Springer Monographs in Mathematics

Feng DaiYuan Xu

Approximation Theory and Harmonic Analysis on Spheres and Balls

Springer Monographs in Mathematics

For further volumes:http://www.springer.com/series/3733

Feng Dai • Yuan Xu

Approximation Theoryand Harmonic Analysison Spheres and Balls

123

Feng DaiDepartment of Mathematical

and Statistical SciencesUniversity of AlbertaEdmonton, AB, Canada

Yuan XuDepartment of MathematicsUniversity of OregonEugene, OR, USA

ISSN 1439-7382ISBN 978-1-4614-6659-8 ISBN 978-1-4614-6660-4 (eBook)DOI 10.1007/978-1-4614-6660-4Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2013934217

Mathematics Subject Classification: 41Axx, 42Bxx, 42Cxx, 65Dxx

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Preface

This book is written as an introduction to analysis on the sphere and on the ball,and it provides a cohesive account of recent developments in approximation theoryand harmonic analysis on these domains. Analysis on the unit sphere appears aspart of Fourier analysis, in the study of homogeneous spaces, and in several fieldsin applied mathematics, from numerical analysis to geoscience, and it has seenincreased activity in recent years. Its materials, however, are mostly scattered inpapers and sections of books that cover more general topics. Our goals are twofold.The first is to provide a self-contained background for readers who are interested inanalysis on the sphere. The second is to give a complete treatment of some recentadvances in approximation theory and harmonic analysis on the sphere developed inthe last fifteen years or so, of which both authors are among the earnest participants,and several chapters of the book are based on materials from their own research.

The book is loosely divided into four parts. The first part deals with analysison the sphere with respect to the surface measure dσ , the only rotation-invariantmeasure on the sphere. We give a self-contained exposition on spherical harmonics,written with analysis in mind, in the first chapter, and present classical results ofharmonic analysis on the sphere, including convolution structure, Cesaro summa-bility of orthogonal expansions, the Littlewood–Paley theory, and the multipliertheorem due to Bonami–Clerc in the next two chapters. Approximation on thesphere is discussed in the fourth section, where a recent characterization of bestapproximation by polynomials on the sphere is given in terms of a modulus ofsmoothness and its equivalent K-functional. An introduction to cubature formulas,which are necessary for discretizing integrals to obtain discrete processes ofapproximation, is given in the sixth chapter. A recent proof of a conjecture onspherical design, synonym of equal-weight cubature formulas, by Bondarenko,Radchenko, and Viazovska, is included, for which the necessary ingredient of theMarcinkiewicz–Zygmund inequality is established in the fifth chapter, where theinequality and several others are established for the doubling weight on the sphere.

The second part discusses analysis in weighted spaces on the sphere. Thebackground of this part is a far-reaching extension of spherical harmonics due toC. Dunkl, in which the role of the orthogonal group is replaced by a finite reflection

v

vi Preface

group, the measure dσ is replaced by h2κ(x)dσ , where hκ is a weight function

invariant under a reflection group with κ being a parameter, and spherical harmonicsare replaced by h-spherical harmonics associated with the Dunkl operators, a familyof commuting differential–difference operators that replace partial derivatives. Thestudy of h-spherical harmonic expansions started about fifteen years ago. Manydeeper results in analysis were established only in the case of the group Z

d2 ,

for which hκ is given by hκ(x) = ∏di=1 |xi|κi . In order to avoid heavy algebraic

preparations, we give a self-contained exposition of Dunkl’s theory in the case ofZ

d2, which is composed to highlight its parallel to the theory of spherical harmonics.

Most results on ordinary spherical harmonic expansions can be extended to h-spherical harmonic expansions, including finer Lp results on projection operatorsand the Cesaro means, maximal functions, and multiplier theorem, as well as acharacterization of best approximation that was developed by many authors. Wegive complete proofs of these results, which are more challenging than proofs forclassical results for dσ , and in fact, in some cases, simplify those proofs for classicalresults when the parameters κ are set to zero.

The third part deals with analysis on the unit ball and on the simplex. There areclose relations between analysis on spheres and that on balls of different dimensions,which enables us to utilize the results in the part two to develop a parallel theoryfor approximation theory and harmonic analysis on the unit ball. There is also aconnection between analysis on the ball and that on the simplex, which carries much,but not all, of analysis on the ball over to the simplex. These results are composedin parallel to the development on the sphere.

The fourth part consists of one chapter, the last chapter of the book, whichdiscusses five topics related to the main theme of the book: highly localizedpolynomial frames, distribution of nodes of positive cubature, positive and strictlypositive definite functions, asymptotics of minimal discrete energy, and computer-ized tomography.

Analysis on the sphere has seen increased activity in the past two decades. Thereare other related topics that we decided not to include, for example scattered datainterpolation, applications of spherical radial basis functions (zonal functions), andnumerical or computational analysis on the sphere. These topics are more closelyrelated to the applied and computational branches of approximation theory. Ourchoices, dictated by our own strengths and limitations, are those topics that areclosely related to the main theme—approximation theory and harmonic analysis—of this book.

We keep the references in the text to a mininum and leave references andhistorical remarks to the last section of each chapter, entitled “Notes and FurtherResults,” where we also point out further results related to the materials in thechapter. Some common notation and terminology are given in the preamble at thefront of the book, and there are two fairly detailed indexes: a subject index and asymbol index.

During the preparation of this book, we were granted a “Research in Team”for a week at the Banff International Research Station and two months at the

Preface vii

Centre de Recerca Matematica, Barcelona, where we participated in the programApproximation Theory and Fourier Analysis. We are grateful to both institutions. Wethank especially the organizer, Sergey Tikhonov, of the CRM program for his helpin arranging our visit. We also thank Professor Heping Wang, of Capital NormalUniversity, China, for his assistance in our proof of the area-regular decompositionof the sphere. The first author is greatly indebted to Professor Zeev Ditzian forhis generous help and constant encouragement. The second author used the draftof the book in a seminar course at the University of Oregon, and he thanks hiscolleagues Marcin Bownik and Karol Dziedziul (Technical University of Gdansk,Poland) and graduate students Thomas Bell, Nathan Perlmutter, Christopher Shum,David Steinberg, and Li-An Wang for keeping the course going. We thank oureditor, Kaitlin Leach, of Springer, for her professional advice and patience duringthe preparation of our manuscript, and we thank the copy editor David Kramer atSpringer for numerous grammatical and stylish corrections. Finally, we gratefullyacknowledge the grant support from NSERC Canada under grant RGPIN 311678-2010 (F.D.) and the National Science Foundation under grant DMS-1106113 (Y.X.)and a grant from the Simons Foundation (# 209057 to Yuan Xu).

Edmonton, Canada Feng DaiEugene, OR Yuan Xu

To My TeacherProfessor Kunyang Wang

F.D.

To LitianWith Appreciation

Y.X.

Contents

Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Space of Spherical Harmonics and Orthogonal Bases. . . . . . . . . . . . . . 11.2 Projection Operators and Zonal Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Zonal Basis of Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Laplace–Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Spherical Harmonics in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . 171.6 Spherical Harmonics in Two and Three Variables . . . . . . . . . . . . . . . . . . 19

1.6.1 Spherical Harmonics in Two Variables . . . . . . . . . . . . . . . . . . . . 191.6.2 Spherical Harmonics in Three Variables . . . . . . . . . . . . . . . . . . 20

1.7 Representation of the Rotation Group .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.8 Angular Derivatives and the Laplace–Beltrami Operator . . . . . . . . . . 231.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Convolution Operator and Spherical Harmonic Expansion . . . . . . . . . . . 292.1 Convolution and Translation Operators on the Sphere . . . . . . . . . . . . . 292.2 Fourier Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 The Hardy–Littlewood Maximal Function .. . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Spherical Harmonic Series and Cesaro Means. . . . . . . . . . . . . . . . . . . . . . 402.5 Convergence of Cesaro Means: Further Results . . . . . . . . . . . . . . . . . . . . 432.6 Near-Best Approximation Operators and Highly

Localized Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.7 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Littlewood–Paley Theory and the Multiplier Theorem . . . . . . . . . . . . . . . . 533.1 Analysis on Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Littlewood–Paley Theory on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 The Marcinkiewicz Multiplier Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4 The Littlewood–Paley Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 The Riesz Transform on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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3.5.1 Fractional Laplace–Beltrami Operatorand Riesz Transform.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5.2 Proof of Lemma 3.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.6 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Approximation on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1 Approximation by Trigonometric Polynomials .. . . . . . . . . . . . . . . . . . . . 804.2 Modulus of Smoothness on the Unit Sphere .. . . . . . . . . . . . . . . . . . . . . . . 854.3 A Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.4 Characterization of Best Approximation .. . . . . . . . . . . . . . . . . . . . . . . . . . . 934.5 K-Functionals and Approximation in Sobolev Space . . . . . . . . . . . . . . 954.6 Computational Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.7 Other Moduli of Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.8 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Weighted Polynomial Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1 Doubling Weights on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 A Maximal Function for Spherical Polynomials. . . . . . . . . . . . . . . . . . . . 1115.3 The Marcinkiewicz–Zygmund Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4 Further Inequalities Between Sums and Integrals . . . . . . . . . . . . . . . . . . 1185.5 Nikolskii and Bernstein Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.6 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Cubature Formulas on Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.1 Cubature Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Product-Type Cubature Formulas on the Sphere . . . . . . . . . . . . . . . . . . . 1326.3 Positive Cubature Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.4 Area-Regular Partitions of Sd-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.5 Spherical Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.6 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7 Harmonic Analysis Associated with Reflection Groups . . . . . . . . . . . . . . . . 1557.1 Dunkl Operators and h-Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 1567.2 Projection Operator and Intertwining Operator .. . . . . . . . . . . . . . . . . . . . 1607.3 h-Harmonics for a General Finite Reflection Group.. . . . . . . . . . . . . . . 1667.4 Convolution and h-Harmonic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.5 Maximal Functions and the Multiplier Theorem . . . . . . . . . . . . . . . . . . . 1727.6 Maximal Function for Zd

2-Invariant Weight . . . . . . . . . . . . . . . . . . . . . . . . . 1797.7 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8 Boundedness of Projection Operators and Cesaro Means . . . . . . . . . . . . . 1898.1 Boundedness of Cesaro Means Above the Critical Index . . . . . . . . . . 1898.2 A Multiple Beta Integral of the Jacobi Polynomials . . . . . . . . . . . . . . . . 1918.3 Pointwise Estimation of the Kernel Functions . . . . . . . . . . . . . . . . . . . . . . 1988.4 Proof of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2008.5 Lower Bound for Generalized Gegenbauer Expansion .. . . . . . . . . . . . 2048.6 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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9 Projection Operators and Cesaro Means in Lp Spaces . . . . . . . . . . . . . . . . . 2139.1 Boundedness of Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.2 Boundedness of Cesaro Means in Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . 217

9.2.1 Proof of Theorem 9.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.2.2 Proof of Theorem 9.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

9.3 Local Estimates of the Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . 2259.3.1 Proof of Theorem 9.1.2, Case I: γ < σκ − d−2

2 . . . . . . . . . . . . 2269.3.2 Proof of Theorem 9.1.2, Case II: � = ��− d−2

2 . . . . . . . . . . 2349.4 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

10 Weighted Best Approximation by Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . 24110.1 Moduli of Smoothness and Best Approximation . . . . . . . . . . . . . . . . . . . 24110.2 Fractional Powers of the Spherical h-Laplacian . . . . . . . . . . . . . . . . . . . . 24510.3 K-Functionals and Best Approximation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24810.4 Equivalence of the First Modulus

and the K-Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.5 Equivalence of the Second Modulus and the K-Functional . . . . . . . . 25410.6 Further Properties of Moduli of Smoothness . . . . . . . . . . . . . . . . . . . . . . . 26110.7 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

11 Harmonic Analysis on the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.1 Orthogonal Structure on the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.2 Convolution and Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27211.3 Maximal Functions and a Multiplier Theorem.. . . . . . . . . . . . . . . . . . . . . 27611.4 Projection Operators and Cesaro Means on the Ball . . . . . . . . . . . . . . . 28111.5 Near-Best-Approximation Operators and Highly

Localized Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.6 Cubature Formulas on the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

11.6.1 Cubature Formulas on the Ball and on the Sphere .. . . . . . . 28711.6.2 Positive Cubature Formulas and the MZ Inequality . . . . . . 28911.6.3 Product-Type Cubature Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 291

11.7 Orthogonal Structure on Spheres and on Balls . . . . . . . . . . . . . . . . . . . . . 29311.8 Notes and Further Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

12 Polynomial Approximation on the Unit Ball. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29712.1 Algebraic Polynomial Approximation on an Interval . . . . . . . . . . . . . . 29812.2 The First Modulus of Smoothness and K-Functional .. . . . . . . . . . . . . . 303

12.2.1 Projection from Sphere to Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30312.2.2 Modulus of Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30512.2.3 Weighted K-Functional and Equivalence .. . . . . . . . . . . . . . . . . 30712.2.4 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30912.2.5 The Moduli of Smoothness on [−1,1] . . . . . . . . . . . . . . . . . . . . . 31112.2.6 Computational Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

12.3 The Second Modulus of Smoothness and K-Functional. . . . . . . . . . . . 31312.3.1 Analogue of the Ditzian–Totik K-Functional . . . . . . . . . . . . . 31412.3.2 Direct and Inverse Theorems Using the K-Functional .. . . 319