as a special case include well-knownpolynomial B-splines and share most properties of B-splines (linear independency,local support, smoothness, partition of unity, nonnegativity). A system of dualfunctionals possessing the property of biorthogonality to the system of constructedsplines is created. Then the authors construct approximation functionals, used as thecoefficients at the basis functions in local schemes of approximation. The obtainedresults are illustrated with an example of a polynomial generating vector function,which leads to the construction of B-splines and the de Boor–Fix functionals. Fornonpolynomial generating vector functions, we give formulas for construction ofnonpolynomial splines and the dual de Boor–Fix-type functionals.

In chapter “A Multidimensional Hardy-Littlewood Theorem”, Elijah Liflyandand Ulrich Stadtmueller discuss extensions and generalizations of the followingclassical result due to Hardy and Littlewood: If a (periodic) function f and itsconjugate f are both of bounded variation, then their Fourier series convergeabsolutely. The proof of this theorem strongly relies on a classical theorem of F. andM. Riesz which asserts that the functions considered are absolutely continuous.Nonperiodic analogues of these results claim, first of all, for proper understandingwhat a conjugate of a function of bounded variation is. Luckily, the classical notionof the modified Hilbert transform works well in this setting. The main tool indimension one is the interchange of the modified (or non-modified, if possible)Hilbert transform and derivation. Of course, the theory of the real Hardy space isalso used. For extensions to the case of several dimensions, the choice of variationis crucial, since there are many notions of bounded variation in several dimensions.The classical (and apparently the oldest) notion of Hardy’s variation is chosen. Thisallows the proof to run in an inductive way, more or less along the same lines asin dimension one with respect to each variable. One may expect that involvingdifferent variations will lead to results of other types and will apparently need otherapproaches.

In chapter “The Spurious Side of Diagonal Multipoint Padé Approximants”,Doron Lubinsky surveys some of the convergence theory of Padé and multipointPadé approximation, especially the diagonal case. In particular, some of the classical

xiv Preface

theorems, such as the Nuttall and Pommerenke theorems, Stahl’s theorems, andvarious forms of the Baker–Gammel–Wills conjecture, are discussed. There aresuggestions for directions for future research, and a discussion of the difficultiesof proving convergence, notably that of spurious poles.

In chapter “Spline Summability of Cardinal Sine Series and the Bernstein Class”,W. R. Madych provides a resolution to a conjecture proposed by I. J. Schoebergin 1976: If {f (n) : n = 0,±1,±2, . . .} are the samples of a function f in theBernstein class Bπ , then, as k tends to∞, the cardinal spline interpolants of order 2kof this data, Sk({f (n)}, x), converge to f (x)+ c sin πx, for an appropriate constantc, uniformly on compact subsets of the real axis.

The Bernstein class Bπ consists of entire functions of exponential type ≤ π thatare bounded on the real axis. The sequence of samples {f (n)} does not uniquelydetermine the function f in Bπ .

The chapter begins with the basic background and Schoenberg’s rationale. Next,a new result is presented that significantly extends a theorem that was one of themotivating ingredients of the conjecture. The result concerns the limiting behaviorof Sk({f (n)}, x) as k tends to ∞ that is valid for all data sequences {f (n)} ofpolynomial growth. This is followed by the presentation of (1) a regularity resultfor the cardinal spline summability method, (2) a Tauberian type theorem, and (3)an example of a function in Bπ that is not representable as a convergent cardinalsine series that lead to a negative resolution of the conjecture.

In chapter “Integral Identities for Polyanalytic Functions”, Anastasiia Minenkovaand Olga Trofimenko overview some results from the theory of polyanalyticfunctions. We consider the problem of the mean value of polyanalytic functions ofcertain types. We say that f (z) is areolar monogenic in the unit disk D if and only

if(

∂∂z

)f is an analytic function in D. Therefore, we can say that f (z) is areolar

monogenic if and only if(

∂∂z

)2f = 0 holds in D. Areolar monogenic functions go

back to D. Pompeiu. The further study of these functions is presented in works byM.O. Reade. It gave a push for development of the theory of more general type offunctions, called polyanalytic. The results considered in this chapter are analoguesof the Cauchy, Morera, and Fedoroff theorems for circular domains and polygonaldomains. Also, the so-called two radii theorems are discussed.

In chapter “Pointwise Behavior of Christoffel Function on Planar ConvexDomains”, Andriy Prymak and Olena Usoltseva prove a general lower bound onChristoffel functions, a valuable tool in various areas of analysis and mathematics,on planar convex domains in terms of a modification of the parallel section functionof the domain. For a certain class of planar convex domains, in combination witha recent general upper bound, this allows us to compute the pointwise behavior ofChristoffel functions.

In chapter “Towards Best Approximations for |x|α”, Michael Revers presents asurvey on asymptotic relations for the approximation of |x|α , α > 0 in L∞ [−1, 1]by Lagrange interpolation polynomials based on the zeros of the Chebyshevpolynomials of the first kind.

Preface xv

From the Chebyshev alternation theorem, it follows that for each integer n thebest approximating polynomial of order n to |x|α in the in L∞-norm can be repre-sented as an interpolating polynomial with unknown consecutive nodes in [−1, 1].Thus, if one can find something about the nature of those best approximatinginterpolation nodes in [−1, 1], then we would successfully find an approach fora constructive analytical approximation toward a representation for the Bernsteinconstants Δ∞,α . Along the way, we explore connections of our results togetherwith papers of Ganzburg and Lubinsky, by presenting numerical results, indicating apossible constructive way toward some representations for the Bernstein constants.

In chapter “Fixed Volume Discrepancy in the Periodic Case”, VladimirTemlyakov studies the smooth fixed volume discrepancy in the periodic case. Itis proved that the Frolov point sets adjusted to the periodic case are optimal in acertain sense order of decay of the smooth periodic discrepancy. The upper boundsfor the r-smooth fixed volume periodic discrepancy for these sets are established inthis chapter.

In chapter “Approximation by Trigonometric Polynomials in Stechkin MajorantSpaces”, Sergey S. Volosivets considers the Stechkin majorant spaces Ep(ε) suchthat f ∈ Ep(ε) has best trigonometric approximations En(f )p in L

p2π , 1 ≤ p ≤ ∞,

satisfying the inequality En(f )p ≤ Cεn, n ∈ Z+, where C does not depend onn, εn ↓ 0. We prove that the trigonometric system is a basis in these spaces. Thegeneral estimates of best approximation in Ep(ε) including Jackson and Bernsteininequalities are established. For τn(f )(x) = ∑n

k=0 ankSk(f )(x), where Sk(f ) arepartial Fourier sums of f and {ank : n ≥ 0, 0 ≤ k ≤ n} satisfies certain conditionof generalized monotonicity type, some bounds for the degree of approximation‖f − τn(f )‖Ep(ε) are obtained. The sharpness of such results is proved under somerestrictions. Also, some applications of obtained results to the approximation inHölder–Lipschitz spaces are given.

In chapter “On Multivariate Sampling of a Class of Integral Transforms”, AhmedZayed discusses some linear integral transformations that play an important rolein signal processing and optical systems. One of the most important integraltransformation used in applications is the Fourier transform. The Whittaker–Shannon–Kote’lnikov sampling theorem provides a tool for the reconstruction ofbandlimited functions from their samples at a discrete set of points. A functionis bandlimited if its Fourier transform is supported on a finite interval symmetricabout the origin. Sampling of n-dimensional transforms are, generally, more difficultto obtain because the samples depend on the geometry of the region I. Anothergroup of transformations that are closely related to the Fourier transform includesthe fractional Fourier transform, the special affine Fourier transform, and thelinear canonical transform. These transforms also have applications in electricalengineering and optics. In this chapter, we give a brief introduction to thesetransforms and then obtain multivariate sampling theorems for their extensions tohigher dimensions.

xvi Preface

Acknowledgements This volume would not have been possible without the contri-butions of all of our authors. We are grateful for the time and effort these individualsplaced in writing their manuscripts.

All the articles were peer reviewed, so we are grateful to our referees, who tookthe time to help make this volume a reality. We used their well-placed commentsand words to describe in the preface the articles in this volume. Thank you to all ofour reviewers!

We would like to thank Sherry, Yingkang’s wife, who supported the project andgave us beautiful photos to share.

We are also indebted to our editor at Springer, Luca Sidler, who was accommo-dating and patient with us as we put together the volume.

And last but not least, we should mention the National Science Foundation whogenerously supported the conference. Without this financial support the projectwould never have come to fruition and our lasting remembrance of Yingkang wouldbe incomplete.

Statesboro, GA, USA Martha AbellStatesboro, GA, USA Emil I. IacobStatesboro, GA, USA Alexander M. StokolosStatesboro, GA, USA Sharon TaylorBarcelona, Spain Sergey TikhonovStatesboro, GA, USA Jiehua Zhu

Contents

Part I Yingkang

Remembering Professor Yingkang Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Martha Abell

Remembrances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

On Some Properties of Moduli of Smoothness with Jacobi Weights . . . . . . . 19Kirill A. Kopotun, Dany Leviatan, and Igor A. Shevchuk1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 The Polynomials of Best Approximation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Further Properties of the Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Part II Approximation Theory, Harmonic and Complex Analysis,Splines and Classical Fourier Theory

Special Difference Operators and the Constants in the ClassicalJackson-Type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Alexander G. Babenko and Yuriy V. Kryakin1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Whitney’s Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Bohr–Favard Inequality and Best Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Bohr–Favard Difference Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Integral Approximation of the Characteristic Function . . . . . . . . . . . . . . . . . . . . 386 Neumann Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Operators W2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Jackson–Stechkin Inequality for W2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Bernstein–Nikolskii–Stechkin Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210 Approximation by Algebraic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xvii

xviii Contents

Comparison Theorems for Completely and Multiply MonotoneFunctions and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Vladyslav Babenko, Yuliya Babenko, and Oleg Kovalenko1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Definition of Considered Functional Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Kolmogorov’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Statement of the Problem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Connections Between Stated Problems and Considered Classes . . . . 564.4 Some Properties of X-Perfect Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 Solution to Kolmogorov’s Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Some Other Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 On the Smoothest Hermite–Birkhoff Interpolation . . . . . . . . . . . . . . . . . . 615.2 On Sharp Estimates for Intermediate Moments . . . . . . . . . . . . . . . . . . . . . . 625.3 On Extremal Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Concerning Exponential Bases on Multi-Rectangles of Rd . . . . . . . . . . . . . . . . . . 65Laura De Carli1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.1 Bases and Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.2 Exponential Bases on L2(Q0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.3 Stability of Riesz Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.4 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5 Three Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 Corollaries and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1 A Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2 Two Cubes in R

d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 Spectral Domains in R

d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4 Extracting Riesz Bases from Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Estimating the Frame Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Appendix: Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Hankel Transforms of General Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . 87Alberto Debernardi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913 Abel–Olivier Test for GM Functions and Sequences . . . . . . . . . . . . . . . . . . . . . . 924 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Contents xix

Univalence of a Certain Quartic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Jimmy Dillies1 In Memoriam .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.1 Decomposition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.2 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Finding, Stabilizing, and Verifying Cycles of Nonlinear DynamicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Dmitriy Dmitrishin, Ionut E. Iacob, Ivan Skrinnik, and Alex Stokolos1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092 Closed Loop Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.1 Characteristic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.2 Geometric Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.3 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2.3.1 Case γ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.3.2 Case γ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.4 Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.4.1 Construction of the Polynomials q (z) . . . . . . . . . . . . . . . . . . . . . . . 1142.4.2 Construction of the Polynomials p (z) . . . . . . . . . . . . . . . . . . . . . . . 115

3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.1 Hénon Map, n = 1, . . . , 1200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2 Elhadj–Sprott Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.3 Ikeda Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.4 Lozi Map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.5 Holmes Cubic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.6 Numerical Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Finding Orbits of Functions Using Suffridge Polynomials . . . . . . . . . . . . . . . . . . 127Dmitriy Dmitrishin, Paul Hagelstein, and Alex Stokolos1 Introduction and Statement of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 Conclusions and Further Directions of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 132References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

The Sharp Remez-Type Inequality for Even TrigonometricPolynomials on the Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Tamás Erdélyi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362 New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xx Contents

3 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

The Lebesgue Constants of Fourier Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Michael I. Ganzburg and Elijah Liflyand1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472 Lebesgue Constants Generated by the Homothety of a Fixed Set . . . . . . . . . 149

2.1 Cubic Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492.2 Spherical Partial Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492.3 Hyperbolic Partial Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3 Polyhedral Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.1 General Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.2 Intermediate Growth .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.3 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4 Partial Increasing of Lebesgue Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Liouville–Weyl Derivatives of Double Trigonometric Series . . . . . . . . . . . . . . . . 159Ainur Jumabayeva and Boris Simonov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

1.1 The One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1591.2 The Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Inequalities in Approximation Theory Involving FractionalSmoothness in Lp , 0 < p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Yurii Kolomoitsev and Tetiana Lomako1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

2.1 Inequalities for the Best Approximation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1872.2 Inequalities for the Moduli of Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . 1892.3 The Direct and Inverse Approximation Theorems . . . . . . . . . . . . . . . . . . . 1902.4 On Decreasing of the Fractional Modulus of Smoothness. . . . . . . . . . . 192

3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1933.1 Properties of the Fractional Moduli of Smoothness. . . . . . . . . . . . . . . . . . 1933.2 Inequalities for Trigonometric Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1943.3 Approximation of a Function and Its Derivatives . . . . . . . . . . . . . . . . . . . . 195

4 Proofs of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

On de Boor–Fix Type Functionals for Minimal Splines . . . . . . . . . . . . . . . . . . . . . 211Egor K. Kulikov and Anton A. Makarov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112 Preliminary Notation and Some Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

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3 On Approximation Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154 On Representation of Nonpolynomial Splines of Lower Orders . . . . . . . . . . 2165 On Realizations of Approximation Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

A Multidimensional Hardy–Littlewood Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227Elijah Liflyand and Ulrich Stadtmüller1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2272 The Hardy–Littlewood Theorem on the Real Line . . . . . . . . . . . . . . . . . . . . . . . . . 2293 Multidimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323.2 Vitali’s Variation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2333.3 Hardy’s Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343.4 Product Hardy Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343.5 Absolute Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4 Integrability of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364.1 Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2374.2 Conditions for Absolute Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.3 Hardy–Littlewood Type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

The Spurious Side of Diagonal Multipoint Padé Approximants . . . . . . . . . . . . 241Doron S. Lubinsky1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2412 Some Padé History and Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2453 Convergence.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2464 Spurious Poles and Varying Interpolation Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . 2545 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Spline Summability of Cardinal Sine Series and the Bernstein Class . . . . . . 261Wolodymyr R. Madych1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612 Definitions and Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

2.1 The Classes Eσ and Bσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2622.2 The Paley–Wiener Spaces PW(m), m = 0, 1, 2, . . . . . . . . . . . . . . . . . . . . 2632.3 Cardinal Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

3 Piecewise Polynomial Cardinal Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2653.1 Definitions and Essential Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2653.2 Spline Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2663.3 The Bernstein Class Bπ and Spline Summability . . . . . . . . . . . . . . . . . . . . 267

3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2673.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2673.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

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3.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2683.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2683.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

4 The Behavior of Sk({cn}, x) as k Tends to ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2694.1 General Results and Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2694.2 A Tauberian Type Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2704.3 Even Functions in Bπ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2714.4 Odd Functions in Bπ with Interesting Properties . . . . . . . . . . . . . . . . . . . . 2714.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Integral Identities for Polyanalytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Anastasiia Minenkova and Olga Trofimenko1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2792 Mean Values for Circular Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823 Integral Identities for Polygonal Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

Pointwise Behavior of Christoffel Function on Planar ConvexDomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293Andriy Prymak and Olena Usoltseva1 Introduction and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2932 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Towards Best Approximations for |x|α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303Michael Revers1 The Bernstein Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3032 Results on Polynomial Interpolations for |x|α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3073 On the Way to Best Approximation Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 309References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Fixed Volume Discrepancy in the Periodic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Vladimir N. Temlyakov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3152 Point Sets Based on the Frolov Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213 A Lower Bound for the Smooth Periodic Discrepancy . . . . . . . . . . . . . . . . . . . . 3244 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

Approximation by Trigonometric Polynomials in StechkinMajorant Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331Sergey S. Volosivets1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Definitions and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3343 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

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On Multivariate Sampling of a Class of Integral Transforms . . . . . . . . . . . . . . . 347Ahmed I. Zayed1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3472 The Fractional Fourier Transform.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3503 Generalizations of the Fractional Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 3524 The Linear Canonical Transform.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3545 Sampling of the Two-Dimensional Linear Canonical Transform.. . . . . . . . . 355References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Applied and Numerical Harmonic Analysis (94 volumes) . . . . . . . . . . . . . . . . . . . 369

Part IYingkang

Yingkang Hu, 1949–2016. Photo courtesy of Xia Lu, photographer unknown

2 I Yingkang

Travel was one of Yingkang’s passion. Photo courtesy of Xia Lu, photographer unknown

I Yingkang 3

Somewhere in China. Photocourtesy of Xia Lu,photographer unknown

4 I Yingkang

Fishing was Yingkang’s yet another passion. With Florida Gar in hands. Photo courtesy of Xia Lu,photographer unknown

I Yingkang 5

And, certainly, art. . . . Photocourtesy of Xia Lu,photographer unknown

Remembering Professor Yingkang Hu

Martha Abell

After his sudden passing on March 11, 2016, colleagues of Professor Yingkang Hufrom around the globe came together at The International Conference on Approx-imation Theory, May 8–11, 2017, in Savannah, Georgia, to honor his memory asa leader in the field and to reflect on the tremendous impact he left on all aroundhim. The conference provided a great opportunity to think back on Yingkang’sacademic accomplishments and contributions over a near 30-year career. In addition,conference attendees were able to reminisce about their interactions with Yingkangon both professional and personal levels. Many who knew Yingkang providedwritten memorials in his honor in advance of the conference, which were assembledin a booklet distributed on May 9 at the “Afternoon in Honor of Yingkang Hu.”Several words stand out when thinking back on Yingkang’s life as evidenced in thememorials, colleague, teacher, scholar, mentor, and friend. Colleagues and studentsalike remember him as an inspirational scholar, a great teacher, and a humble servantto the profession.

Yingkang was a wonderful colleague because of his supportive and collegialnature. He was always willing to do his part as a departmental citizen, whetheras a cheerful committee member or as the committee chair. As one of the moresenior faculty members in the department, Yingkang was asked to chair our searchcommittees many times and it was there where he was able to leave a lasting legacyby identifying strong candidates who were excellent teachers and scholars. In fact,many of our younger departmental colleagues fondly remember their first contactwith Georgia Southern University through Yingkang when he called them for aphone interview or with an invitation for a campus interview. Many others reflectback on how he took time from his schedule to show them around the community

M. Abell (�)Georgia Southern University, Statesboro, GA, USAe-mail: martha@georgiasouthern.edu