Introduction to Fourier series [1 ed.] 0824796101, 9781003067443

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PURE

AND

APPLIED

MATHEMATICS

A Series of Monographs and Textbooks

INTRODUCTION TO FOURIER SERIES

0 .5

-4

2

-2

- 0 .5

Rupert Lasser

INTRODUCTION TO FOURIER SERIES

PURE AND APPLIED MATHEMATICS A Program of Monograpm, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newarlc, Delaware

Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passmmi University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

Gian-Carlo Rota Massachusetts Institute of Technology

Marvin Marcus University of California, Sanla Barbara

David L. Russell Virginia Polytechnic Institute and State University

W. S. Massey Yale University

Walter Schempp Universitiil Siegen

Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATIIEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (19701 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood. trans.) (19701 4. 8. N. Pshenichnyi, Necessary Conditions for an Extremum IL. Neustadt, translation ed.; K. Makowski, trans.) (19711 5. L. Narici et al•• Functional Analysis and Valuation Theory (19711 6. S.S. Passman, Infinite Group Rings (1971) 7. L. Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971, 19721 8. W. Boothby and G. L. Weiss. eds., Symmetric Spaces 119721 9. Y. Matsushima, Differentiable Manifolds IE. T. Kobayashi, trans.I (19721 10. L. E. Ward, Jr., Topology (19721 11. A. Babakhanian, Cohomological Methods in Group Theory I 19721 12. R. Gilmer, Multiplicative Ideal Theory 119721 13. J. Yeh, Stochastic Processes and the Wiener Integral 119731 14. J. Baffos-Neto, Introduction to the Theory of Distributions (19731 15. R. Larsen, Functional Analysis 119731 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles 119731 17. C. Procesi, Rings with Polynomial Identities ( 1973) 18. R. Hermann, Geometry, Physics, and Systems (19731 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (19731 20. J. Dieudonn~, Introduction to the Theory of Formal Groups (19731 21. I. Vaisman, Cohomology and Differential Forms 119731 22. 8.-Y. Chen, Geometry of Submanifolds 119731 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 19751 24. R. Larsen, Banach Algebras 119731 25. R. 0. Kujala and A. L. Vitter, eds., Value Distribution Theory: Part A; Part 8: Deficit and Bezout Estimates by Wilhelm Stoll 119731 26. K. 8. Stolarsky, Algebraic Numbers and Diophantine Approximation 119741 27. A. R. Magid, The Separable Galois Theory of Commutative Rings 119741 28. 8. R. McDonald, Finite Rings with Identity (19741 29. J. Satake, Linear Algebra (S. Koh et al., trans.I (19751 30. J. S. Golan, Localization of Noncommutative Rings (19751 31. G. Klambauer, Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology (19761 33. K. R. Goodearl, Ring Theory (19761 34. L. E. Mansfield, Linear Algebra with Geometric Applications ( 19761 35. N. J. Pullman, Matrix Theory and Its Applications (19761 36. 8. R. McDonald, Geometric Algebra Over Local Rings (19761 37. C. W. Groetsch, Generalized Inverses of Linear Operators 119771 38. J. E. Kuczkowski and J. L. Gersting, Abstract Algebra I 19771 39 . C. 0. Christenson and W. L. Voxman, Aspects of Topology 119771 40. M. Nagata, Field Theory 119771 41. R. L. Long, Algebraic Number Theory ( 19771 42. W. F. Pfeffer, Integrals and Measures (19771 43. R. L. Wheeden and A. Zygmund, Measure and Integral ( 19771 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (19781 45. K. Hrbacek and T. Jech, Introduction to Set Theory (19781 46. W. S. Massey, Homology and Cohomology Theory (19781 47. M. Marcus, Introduction to Modern Algebra 119781 48. E. C. Young, Vector and Tensor Analysis (19781 49. S. 8. Nadler, Jr., Hyperspaces of Sets (19781 50. S. K. Segal, Topics in Group Kings (19781 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis ( 19781 52. L. Corwin and R. Szczarba, Calculus in Vector Spaces 11979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals (19791

54. 55. 56. 57. 58. 59.

60. 61 . 62. 63. 64. 65. 66. 67 . 68. 69. 70.

71 .

72. 73. 74. 75. 76. 77. 78. 79. 80. 81 . 82 . 83. 84. 85. 86. 87. 88. 89. 90. 91 . 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103.

J. Cronin, Differential Equations ( 19801 C. W. Groetsch, Elements of Applicable Functional Analysis 11980) /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry 119801 H. I. Freedan, Deterministic Mathematical Models in Population Ecology (19801 S. B. Chae, Lebesgue Integration (19801 C. S. Rees et al., Theory and Applications of Fourier Analysis ( 19811 L. Nachbin, Introduction to Functional Analysis IA. M. Aron, trans.I (19811 G. Orzech and M. Orzech, Plane Algebraic Curves 119811 R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis

(19811

W. L. Voxman and R. H. Goerschel, Advanced Calculus ( 1 9811 L. J. Corwin and R. H. Szczarba, Multivariable Calculus ( 19821 V. I. lstrltescu, Introduction to Linear Operator Theory 119811 R. D. Jii~inen, Finite and Infinite Dimensional Linear Spaces ( 19811 J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry 119811 D. L. Armacost, The Structure of Locally Compact Abelian Groups (19811 J . W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1981) K. H. Kim, Boolean Matrix Theory and Applications 119821 T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments ( 19821 D. B.Gauld, Differential Topology (19821 R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (19831 M. Carmeli, Statistical Theory and Random Matrices 119831 J. H. Carruth et a/., The Theory of Topological Semigroups I 19831 R. L. Faber, Differential Geometry and Relativity Theory 119831 S. Barnett, Polynomials and Linear Control Systems 119831 G. Karpilovsky, Commutative Group Algebras 119831 F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings 119831 /. Vaisman, A First Course in Differential Geometry 119841 G. W. Swan, Applications of Optimal Control Theory in Biomedicine (19841 T. Petrie andJ. D. Randall, Transformation Groups on Manifolds 11984) K. Goebel and S . Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (19841 T. Albu and C. Ntlsttlsescu, Relative Finiteness in Module Theory 119841 K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1984) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings 119841 B. R. McDonald, Linear Algebra Over Commutative Rings (19841 M. Namba, Geometry of Projective Algebraic Curves 11984) G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) M. R. Bremner et al., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (19851 A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces ( 1985) A. J. Jerri, Introduction to Integral Equations with Applications (1985) G. Karpilovsky, Projective Representations of Finite Groups (1985) L. Narici and E. 8eckenstein, Topological Vector Spaces (1985) J. Weeks, The Shape of Space (19851 P.R. Gribik and K. 0 . Kortanek, Extremal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis

(19861

G. D. Crown et al., Abstract Algebra ( 1986) J. H. Carruth eta/., The Theory of Topological Semigroups, Volume 2119861 R. S. Doran and V. A. Belfi, Characterizations of C"-Algebras (19861 M. W. Jeter, Mathematical Programming (19861 M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications I 19861 104. A . Verschoren, Relative Invariants of Sheaves I 19871 105. R. A. Usmani, Applied Linear Algebra (19871 106. P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p

> 0 (1987) 107. J. A. Reneke et al., Structured Hereditary Systems ( 1987) 108. H. Busemann and 8 . 8. Phadke, Spaces with Distinguished Geodesics ( 19871 109. R. Harte, lnvertibility and Singularity for Bounded Linear Operators I 19881

110. G. S. Lsdde et sf., Oscillation Theory of Differential Equations with Deviating Arguments 119871 111. L. Dudkin et al., Iterative Aggregation Theory ( 19871 112. T. Okubo, Differential Geometry (19871 113. D. L. Stanc/ and M . L. Stancl, Real Analysis with Point-Set Topology 119871 114. T. C. Gard, Introduction to Stochastic Differential Equations I 19881 115. S.S. Abhyankar, Enumerative Combinatorics of Young Tableaux 119881 116. H. Strede and R. Fsmstemer, Modular Lie Algebras and Their Representations 119881 117. J. A. Huckaba, Commutative Rings with Zero Divisors (19881 118. W. D. Wallis, Combinatorial Designs (19881 119. W. Wies/aw, Topological Fields 119881 120. G. Ka,Pitovsky, Field Theory (19881 121. S. Csenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings ( 1 9891 122. W. Kozlowski, Modular Function Spaces 119881 123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps 119891 124. M. Pavel, Fundamentals of Pattern Recognition 11 9891 125. V. Lskshmiksntham et Bl., Stability Analysis of Nonlinear Systems (19891 126. R. Sivsrsmakrishnan, The Classical Theory of Arithmetic Functions 119891 127. N. A. Watson, Parabolic Equations on an Infinite Strip ( 19891 128. K. J. Hastings, Introduction to the Mathematics of Operations Research I 19891 129. 8. Fine, Algebraic Theory of the Bianchi Groups (19891 130. D. N. Dikranjan et al., Topological Groups 119891 131 • J. C. Morgan II, Point Set Theory I 19901 132. P. Biler and A. Witkowski, Problems in Mathematical Analysis 119901 133. H. J. Sussmsnn, Nonlinear Controllability and Optimal Control ( 19901 134. J. -P. Florens et sf., Elements of Bayesian Statistics ( 19901 135. N. Shell, Topological Fields and Near Valuations 119901 136. B. F. Doolin snd C. F. Martin, Introduction to Differential Geometry for Engineers 119901 137. S. S. Hofland, Jr., Applied Analysis by the Hilbert Space Method I 1 9901 138. J. Oknirlski, Semigroup Algebras I 19901 139. K. Zhu, Operator Theory in Function Spaces 119901 140. G. B. Price, An Introduction to Multicomplex Spaces and Functions 119911 141. R. B. Darst, Introduction to Linear Programming I 19911 142. P. L. Sschdev, Nonlinear Ordinary Differential Equations and Their Applications 11991) 143. T. Husain, Orthogonal Schauder Bases I 19911 144. J. Foran, Fundamentals of Real Analysis (19911 145. W. C. Brown, Matrices and Vector Spaces (19911 146. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces I 19911 147. J. S. Golan and T. Head, Modules and the Structures of Rings 11991) 148. C. Small, Arithmetic of Finite Fields 119911 149. K. Yang, Complex Algebraic Geometry (19911 150. D. G. Hoffman eta/., Coding Theory 119911 151. M. 0. Gonz"ez, Classical Complex Analysis ( 19921 152. M . 0. Gonztllez, Complex Analysis (19921 153. L. W. Baggett, Functional Analysis 119921 154. M. Sniedovich, Dynamic Programming 119921 155. R. P. Agarwal, Difference Equations and Inequalities (19921 156. C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis I 19921 15 7. C. Swartz, An Introduction to Functional Analysis Cl 9921 158. S. B. Nadler, Jr., Continuum Theory 119921 159. M.A. Al-Gwaiz, Theory of Distributions 119921 160. E. Perry, Geometry: Axiomatic Developments with Problem Solving 119921 161. E. Castillo and M . R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering 119921 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis 11992) 163. A. Charlier et al., Tensors and the Clifford Algebra I19921 164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations 119921 165. E. Hansen, Global Optimization Using Interval Analysis (19921

166. 167. 168. 169. 170.

171.

172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199.

S. Guerre-Delabriee, Classical Sequences in Banach Spaces (19921 Y. C. Wong, Introductory Theory of Topological Vector Spaces (19921 S. H. Kulkarni and 8. V. Umsye, Real Function Algebras 119921 W. C. Brown, Matrices Over Commutative Rings 119931 J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (19931 W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) E. C. Young, Vactor and Tensor Analysis: Second Edition 11993) T. A. Bick, Elementary Boundary Value Problems 119931 M. Pavel, Fundamentals of Pattern Recognition: Second Edition (19931 S. A. Albeverio et al., Noncommutative Distributions 119931 W. Fulks, Complex Variables 119931 M. M. Rao, Conditional Measures and Applications (19931 A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes ( 1994) P. Neittaanmlki and 0. Tibs, Optimal Control of Nonlinear Parabolic Systems (19941 J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition 119941 S. Heikki/8 and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations I 1994) X. Mao, Exponential Stability of Stochastic Differential Equations (19941 8. S. Thomson, Symmetric Properties of Real Functions (19941 J. E. Rubio, Optimization and Nonstandard Analysis (19941 J. L. Bueso et al, Compatibility, Stability, and Sheaves (19951 A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (19951 M. R. Dame/, Theory of Lattice-Ordered Groups I 19951 Z. Naniewicz and P. D. Panagiotopoulcn, Mathematical Theory of Hemivariational Inequalities and Applications I 19951 L. J. Corwin and R.H. Szczarba, Calculus in Vector Spaces: Second Edition 119951 L. H. Erbe et a/., Oscillation Theory for Functional Differential Equations (19951 S. Agsian et a/., Binary Polynomial Transforms and Nonlinear Digital Filters 119951 M. /. Gil', Norm Estimations for Operation-Valued Functions and Applications 119951 P.A. Grillet, Semlgroups: An Introduction to the Structure Theory (19951 S. Kichensssamy, Nonlinear Wave Equations 119961 V. F. Krotov, Global Methods in Optimal Control Theory (1996) K. I. Beidar et a/., Rings with Generalized Identities 119961 V. /. Amautov et al., Introduction to the Theory of Topological Rings and Modules (19961 G. Sierlcsms, Linear and Integer Programming I 1996) R. Lasser, Introduction to Fourier Series I 19961 Additions/ Volumes in Preparation

INTRODUCTION TO FOURIER SERIES Rupert Lasser Medical University of Liibeck Liibeck, Germany

0

CRC Press

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Library of Congress Cataloging-in-Publication Data Lasser, Rupert. Introduction to Fourier series I Rupert Lasser. p. cm. - (Monographs and textbooks in pure and applied mathematics ; 199) Includes bibliographical references (p. ) and index. ISBN 0-8247-9610-1 (hardcover : alk. paper) 1. Fourier series. I. Title. II. Series. QA404.L33 1996 515'.2433-dc20

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PREFACE On December 21, 1807 Jean Baptiste Joseph de Fourier (176S.1830) addressed the French Academy and made the claim that an arbitrary function defined on a finite interval could be represented as an infinite summation of cosine and sine functions. The mathematicians of that time - Fourier was an engineer - questioned the validity of Fourier's theorem. Their objections were not based on integrability of the function or on the sense of convergence. They believed that any combination of cosine and sine functions could give only infinitely differentiable functions. They could not reconcile Taylor's expansion with Fourier's expansion. Of course Taylor's concept and Fourier's concept in spite of being extremely different are not in contradiction. The Taylor series

f:

/Cn)(Zo) z"

n=O

nl

is determined by the local behaviour of/, namely by the derivatives /(zo), and the expansion gives a continuation in the vicinity of Zo· The Fourier series ft3-0()

is determined by the global behaviour of /, namely by the Fourier coeffi-

cients lc.n), and the expansion gives the function in the entire range (0, 21r(.

Consider an analytic function / and expand it in a Taylor series about zo = 0. If the radius of convergence is greater than one Fourier and Taylor series coincide on the unit circle and represent the function /(eit). When the radius of convergence is equal to one the Taylor series will not converge at some or all points on the unit circle, even when f is defined on the unit circle. The Fourier series does in general converge and represent the function I on the unit circle. Thus Taylor series break down on the unit circle, but Fourier series are still valid. The essential ideas of Fourier series are the topic of this book. One can say (very roughly) that Fourier's expansions rest on the orthogonality of the functions z --t z" = eint, n E Z, iii

iv

PREFACE

with respect to the Lebesgue measure on [O, 211'[. But our considerations are not restricted to the calculus in the corresponding Hilbert space. Expansions of functions belonging to arbitrary homogeneous Banach spaces can be investigated as well as pointwise convergence and various other topics. Our principal intention in writing this book has been to provide an introductory, self-contained, concise treatise on Fourier series, making use of some techniques of functional analysis and measure theory. We emphasize the concept of approximate identities for homogeneous Banach spaces on T (especially that of the Fejer and the Poisson kernel) and we give some applications in particular for weakly stationary stochastic processes. The text evolved from lectures given in a one year course at the Technical University of Munich. The elements of functional analysis (with proofs) used in the text are gathered in the appendix. Nevertheless it would be advantageous if the reader were familiar with basic facts concerning Banach spaces. The same is true for basic notions of measure theory, in particular for Lebesgue integration. Convergence theorems of measure theory often used in the book ca.n be found in the appendix, too. Concerning Lebesgue measure our preferred reference is (18J. This book cannot and should not cover completely such an immense theme as Fourier series. For example we do not investigate the concept of maximal functions. There are two well-known encyclopaedic monographs on classical Fourier analysis, the monumental treatises by A. Zygmund [24] and N.K. Bari [3). Of course there are further excellent books on Fourier analysis focusing on certain aspects of the subject. The reader may find many of them in the references (6], [7), [9), (13), [14] and [22). The present book is meant as an introduction to Fourier transforms on T, i.e., to Fourier series, in a clear and compact form for students with some basic knowledge of real and functional analysis and of measure theory. The first two chapters deal with basic facts of Fourier transforms of functions which are elements of homogeneous Banach spaces on T. Here we have chosen access by means of the Fejer kernel. Chapter 3 shows the pointwise convergence of Fejer and Poisson series of integrable functions. Hilbert space theory is used in Chapter 4. The concept of conjugation and projection is studied in Chapter 5 to derive results on norm convergence of the Fourier series. The main facts concerning local convergence are given in Chapter 6. These six chapters can be understood as the most elementary account of Fourier series. They should give a first general idea of Fourier analysis on T. All other sections deal mainly with approximation and convergence problems, Chapter 7 is devoted to representations by

PREFACE

v

Fourier coefficients. In particular Herglotz's theorem is proved as well as Wiener's characterization of continuous measures . The Hilbert transform is introduced and its continuity as an operator on V(T), 1 < p < oo, is derived in Chapter 8. The investigation of approximate identities is continued in Chapter 9, where the Rogosinski kernel, the Jackson kernel and many others are studied. Chapter 10 deals with triangular summation schemes""·"' n e N0 , k = 0, ... , n. The concavity (or convexity) of the sequences ""·" is fairly useful to derive approximate identities. In the next chapter the performance of the approximate identities is compared with the elements of best approximation. Among others Jackson's theorems are given and examples of Zygmund approximation sequences are introduced. Chapter 8 to 11 so far concentrate on more advanced problems of approximation. The following three chapters are devoted to H"-spaces and related topics. In Chapter 12 Poisson kernels and harmonic functions are studied, the Beurling-Helson Theorem is given as well as the theorem of F. and M. Riesz characterizing those measures µ with J'(n) = 0 for n < 0. In Chapter 13 the conjugation of approximate identities is discussed and in Chapter 14 the SzegO-Kolmogorov theorem related to SzegO's extremum problem - is derived. A dual point of view, in fa.ct starting from l 1 (Z), is presented in the chapter on absolute convergence containing also results on spectral synthesis. In Chapters 16 and 17 the major facts of the Fourier transform and Plancherel transform on R are given. Applying these results the Poisson summation formula is proved in Chapter 18. Its usefulness is stressed by deriving the Whittaker-Shannon sampling theorem on R. A list of exercises is added to each chapter. Most of them are supplementary to the text or simply present interesting results on the subject. The problems range in difficulty from simple to hard. During the writing of this book I benefitted from the advice and constructive criticism of many colleagues, and thank them all. In particular I thank Frank Filbir and Roland Girgensohn for making valuable suggestions for improvement. Special thanks are due to Manfred Tasche for the meticulous care with which he examined the whole manuscript; his comments were extremely helpful. I am especially indebted to Martina Weber who performed the typesetting perfectly without losing her sense of humour. Rupert Lasser

t?\ Taylor & Francis ~ Taylor & Francis Group http://taylorandfrancis.com

CONTENTS

Preface 1 2 3 4

iii

Fourier Coefficients Approximate Identities Approximate Identities and Pointwise Convergence Square Integrable Functions

5 Convergence of Fourier Series in Norm 6 Local Convergence

7 Characterization of Fourier Coefficients 8 Hilbert Transform 9 Characterizations of Approximate Identities 10 Triangular Schemes 11 Elements of Best Approximation 12 Poisson Integrals and Hardy Spaces 13 Conjugation of Approximate Identities 14 SzegO-Kolmogorov Theorem 15 Absolute Convergence of Fourier Series 16 Fourier Transform on R 17 Plancherel Transform on R 18 Poisson Summation Formula

1 11

25

37

49

63 75 93 107 121 137 161 177 191

203 225

241

253

Appendices Appendix A: Measure Theory Appendix B: Banach Spaces Appendix C: Banach Algebras

267 271 277

References

281

Index

283

vii

Q

~

Taylor & Francis Taylor & Francis Group http://taylora ndfra ncis.com

1

FOURIER COEFFICIENTS We start with the computation of the coefficients of the Fourier expansion of a function on the torus. It is most instructive to begin with linear combinations of the basic functions z - z"', where m E Z and z E T. Thus consider a trigonometric polynomial, i.e.

P(z)

=

L N

m=-N

BmZ"', z ET,

with complex coefficients a-N• .. . , aN. The degree of Pis N if aN a_N :/:

0.

:F 0 or

Then we can determine the coefficients by

t

1 1 - { P(z)z-ndz = am 211' m•-N 211' jT where we used f kd _ {2fr iktdt _ { 211' JT z z - lo e - 0

{ z"'-ndz

jT

for for

k k

= an,

=0

':f:. 0

the orthogonality of the family {zn : n E Z}, a basic property in Fourier analysis. (Note that T is not the Cauchy integral of complex analysis.) Extending this to a formal expansion we introduce

J

Definition ( 1.1) : call

Let / E L 1 (T), where Tis the torus. For n E Z we

f(n) :=

2~ £J(z)z-"dz

then-th Fourier coefficient of/. The formal series

S(/)(z)

=

L 00

n=-oo

is called Fourier series off E L 1 (T).

1

f(n)z"

CHAPTER I

2

One of the main problems in Fourier analysis is to discover conditions guaranteeing the convergence of S(/) and to investigate whether f = S(/) is valid. We see that our task is a problem of approximation. Given a function f on the torus T, we consider trigonometric polynomials built up by Fourier coefficients (or weighted Fourier coefficients) and ask whether the Fourier polynomials approximate the function /. The weights are introduced to improve certain features of the common Fourier series. We will also investigate other approximation procedures such as by Poisson series or Gaussian series. They are limits of certain weighted (infinite) Fourier series. The general approach will be that of approximate identities. Of course, our problem depends heavily on whether we focus on local or global aspects. For global approximation properties in general the corresponding norm is important. The reader will subsequently find many figures showing the approximation of various functions by (weighted) Fourier polynomials. The following properties of the Fourier coefficients can be derived directly from their definition.

Theorem (1.2) : (1)

(!

(2)

""::L

Let f,g E L 1 (T), a EC. For all n E Z we have:

+ gf(n) = J(n) + g(n), (o/f(n) = ?J(n).

(4)

x--= -/(-n), where /(z) = /(z). (L•• /f(n) = J(n)z;n, where L •• J(z) = /(zz; 1), (e1df(n) = J(n - k), where e1c(z) = z1c.

(5)

IJ(n)I ~

(3)

/(n)

z0 ET.

11/111.

It is very important -as we will see soon- to have an operation in the Banach space L 1 (T) such that L 1 (T) becomes a Banach algebra.

Theorem (1.3) :

Let f,g

is absolutely integrable.

e L 1 (T).

For almost all z

eT

the function

FOURIER COEFFICIENTS

Denoting

2~ £/(z(- )g(()d(, 1

h(z) = we have

he L1(T) and llhll1

3

11/111119111· Also,

~

h(n) = f(n)g(n) is valid for all n

e Z.

PROOF. First note that we can assume / and g Borel measurable. The function F(z, () = /(z(- 1)g(() is Borel-measurable. Moreover, for almost all (the function z - F(z, () is a constant multiple of Le f, hence integrable and

The theorems of Tonelli and Fubini imply the integrability of {z, () F(z,(). Moreover, ( - F(z,() is integrable for almost all z ET and

2~ £1h(z)ldz = 2~ £1 2~ £ F(z,()d(ldz ~ 4! ££1F(z,()ldzd( 2

= ll/ll1119lli· Remains to prove: h(n)

h(n)

= 2~

= f(n)g(n).

In fact, it follows

£ h(z)z-ndz = ! £ £/(zC )(zC )-ng(()(-"dz

= 2~

£

f(z)z-ndz

1

1

2~ £g(()Cnd( = f(n)g(n). •

4

2

d(

Definition (1.4) : Let /, g e L 1(T). The function h of Theorem (1.3) is denoted by / • g and is called the convolution of I and g. A direct (though somewhat cumbersome) calculation shows that the convolution is a commutative, associative and (with the addition) distributive operation in L 1 (T). Hence, L 1 (T) is a commutative Banach algebra. In

CHAPTER I

4

particular, Theorem (1.3} says that the mapping -: (L 1 (T}, •) - {i°0 (Z), ·) called Fourier transformation is an algebra homomorphism (' ·' is the componentwise multiplication in l 00 (Z).). The following useful result shows that the convolution of an arbitrary f E L 1 (T) and a trigonometric polynomial again yields a trigonometric polynomial.

Theorem (1.5) :

Let

f

E L 1(T) and P(z)

Then

I. P(z) =

L N

m•-N

=

N

L

m•-N

amzm.

amf(m)zm.

PROOF. N

L amf(()(zc rd( 271' jT m•-N

f • P(z) = ..!.. f

= =

t

mm-N

amzm

1

2~ L/(()Cmd( T

N

L:

m•-N

amlzm. •

When we look at the functions on T as 271'-periodic functions on R, differentiation and integration act on the Fourier coefficients as follows:

Theorem {1.6) : (1) Let

f

E L 1 (T),

f(O)

= 0.

Then, by F(e") :=

Lt f(e'-r)dT

a con-

tinuous function on Tis defined fort E IO, 271'1, and F(n) = ,~f(n) for n #: 0 is valid. (2) Let

J be a differentiable function on T. f(n)

If J' is integrable, then

= ;_(J')ln) for all n #: 0 holds. in

FOURIER COEFFICIENTS

5

PROOF. The continuity of F(z) in (1) is obvious for z 1' 1. The continuity in z = 1 follows from

F{ei2ir) - F(e'0 )

= fo

2 1r

/{e'")dr

= 27rf(O) = 0.

Integration by parts yields (see [11; p. 287]):

Finally {2) holds by (1), compare [18, Theorem 7.21}. Note that (!'no) = 0

is valid by the 27r-periodicity of /.

Corollary

(1.z>:

entiable, then

If f is a function on T and twice continuously differ-

L 00

n--oo

PROOF. Note that

+

I

lf{n)I = (f'!'Cn) in

lf(n)I < oo •

I=I (f"nn) I~ 2-111 111· + -n2

n2

11

Often we extend our investigations to certain linear subspaces of L 1(T) that bear a norm of their own.

Definition (1.8) : A homogeneous Banach space on T is a linear subspace B of L 1{T} equipped with a norm II lls satisfying the following properties: (1) The function en(z) = zn belongs to B for all n E Z.

CHAPTER I

6

II llB· With f E B and z E T also Lal belongs to B, and llLa/lls = 11/lls

(2) B is (3)

a Banach space with respect to II lls, and II 111

$

holds.

(4) The mapping z - Lal (z ET) is continuous with respect to for all f EB.

11 lls

Examples:

{1) B

= C(T) = {/: T- C; /continuous on T} with

11/lloo =

max l/(z)I is a homogeneous Banach space on T. zET

(2) Let 1 $ p < oo. The space V'(T) with Banach space on T. (3) Let A(T) denote { f E C(T):

11/llA(T) =

11/11,,

is a homogeneous

1E 11 (Z)} with the norm

L 00

n::a-oo

lf 0, / E V(T), exists a continuous function g on T such that II/ - gll,, < E. The mapping z - Lzg is continuous with respect to II II,,, since g is uniformly continuous. Then z - Lal is also continuous with respect to II II,,. The other properties

FOURIER COEFFICIENTS

7

characterizing homogeneous Banach spaces obviously hold for any of the examples above. Note that L00 (T) with

11/lloo =

ess sup l/(z)I aET

is not a homogeneous Banach space on T. Indeed, the mapping z-+ L,f is not continuous in e-•to with respect to 1111 00 , where we define /(e't) = 0 for t e (0, to) and /(e't) = 1 for t e )t0 , 21"[ and t 0 e JO, 21"[ fixed. In Chapter 15 we will introduce another homogeneous Banach space B = U(T). Furthermore in Chapter 11 it is shown that certain Lipschitz spaces are homogeneous Banach spaces on T.

EXERCISES (1) Let a trigonometric polynomial P be given in the real form

P(t) =

1

n

2ao + :~)aA: cos kt+ bk sin kt) A:ml

as well as in the complex form

P(t) =

L n

CA:e'kt

A:--n

Determine the relationship between a1c, b1c and c1c. Which condition on the CA: 's corresponds to the case that the coefficients ak, b1c are real? (2) Determine the coefficients a1c, b1c by writing the (formal) Fourier series 00

L:

/(k)e'"t

lc•-00

off

e L 1(T)

in the form 1

00

L(a1c cos kt+ b1c sin kt). 2ao + lc=l

CHAPTER I

8

Show that the coefficients a1c and b1c are real, if and only if / is real-valued. What are the conditions on a1c and b1c, if f is an even

or odd function?

(3) Prove Theorem (1.2).

(4) Show that the convolution is a commutative, 88SOciative and distributive operation in L 1 (T). (5) Let /,g: T- C be two continuous functions. Show that/• g(z) is the uniform limit of the Riemann sums

where 21rj t ti = m+-1 , ..;

= eilj , J. = 0, ... , m + 1 .

Also, prove that / • g is continuous. (6) Determine the Fourier coefficients of the following 211'-periodic functions: e(t)

={ ~

/(t} =

{~I

g(t)

={ ~-

h(t)

=

t

ltl

for

ltl < 4

for

4~ltl~11'

for for for

-l 0, then there exists an a e Q

L" I/(ze'T") + /(ze-'T") - I 0

2

a dT

1 r" +;;Jo la - /(z)ldT ~ E,

provided h is chosen small enough.

+

Remark: The proof of Lemma (3.2) shows that the asymmetric relations

also hold for almost all z

e T.

Combining Theorem (3.1) and Lemma (3.2), we obtain:

Corollary (3.3) :

zeT.

Let

I e

L 1(T), then '7n(/)(z) - /(z) for almost all

The convergence of '7n(/)(z) implies a statement on convergence with respect to Abel summability as we will see immediately. One says that a series

APPROXIMATE IDENTITIES AND POINTWISE CONVERGENCE 00

00

"a1c is Abel summable to S, if r-1Um "t""' a1cr• L., L., 1c-o 1c- o called Ceslro summable to S, if n-.oo lim II1I < ~- This proves the assertion. t

we get

2

0 so that 0 < 1 - r < 6 implies

Theorem (3.5) : Let / E L 1 (T). Let (Pr)re(o,t( denote the Poisson kernel on T. Then for almost all z ET:

lim Pr* /(z)

r-t-

= r-tlim

00

"'""" f(k)rl'=lzk ~

lt•-00

PROOF. Applying Lemma (3.4) for ao = /(0) and a1c the assertion follows from Corollary (3.3). t

= /(z).

= /(k)~+ /(-k)z- 11 ,

The Lebesgue Corollary (3.3}, combined with the following Lemma (3.6), yields a result on Fourier series convergence provided the Fourier coefficients satisfy a specific asymptotic condition.

APPROXIMATE IDENTITIES AND POINTWISE CONVERGENCE

(Hardy)

Lemma (3. 6) :

L

00

If the series

is Cesaro summable to S and if an

Bk

k=O

then the series

31

L 00

Bk

= 0( -k) 85 n -+ oo,

converges to S.

k-0

PROOF. Denote as previously

Consider for k, n O'n,A:

=

Sn

e N,

k

< n,

+ Sn+l + ·••+ Sn+k-1 k

=

(n + k)O'n+A:-1 k

= (1 + ~) O'n+k-1 - ~O'n-1· -+ oo with f :5 C (C a constant)

So, clearly, for n, k when Un-+ S. Additional transformations yield O'n,A:

=

n+A:-1

"L.J

J•O

k

+

.

n

jaO

= Sn + Hence, from

.

n-k-01 -J kn-3 BJ - " L.J

L

n+A:-1 (

.

1- J

j•n+l

=

n+k-1

~n

"L.J B;

J•O )

- nO'n-1

O'n,k -+

+

n+k-1

S follows .

"L.J -k-BJ n-J

j•n+l

B;.

an = 0( ~) we get with a convenient constant M: lun,A: - Snl

:5

L

n+A:-1 j•n+l

IB;I

:5 M

n+A:-1 l

k _ l

L -:- :5 M-n-.

j•n+lJ

Let 6 > O, and denote E = d, and k = (m) + 1 (where (xj stands for the integer part of x). This gives us first ~ :5 E and second I :5 ~- The results above imply first lun,A: -Snl :5 ME = ~ and second that there exists no E N such that lun,lc - SI < for all n ~ no and k = (m) + 1 85 before. Combined this reduces to ISn - SI < 6 for n ~ no. t

!

CHAPTER III

32

Theorem (3.7) : Let f E L 1(T) with /(n) = O(~) as lnl - oo. Then Sn(/)(z) converge for almost all z ET to /(z). More precisely, for given

z E T the sequences Sn(/)(z) and O'n(/)(z) converge towards the same limit or both sequences diverge. PROOF. Let ao = f= ~ hf(z)g(z)dz

2

is a scalar product turning L2 (T) into a Hilbert space (see Appendix B).

This fact will be used in this chapter intensively. If one assumes that the function f belongs to L 2 (T) - which is often done in applications - Hilbert space theory is an appropriate tool to investigate the Fourier series of J. Therefore we start with some basic facts concerning Hilbert spaces.

Lemma (4.1) : If {xnl:'- 1 is an orthonormal subset of a Hilbert space H (over C), then

is valid for all complex numbers

a 1 , ••• ,aN.

PROOF. The orthonormality yields

=

L4nLa1i: = Ll4nl N

N

N

n-1

11:-1

n•l

37

2

•

t

CHAPTER IV

38

Let {xn}neN be an orthonormal subset of a Hilbert

Corollary (4.2):

L: ltinl 2 < oo, the series 00

space H. Then for any sequence (Bn)nEN with

L OnXn converges in H. 00

n=-1

PROOF. We have to show that

is a Cauchy sequence in H . But for N

>M

as M-oo. •

Lemma {4.!l) : Let H be a Hilbert space and {xn}:.1 an orthonormal subset of H. For x e H denote 4n = < x,xn >. Then

PROOF. A direct computation gives 2

N

X- LBnXn n .. 1

= llxll

2

=

LBn < Xn,X > - LOn < x,xn > + E1anl

nsl

= 11x11 2 -

N

E 10n1 n=l

2

• •

2

SQUARE INTEGRABLE FUNCTIONS

39

Lemma (4.J) : Let {zo}aeA be an orthonormal subset of a Hilbert space Hand let x EH. Then< x,x 0 > ~ 0 only for countable et EA.

PROOF. Because of Lemma (4.3) the set An

={

Q

E A:

I < X, Xa > 12 > 11~1 } 2

contains at the most n-1 elements. Since {et

EA:< x,xa > ::/: 0} ~

UAn, 00

n•l

the assertion follows. •

Corollary (4.5) : (Bessel's inequality) Let {xa}aeA be an orthonormal subset of a Hilbert space H. For x e H let a 0 = < x, x 0 > (a0 are called generalized Fourier coefficients). Then we have

L laal aEA

2

~ llxll 2 •

PROOF. The assertion follows from Lemma (4.3) and Lemma (4.4) combined. •

We denote an orthonormal subset {xa}oeA of a Hilbert Definition (4.6) : space orthonormal basis of H, if the condition < x,x0 > = 0 for all a e A implies x = 0. (Occasionally, orthonormal bases are termed complete orthonormal ayatema.)

Lemma (,4. V : Let {xa}oeA be an orthonormal subset of a Hilbert space H . The following assertions are equivalent: (1) {zo}aeA is an orthonormal basis.

(2)

llzll 2 = E

(3)

X

=E

oEA

aEA

I< x,x0

< x, Xa

> 12 for all x EH (Parseval).

> Xa for all X E H .

40

CHAPTER IV

PROOF. Note that by Lemma (4.4) summation in (2) and (3) is actually done by countable subsets of A. The equivalence of (2) and (3) is directly

implied by Lemma (4.3). Evidently, (3) also implies (1). Tu demonstrate

that (1) also implies (3) note that with Corollary (4.2) and Corollary (4.5) the series

L

xa

a EA

converges in H . Setting

y=

L

aEA

xa

we find < y, Xa > = < x, x 0 > for all a E A. This means, for all a EA. By (1) the element y is equal to x. t

Remark: Note that the series

LI < x,x > I
Xa

< y-x, x 0 > = 0

converges absolutely (i.e.

oo, see Exercise 2). Hence, the order in which we

perform the summation is of no importance.

Lemma U.8} : (Parseval) Let {xa}aeA be an orthonormal basis of the Hilbert space H. For x, y we have

=L

aEA

x 00

X

0 •

0 • •

k-=l

Therefore, N

< x,y > = N-oo lim < L < x,x0 • > x 0 .,y > k=l

N

= N_.ooL.J lim ~ < x,.:ra• >< Xa••Y >, k•l

EH

SQUARE INTEGRABLE FUNCTIONS

where we used the continuity of the function x

41

-< x, z > for each z E H. +

Next we turn to the Hilbert space H = L 2 (T).

Theorem (4.9) :

H

The set {e.t}nez is an orthonormal basis of

= L 2 (T}, where e,.(z) = z".

PROOF. Clearly, {e.t}nez is an orthonormal subset of L 2 (T). Consider a function/ e L 2 (T) such that 0

= < /,en > = 2~

£

/(z)z-"dz

= lc.n)

for all n E Z. The uniqueness Theorem (2.8} implies /

= 0 in L 2 (T). +

Corollary (4.10) :

t l/(n)l2. 2~ hl/(z)l 2dz = n•-oo (2) If /,g E L (T), then ~ hf(z)g(z)dz = f, lc.n)g(n). 2 n•-oo (1) If/ E L 2(T}, then

T

2

T

L 00

(3) If (an)nez is a sequence of complex numbers with then there exists /

e L 2 (T)

with an = /(n).

n=-oo

l11nl 2 < oo,

PROOF. The assertions follow from Theorem (4.9), Lemma (4.7) and Lemma (4.8) as well as Corollary (4.2}.

+

Given /, g E L 2 (T}, by Theorem (1.3} the function f • g(z) is defined for almost all z ET. However, as we assume/ and g to be square integrable, the Cauchy-Schwarz inequality yields that the function /(ze- 1)g(e) also becomes integrable for all z E T. Thus, / • g(z) is determined for all z e T. Continuity of the mapping z - L 11 / also gives us

e-

I/• 9('7z- 1 )

-

I• g(11)I ~ 2~ ~

£

1

1

ll('lz-•e- >- /(11e- >I lg(e)ld(

llL.J -

/l121lgl12 ,

CHAPTER IV

42

i.e., the uniform continuity of/• g. In particular, / • g Corollary (4.10) implies:

Corollacy (4.11)

tinuous, and

For /,g

:

II/• 9112 =

e L 2(T)

e L 2 (T)

and

the function/• g is uniformly con-

Ct~ i/(n)l'li(n)l2) !

PROOF. Making use of Corollary (4.10} and Theorem (1.3) we get

II/• 911~ =

L

00

nxs-oo

1(1 • gnn)l 2 =

L 00

n=:-oo

lf(n)l 2 li(n)l 2 • •

Remark: (1) From Theorem (4.9) we can derive that any orthonormal basis in L 2 (T) is countably infinite. If {/a}ae.4 is a second orthonormal basis of L 2 (T), then consider An= {a EA:< en,/0 > ':/: O}. By Lemma (4.4) the sets An are at most countable. For a E A\ Unez An we have < en, /a > = 0 for all n E Z. This means / 0 = 0, a contradiction. Hence, A = UnezAn Is countable. By similar argumentation we get that A cannot be finite. Thus, A is countably infinite.

(2) Corollary (4.10} says that the Fourier transformation is an isometric isomorphism between L2 (T) and l 2 (Z). This mapping is also called Plancherel iaomorphUm. Corollary (4.10) further implies convergence of the partial sums Sn(/) to/ provided/ E L2 (T}, where Sn(/)(z)

=

L n

f(k)z".

lc=-n

The problem of the pointwise convergence of Sn(/)(z) to /(z) when f e L2 (T) has a long history. The problem was not solved before 1965, when Carleson demonstrated that Sn(/)(z) - /(z) for almost all z e T. Proving this is, however, outside the scope of this book, see e.g ll2J. Let us examine an important subspace of L 2 (T), the space H 2 (T). It is defined by H 2 (T) = e L2 (T}: /(n) = 0 for n < 0}.

{!

43

SQUARE INTEGRABLE FUNCTIONS

H 2 (T) is a closed linear subspace of L 2 (T). We show how to identify H 2 (T) with the so-called Hardy space H2(U). As usual by U we denote the set {z EC: lzl < 1}.

Definition (4.12) : space H2(U), if

A holomorphic function Fon U belongs to the Hardy

E 1Bnl 00

2

< oo holds, where F(z) =

E :

1 = R-oo lim R 2

JR f(x)g(x)dx a scalar product is de-R

fined on L. Prove that L is not complete with respect to the norm induced by this scalar product.

CHAPTER IV

46

(5) Applying Corollary (4.10) prove

L

00

A:=l

!

1 2 k2 = ~ .

(6) Let 1 ~ p ~ oo and ~ + = 1. Show that for f E Il'(T), g E £9(T), the function / • g is continuous. Prove the inequality II/• 9lloo ~ 11111,,llgllq· (7) Let (a)

f, g E L 2 (T).

f • g(z) =

Prove:

L

00

/(n)g(n)z" for all z E T, where the series

n=-oo

converges absolutely and uniformly.

(b) f g e L 1 (T) and (/gnn) (8) (a) Prove for N

=

L 00

k=-oo

/(k)g(n - k).

e N and z = e2fri/N:

N

_!_LznA:={l fork=O N

(b) Let

0 for k

n=- 1

f e L2 (T).

For

9N(()

= 1, ... , N

Ne N define 9N

- 1.

E L 2 (T) by

N

= ~ L /((e2•~n/N). n•l

Prove that in constant lco).

L 2 (T)

the function 9N converges to the

(9) Let f E H 2 (T) and assume that f does not vanish almost everywhere. Prove that /(z) =F 0 for almost all z E T. (Hint: We may assume without loss of generality /(0) =F O. Consider the closed hull A of the set {(1 + b1e1 + ~e2 · · · + bnen)/ : n e N, bi e C} in L 2 (T) and prove that the element g e A with minimal distance to 0, is constant almost everywhere.)

(10) A path "(: (0, 1} -+ C, "((t) = x(t) + iy(t), is called closed if "f(O) = 'Y(l). It is called smooth if "( is continuously differentiable. It is termed simple if "((t1) =F 'Y(t2) is true for ti =F t2, lt1 - t21 < 1. Its length is 1, in case

L('Y) :=

fo

1

h'(t)ldt = 1.

SQUARE INTEGRABLE FUNCTIONS

47

Prove that among all closed, smooth, simple paths of length 1 the circle encloses the biggest area (isoperimetric problem). (Hint: The area A, surrounded by the path -y(t) = z(t) + iy(t) is given by A, = J~ (z(t)y(t) - x'(t)y(t)}dt .)

!

C\ Taylor & Francis ~Taylor & Francis Group http://tayl o ra ndfra nci s.com

CONVERGENCE OF FOURIER SERIES IN NORM In Chapter 2 we saw that""(/) is converging in any homogeneous Banach space B to / E B as n - oo. Having in mind the relation between n-th partial sums

L n

Sn(/)(z) =

f(k)z"

of the Fourier series of f and the n-th Cesaro means

CTn(/) given by

=

t (1- nl!l l)

f(k)z"

lc•-n

CTn(/}

1

= n+1L n

lc=O

S1c(/},

one is quite naturally led to the question of convergence of Sn(/) .

In Chapter 4 we considered the special case B = L2 (T} to realize that Sn(/) converges towards / in the L2 (T)-norm. In this paragraph now it is our objective to investigate the convergence of Sn(/) in homogeneous Banach spaces B. To begin with let us study a general approach to the problem of convergence in norm.

Therorem (5.1) : Let B be a Banach space with norm 1111· Also, let (Tn)"eN be a family of continuous linear operators of B into itself. For a dense subset M of B we assume that lim llTng - gll = 0 for all g e M. n-oo

Then the following two statements are equivalent: (1) For all/ e B we have lim llTn/ - /II= 0. n-oo

(2) llTn/11 :5 Cllfll for all / e B, n E N, where C is a constant independent of / and n. PROOF. Assuming (1), there exists for each / E B a constant C1 such that llTn/ll S Ct for all n EN. The principle of uniform boundedness (see 49

50

CHAPTER V

Appendix B) yields a bound C such that llTn/11 :S Oii/ii for all / E B and neN. Conversely, assume (2). For f E B and E > 0 there exists g E M with II/ - gll < e. Then we have

llTnf -

/II :S llTnf - Tngll + llTng -

gll + Ilg - Ill :S (1 + C)e + llTng - 911·

Hence, {1) follows, since lim llTng - gll n-oo

f

In order to show Sn(/)-+ sufficient to prove

llSnll 8

= 0 holds for all g EM. t

for all/EB by means of Theorem {5.1), it is :

=

sup llSn(/)lls :SC. 11nss1

The norm llSnll 8 is the operator-norm of Sn· Note that Sn maps B into itself. Since Sn(/)= Dn • f, we have

llSnll 8 =

sup llDn •Ills~ llDnlll· n11ss1

The numbers Ln = llDnlh are called Lebesgue constants. The estima.tion llSnll 8 S Ln is, in general, not very useful to get a convergence result. However, to obtain divergence the asymptotic behaviour of Ln can be applied. In fact, we can prove that the sequence of Lebesgue constants Is unbounded.

Lemma {5.2} :

The Lebesgue constants Ln

the inequalities Ln

> !rs In(n)

PROOF. We have

Ln

= ..!._ ("' 211'

lo

Now set r

= n and (D..}lk) = 1 for lkl S n it follows that Sn(FN)

= O'N(Dn) for n $. N.

Thus, for n $. N :

Since O'N(Dn) - Dn for N - oo with respect to 11111 1 we obtain llDnll1 $. llSnllL'(T)·

We have already shown that llDnll1 ~ llSnllL'(T) and so the assertion is proved. •

Lemma (5.2) and Lemma (5.3) imply llSnllL'(T) Theorem (5.1) yields:

oo as n -

00 1

and

52

CHAPTER V

Theorem (5.4) : The L 1 (T)-norm convergence of every Fourier series does not hold, i.e., there exists a function f E L 1(T) such that (Sn/)neN

is

not convergent in the 11111-norm.

PROOF. Since the set of all trigonometric polynomials is dense in L 1 (T), Theorem (5.1) gives the existence of a function f E L 1 (T) whose Fourier series does not converge to/. Assuming that Sn(/) tends tog e L 1 (T), the Fej~r sums Un(/) converges tog, too. According to Theorem (2.6) we have then g =/,a contradiction.

+

The same results hold for B = C(T). In fact, we have

Lemma [5. 5 J :

The equality llSnllC{T)

= Ln is valid for all n e N.

PROOF. Let e > 0. Consider small neighbourhoods around the zeros of Dn(z) such that the union V of these neighbourhoods has a Lebesgue measure smaller than 2n\t. Now choose a continuous function V'n E C(T) with llV'nlloo $ 1 and IPn(z) = sign(Dn(z)) for all z E T\V. Applying IDn(z)I $ 2n + 1 for all z E T, we obtain

2:

2~ (£ IDn(z)ldz - 2e) .

Therefore, llSnllC{T) 2: Ln holds.

+

Theorem (5.6) : The C(T)-norm convergence of every Fourier series does not hold, i.e., there exists a function f E C(T) such that (Sn/)neN is not uniformly convergent. Moreover, one can prove the following result:

CONVERGENCE OF FOURIER SERIES IN NORM

53

Theorem (5. 7): Let z0 e T. There exists a continuous function / e C(T) such that Sn(/)(z0 ) is divergent. PROOF. Assume that Sn(/)(z0 ) converge for all/ e C(T). Then sup ISn(/)(zo)I ~CJ < oo nEN

for all/ e C(T). Consider for n e N the linear functional 'Yn: C(T) - C, 'Yra(/) = Sra(/)(z0 ). According to the principle of uniform boundedness, there is C ;:::: 0 such that

for all n

e N and I e C(T).

Since

Sra(/)(z)

= Sra(L••.-•f(zo))

for all z ET, we find that llSra(/)11 00 S Cll/lloo in contradiction to Theorem (5.1), Lemma (5.2) and Lemma (5.5). t Let us now characterize norm convergence of the Fourier series in homogeneous Banach spaces B by means of conjugation and projection of the Fourier series. With these concepts we will finally succeed in proving norm convergence of the Fourier series in .V(T), 1 < p < oo, see Chapter 8. Given

f e L 1(T), consider the formal conjugate Fourier series

L

00

S(/)(z) =

n•-oo

(Note that by convention sign(O)

(-i) sign(n)/(n)z" .

= 0.)

A function j e L 1 (T) with Fourier coefficients ( -i) sign(n)/(n) is called the conjugate of/. If for f e L 1 (T) there is an integrable function Pf on T whose n-tb Fourier coefficient equals 0 for n < 0 and equals /(n) for n;:::: 0, then Pf is called projection of/.

Definition (5.8) : Let B be a homogeneous Banach space on T. (1) We say that B admits conjugation, if for all/ e B there exists a function g e B such that g(n) = (-i) sign(n)/(n) and

CHAPTER V

54

llglls

~

case, g

Mjl/lls, where Mis a constant independent of/. In this

= /.

(2) We say B admits projection, if for all f e B there exists a function he B such that h(n) = 0 for n < 0 and h(n) = f(n) for n 2: 0 and llhlls ~ Mllflls, where M is a constant independent of /. In this case, h =Pf.

It is easily shown that L 2 (T) admits conjugation and projection. In fact, for I e L 2 (T) we have

L 00

n=-oo

1(-i) sign(n)f(n)l 2 = llJM - lf{O)l 2 ~ II/II~

and

L lf(n)l 00

n=O

2

~ II/II~·

According to Corollary (4.10) (3) there exist

j E L2 (T) and Pf E L2 (T).

Lemma (5.9): Let B be a homogeneous Banach space on T. The following assertions are equivalent:

(1) B admits conjugation.

{2) B admits projection. PROOF. Assume (1). For h

f e B we define

= f{0}/2 + (f + ii)/2.

Since j E B, we have h E B and llhlls ~ Mll/lls for a constant M. Apparently, h{n) = f(n) for n 2: 0 and h(n) = 0 for n < 0 holds, i.e. h=Pf. Conversely, assume that (2} is true. With g=

Hf~o)P/- ~)

we have g EB and llglls ~ Mll/lls for a constant check that g(n} = (-i) sign(n}f(n), i.e. g = j. •

M.

Also, we can easily

CONVERGENCE OF FOURIER SERIES IN NORM

55

Theorem (5.10) : Let B be a homogeneous Banach space on T. Given f E B let /n(z) = z-n/(z) and assume that /n EB and 11/nlls = 11/lls for all n e Z. Then the following conditions are equivalent: (1) In B holds norm convergence of the Fourier series, i.e. lim llSn(/) -

n-oo

for all

f

Ills = 0

EB.

(2) B admits conjugation (and equivalently B admits projection). PROOF. First we prove (1)

~

(2). A simple calculation shows 2n

z"SnC/n)(z) where /n(k)

= /(n + k) is used.

= Lf(k)z", k•O

Let

P2n(/)(z)

2n

=L

k=O

/(k)z"

denote the 2n-th partial sum of the formal series of Pf. According to the assumptions we see:

Ma constant. We show now that (P2n(/))neN is a Cauchy sequence in B. For E > 0 exists N E N and a trigonometric polynomial Q of a degree not greater than N such that II/ - Qlls < E holds, and hence

Since for n, m

it follows

> Jt,

we have

llP2n(/) - P2m(/)l1s

< 2Me.

The completeness of B implies an element h E B such that h = lim P2n(/). n-oo

56

CHAPTER V

Then

llhlls 5 Mll/lls

and

h(k) = lim (P2n(/)nk)

n-+oo

= { 0f(k)

for for

k~O

k (1), note the identity

Hence,

llSn(/)lla 5 llP/-nlls + llP/n+1llB $ 2Mllflls,

where Mis the constant obtained by assumption (2). From Theorem (5.1) finally (1) follows. t With Theorems (5.4) and (5.6) we get the following corollary:

L 1(T) and C(T) do not admit conjugation.

Corollary (5.ll) :

Evidently the size of Fourier coefficients fc.n) of/ E L 1 (T) is crucial for the convergence of Sn(/). But the only things we know is that HJil 00 $ 11/111 and that lim f(n) = O. If f E L2(T) we have llJil2 = 11/112, in particular

lnl-+oo

E lf(n)l 2 < oo.

n=--oo

We conclude this chapter by deriving the theorem of

Hausdorff-Young, which gives further information on the magnitude of the fc.n).

Theorem (5.12) : (Hausdorff-Young) Assume that 1 $ p 5 2, and let q ~ 2 such that ~ + ~

1E l

11Jil9

= 1.

II/II,.. (b) If c = (ct)tez E lP(Z), then there exists a function/ e £9(T) such that Cn = f(n) for every n E Z. Moreover 11Jil 9 5 llcll,.. (a) If/ E V(T), then

9 (Z).

More precisely

$

CONVERGENCE OF FOURIER SERIES IN NORM

57

These two statements can be obtained by an application of the Riesz-Thorin theorem, which is a general result of interpolation of continuity of certain linear operators. For the proof of the Riesz-Thorin theorem we refer the reader to (7, 13.4.lJ. Let (X, M, µ) be a measure space. (M is a a-algebra of subsets of X, andµ a positive measure defined on M.) For a µ-measurable function / we define

II/II,,,,. = ([ l/(x)I" dµ(x)) l/'P if 1 $ p < oo and

11/lloo,µ

= ess sup l/(x)I = inf{M 2: 0: µ{x EX: l/(x)I 2: M} = O}. zeX

Note that II/II,,,,. may be oo. A function/: X - C is called µ-simple, if it is expressible as linear combination of characteristic functions XM, where M E M with µ(M) < oo. Instead of (X, M, µ) we use the shorter form (X, µ).

Theorem (5.13) ; (Riesz-Thorin) Let (Xi, µi) be two measure spaces, and let S be a linear operator defined on all µ 1-simple functions and taking values in the space of µ2-measurable functions on X2. Given points (a1, {J1}, and (a~.h IJ.J) of the square (0, IJ x [O, I J 888ume that simultaneously hold

and

llS/111/JJ,,µ2 $ M2ll/ll11a2.11t1

then for all interpolating values a= (l ->.)a1 + >.a2 1 /J = (1 - >.)/J1 with 0 < >. < 1 we have

+>.tJ.z

Remark: If a> 0 the operator Scan be extended to L 11°(X1 ,µ1 ), pre-

serving the norm inequality above.

CHAPTER V

58

We have to show the Riesz-Thorin theorem yields the Hausdorff-Young theorem. For part (a) of Theorem (5.12) we consider (X1 1 µ 1) = (T, 2~dz) and (X2 1 µ2) = (Z,dn), where dz is the Lebesgue measure on T and dn is the counting measure on the integers. The Fourier transformation f will be the linear operator S. Further take (a1 1 P1) = (1,0) and (a2,/h) = (~,!). By Theorem (1.2)(5) and Corollary (4.10)(1) we can apply the Riesz-Tborin theorem and o~ tain Theorem (5.12)(a). For the second statement of Theorem (5.12) we interchange the roles of (T, /;dz) and (Z,dn). Now the linear operator S will be c - t, where c = (c1c)1cez and t(e") = E~_ 00 c1ce•A:t. We take again (a1 1 P1) = (1,0) and (a2,P,) = (l, !). For (a1P1) = (1,0) we consider l 1 (Z) with the image A(T). Obviously lltlloo ~ llclh· Using Corollary (4.10){3) we get Theorem (5.12)(b), too .

f

.Remark: Theorem (5.12) cannot be extended to the case p > 2. In Exercise 7 of Chapter 15 we will give an example of a continuous funclf(n)l 9 = oo for each q with 1 ~ q < 2. tion f on T such that However note, that for f E V(T) with p > 2 we have f E L2 (T) and hence

E:°--oo

En--oo l/(n)I 2 < oo. 00

-

EXERCISES (1) Assume the formal Fourier series off E L 1 (T) be given in the form (a)

!ao + lc•l E (a1c cos kt+ b1c sin kt). ()()

Show that the formal conjugate series equals (b)

E (-b1c cos kt+ a1c sin kt). 00

lcml

F\Jrther show that (a) is the real part and (b) the imaginary part of the power series

L c1cz1c ()()

1

z

lc=O

with C-0

(2) Let c let

= ~,

= (Cn)neNo

c1c

= a1c -

= e" ib1c 1 n E N.

be a sequence of complex numbers. Fork E No

59

CONVERGENCE OF FOURIER SERIES IN NORM

= Ck -

2Ck+1

+ Ck+2•

{a) Prove: If {bn)nENo is another sequence, for all n E N we have n

L

n-1

Ckbk

k=O

where Bk=

=L

k•O

Ac"B" + enBn,

E" b1 (partial summation).

J=O

(b) If the Cn are real numbers and satisfy A 2 etr ~ 0 for all k e No, then we call c convex Show: If c is convex and Jim Cn = 0, then it follows n-oo

lim nAen n ...... oo

00

= O, and

2 ~(k L-, + l)A ck+m =Cm

1:-0

is valid for m e No.

(c) Let (Cn)neNo be a convex sequence with lim Cn = 0 and set n-oo C-n = Cn for n e N. Prove (Polya): There exists a nonnegative function / E L 1 (T) such that Cn = /(n) . (Hint: Consider/= (3) Show:

E 00

k•2

00

E (k + l)A2c1i:F1i:.)

k-0

~ is the Fourier series of a function/

e L 1 (T). (Ap-

ply Exercise 2.) Is the formal conjugate series S(/)(z) a Fourier series (compare Exercise 6 of Chapter 3)? (4) Let (Cn)neN. be a sequence of real numbers decreasing to 0. Further let (bn(Z))neNo be a sequence of complex-valued functions defined on an interval [a, PJ. Prove (Abel's lemma): If the partial sums Bn(z) =

IBn(x)I

~

E" b1i:(z) are uniformly bounded, i.e.,

1:-0

M for all n E No and z E

[a,PJ,

the series

converges uniformly on [a, Pl and the limit function

L enbn(z) 00

S(x) =

"""o

E Cnbn(z) 00

n•O

CHAPTER V

60

satisfies IS(x)I ~Meo for all x E [a,/3J. (Hint: Use partial summation. Remark: One should observe the

difference between Abel's lemma and Abel's limit theorem.) (5) Let (Cn)neN. be a sequence of real numbers decreasing to 0. (a) Prove for 0 < 6
0 such that

(21a IFao,,(e")ldt < £

l1a

t

for 0 < h $ho. In particular, we see that

for 0 < h $ ho. Hence,

f

1a12·-1

A12is valid for 0

IF-.• )I t(t + h)

-

dt - 0.

(b} Prove that the condition (L) of Theorem (6.11) is equivalent to the combination of the properties (Ll) and (L2). (8) Let f E L1 (T), Zo ET, and assume that 8

=

fun /(zoe'") + /(zoe-'")

1a-o+

2

exists. Show that rnfnJISn(/)(zo) -

sl -

0 as n - oo .

T

CHARACTERIZATION OF FOURIER COEFFICIENTS The problems we dealt with so far were approximation-theoretic in nature. We turn next to tasks of the following type: Given is a sequence (Cn)neZ· Which conditions guarantee Cn = f(n) for all n e Z, where /EB and B is a homogeneous Banach space on T? That means we are investigating representation problems. In the sequel denote tln,k real numbers -sometimes called weights- for n E N, k = -n, ... , n, forming a triangular scheme. We assume throughout that tln,o = 1 , tln,-• = tln,k fork= 1, ... , n.

These weights tln,k define trigonometric polynomials An(z) on T by (1)

••-n

Given a sequence c = (Cn)nez define for n EN

L ft

Ant(z) =

(2)

Of special interest are On,•

respectively tln •

•

•--n

tln,kC•z•.

= 1 for k = -n, ... , n ,

lkl = 1- n+-1 fork= -n, ... ,n.

In the first case we have An(z) = Dn(z), the Dirichlet kernel, in the second case An(z) = Fn(z) , the Fejer kernel. Further triangular schemes will be introduced in Chapter 10. If An(z), n E N, make up an approximate identity for B and if we assume that c = (Cn)nez is given by Cn = f(n) for / e B, then by Theorem (1.5) the sequence (An~)neN converges in B to/. The converse implication is also valid. In fact, we have: 75

CHAPTER VII

76

Theorem (7.1) : Let B be a homogeneous Banach space on T. AJJsume that the sequence (An)neN of trigonometric polynomials defined in (1) is an approximate identity for B. Fbr a sequence c (Cn)nez the following conditions are equivalent:

=

(1) c =

f

for / E B.

(2) (An$.

The following characterization gives a very general approach to Fourier transforms of elements of B•.

Theorem (7.3) : Let B be a homogeneous Banach space on T. Let c = (Cn)nez be a sequence of complex numbers, and M ~ O. The following conditions are equivalent: (1) (2)

There exists

l{J E

B• with ll'PllB• ~ M and $(n)

= c,. for all n e Z.

For all trigonometric polynomials P we have

llc~oo P(k)~I ~ MllPllB· PROOF. For a trigonometric polynomial P =

P(k) Hence, lfJ(P)

=0

lkl > N

for

= E 00

....

lc•-00

and

P(k)

N

E

k•-N

= b1:

b1:e1c we have

for

--

lkl ~ N.

P(k)$(k) for all linear functionals on B. If 'I'

with ll'Plls• ~ M and $(n)

L 00

e B•

= Cn , then we get

P(k)~

= lip(P)I ~ MllPllB·

Conversely, assume (2). Let Trig(T) be the linear space of all trigonometric polynomials. Define

~(P)

=

L 00

k•-00

P(k)c1c,

CHARACTERIZATION OF FOURIER COEFFICIENTS

79

where Pe Trig(T). The linear functional rjJ defined on Trig(T) is continuous with respect to 11 lls· According to Corollary (2.11), Trig(T) is dense in B. Hence, by

ip(/) := lim iP(Pn)

for / = lim Pn

n~~

a continuous linear functional satisfies

ll'Plls• :s; M

and

is defined on B (see Exercise 1), which

ip

~(n)

n~oo

= ip(e,.) = en

for all

n

e Z. t

As before let t)

sin (n!)2 sin ((nsin+ 1) ) = cos! _ cos((n + })t). sin i sin l

t c~ ((k + ~) t) 1c-o

(1- nl~\) z'.

e-in! - ein! sin(ni) -- e"ein! e• 0 and applying Riesz' representation theorem there exists an element

1

'I'

e £9(T),

llv>ll 9 S 1 such that llH(f)ll,, -ES

If.

It. JT

H(f)cp(z)dzl,

and hence, ko e N with llH(/)11,, - 2£ S JTH(/n,)(z)cp(z)dzl for all k ~ ko. Since H(/n.)(Z)v>(z)dzl S llH(/n.>11 11 S cll/n, 11 11 holds and by the convergence of llfn.11 11 to 11/11,, we get llH{f)ll,,-2£ :5 ell/II,,. Therefore, llH(/)11 11 S ell/11 11 is valid, i.e. f E Ee, which was to be shown.

If. JT

= H(u) + iH(v), we get H(/) e V(T) and llH(f)ll,, :5 llH(u)ll,,+llH(v)ll,, S c(llull,,+llvll,,) S 2cll/ll,,, where we have used llRe/11,, :5 11/11,, and Him/II,, S 11/11,,.

(2) Since H(/)

(3) As in (2) we have H(/) = H(f+) - H(r), 111+11,, S 11111,. This yields statement (3). t

111-11,

S 11/11,, and

CHAPTER VIII

100

Assume that the trigonometric polynomial /{z) = /(z) > 0 for all z E T. Denote h(z) = /(z) -

/{z)

= E n

••-n

+

n

E

a•z• satisfies

••-n i/(z), where

(-i)sign(k)atz• is the conjugate polynomial of/. Since

real-valued on T, we have O-•

" 2Re{a1cz") ao + E ·-·

-

and /{z)

f

is

=a•. In particular, we get ao E R, /(z) = = E" 21m{a.z•). Moreover, by ao = /(0)

-

·-·

lr=l

n

ao > 0 holds. Also, h(z) = ao + 2 E a1cz• is valid. For z E T we have Reh(z) = /{z) > 0. Hence, for p ER the function h"{z) = (h(z))" is defined on T, where we have chosen for w" = exp{pln{w)), w E C\)-oo,OJ,

even

the principal branch of the logarithm.

Below we use that w - w" is in C\] - oo, OJ holomorphic and thus can be uniformly approximated by polynomials P{w) on compact subsets K ~ C\J- oo,OJ.

Lemma (8. 5):

Let p e R and let /(z)

polynomial such that /(z) > 0 for all z for z ET. Then

n

= E

lr=-n

E T.

a•z" be a trigonometric

Denote h(z) = /(z)

+ if(z)

2~ fr h"(z)dz = ( 2~ fr h(z)dz),, = ag

is valid. PROOF. For p E N the function h"{z) is of the form h'{z)

Ec1cz•.

·-·IT

Therefore, we get /; IT h"(z)dz

= ag.

=

ag +

By linearity we obtain

i; P(h(z))dz = P(ao), where P(w) is a polynomial in w E C. Now choose a compact disc K containing h(T) as well as ao > 0 such that Kn J- oo, OJ is empty. Approximate w - w" uniformly on K by polynomials. Hence, the statement follows. t

HILBERT TRANSFORM

Theorem (8.6):

101

(M. Riesz)

e V(T).

Let 1 < p < oo. For all / E V(T) we have H(/) exists a constant c,. such that

Moreover, there

llH(f)ll, :S c,11/11, for all / E V(T). PROOF. (1) In a first step we show that we merely have to prove the statement for trigonometric polynomials /(z) =

E"

k•-n

atzk with /(z)

>0

for

all z ET. Therefore, assume that in the notation of Lemma (8.4) any positive trigonometric polynomial is an element of Ee. Let / E C(T),/ > 0. Since Tis compact, we have rnin/(z) 2: 6 > 0 for :rET

an appropriate 6 > O. By Corollary (2.11) approximate f uniformly 6

by trigonometric polynomials g with ming(z) 2: zeT

2

> 0.

(Note that

the approximating Fejer polynomials are real-valued, since / is realvalued.) These polynomials approximate/ also in the V(T)-norm. By Lemma (8.4)(1) the set Ee is closed, and hence f E Ee follows. If/ E C(T), / 2: O, by approximation with positve continuous functions we obtain f e Ee as well. Then by Lemma (8.4)(2) and (3) each/ E C(T) is an element of Efc. Finally, using again Lemma (8.4)(1) we have furnished V(T) = Efc. {2) In a second step we prove for 1 < p $ 2 that positive trigonometric polynomials / satisfy the inequality llH(/)11, ::S ell/II,. Since we assume 1 < p $ 2 there exists 6 = 6, with 0 < 6 < 1 < ~ < Pi $ 1r. Set a=~ and {J = Hence, a< 0 and {J > 0. Alao, we have for It!< f:

) cos s ds

41r lo 41r lo 2 sin t [ " 11' lo k1(e•) sins ds. 4

The last two integrals are as functions of z = e" elements of B, that converge in B with j --+ io to cost and sin t . Because of II II 1 ~ II II s this is also valid in L 1{T). Therefore we have

lim k1 • 11(1) =

i-Jo

!2 (1 -

cos2 t - sin2 t) = 0

in L 1 (T). Since the left side is independent oft condition (3) follows. The positivity of k1 yields for 0 < 6 < 1r:

k1 •11(1) =

2~

Lh

k1(e-u) (sin i f ds 2':

2':

(8. !)2

12•-•

~: li

2~

J.h-I

2

kJ(e-") (sini) ds

k1(e-")da.

Hence property (53) for summation kernels on T follows from (3), and consequently (3) implies (4). Finally (4) implies condition (1) by Theorem

{2.10). •

Now by Theorem (9.5) we can easily derive the following equivalence.

CHARACTERIZATIONS OF APPROXIMATE IDENTITIES

Corollary (9.6):

111

Let B be a homogeneous Banach space on T, and let

(kJ)Jel be a family of continuous nonnegative functions with llk;lh = 1 for all; e I. Then (k;);er is an approximate identity for B if and only if there exists zo e T such that lim kJ • ei (zo) = zo. ;-Jo

PROOF. If (k;)Jel is an approximate identity for B, then according to Theorem {9.5) and Theorem (2.10) it is also one for C(T). Especially,

is valid for all z e T. Conversely, assuming this limit relation for division by zo yields

zo e T,

AB the functions k; are real-valued, splitting in the real and imaginary part gives 1 Jim kJ(e") cost dt = 1 ;-;o 211' 0

1211'

and

lim - 1

.

1211' k;(e") sin t dt = O.

;-Jo 211' 0

By identity (*) in the proof of Theorem (9.5) we obtain Jim k; • 17(1) J-io

= 0.

Hence by Theorem (9.5) the family (k;);er is an approximate identity for B. t

Corollary (9. 7) : Under the assumptions of Corollary (9.6) the following assertions are equivalent: (1) (k;);er is an approximate identity for B .

(2) lim (k;)"'(l) i-io

= 1.

PROOF. One has only to observe that k; • ei(l)

= {k;)"'(l). t

In Chapter 2 we have already introduced some special approximate identities, for example the Fejer kernel, the Poisson kernel, the de la Vallee.

Poussin kernel and the (C, a)-kernel. We have also noted that the Dirichlet

CHAPTER IX

112

kernel is not an approximate identity for L 1 (T) and C(T). Of course, it is important to construct kernels distinguished by an optimal convergence

behaviour. Chapter 11 is devoted to this subject. Here we present further

approximate identities possessing very good convergence properties.

At first we consider a convex combination of appropriate translations of the Dirichlet kernel. For n e N set (Bl) where tn = 2n~i, and (Dn)neN is the Dirichlet kernel. The family (Rn)neN is called Rogosinsld kernel. Immediately we have

L cos(ktn)z". n

Rn(z) =

(82)

lr=-n

Since

D ( it)_ sin((2n + 1)!)

ne

-

sinl

'

the special choice of the tn yields

Rn(e't)=~cos((2n+l)~) (sin(~)- sin(~))·

(83)

We prove that (Rn)neN satisfies condition (3) of Corollary (9.2). Simple substitution and estimating by Isin ,8 - sin al ~ IP - al supplies:

lTf IRn(z)I dz = 2 lof' lcos ((2n + l)t)I

I. (+ .:rt ) - s 1

sm t

,, Isin(tcos+ ((2n + l)t) I \L)sin(t- \L) dt =

~ 2tn lo

in(

!1

1

t-

I

i.) dt 2

+12 + [3.

The last integral has been splitted into the three integrals Ii, !2 and !3 over the subintervals

For t

E

(0, tn] is valid: . ( tn) > . (tn) > 1 sm t+2 _sm 2 - 2n+l'

CHARACTERIZATIONS OF APPROXIMATE IDENTITIF.S

where we have used sin{J ~

I

coe((2n + l)t) sin(t >

#f

!P for fJ E [O, JJ. Furthermore, we have

I= I

sin {(2n + l)(t -

Therefore

11 S 2tn

By sinfj ~

!P for fJ E

Finally for t E [j

[O, j)

T)

sin(t -

i>) I< 2n + l. -

LC.. {2n + 1) dt = 271' 2

we conclude:

-1f, }] is valid:

, (t+-tn) >cos->tn v'3

sm

2

-

2 -

2

and

implying

Thus we have shown that

All (.R,.)lk)

fr l.Rn(z)jdz S 471'

2

= cosktn, we get lim (.R,.)lk)

n-oo

and have the following result:

=1

•

2

•

113

CHAPTER IX

114

Corollary (9.8) : Let B be a homogeneous Banach space on T. Let (.Rn)neN denote the Rogosinski kernel on T. Then (.Rn)neN is an approxi-

mate identity for B, i.e.,

n

lim ~ cos(ktn)J(k)e1: =

n-+oo L.,,

A:•-n

.m the norm of B for every f

E B, where tn =

n

2

f

+ 1·

~

The following figure shows the approximation of the Heaviside function f (considered at the end of Chapter 2) by weighted Rogosinski-Fourier sums (n = 21).

FIGURE 10: APPROXIMATION USING ROGOSINSKI KERNEL

Now fix .>. > 0 and define for n E N and k 0

(l) _ n,A: -

2

= O, ..• , n

(>.),.+1:(>.),.-1:(nl) (n + k}l(n - k)I ((..\),.) 2 '

where (>.)m = .>.(.>.+l) ... (.>.+m-1) and (..\)o mer symbol. Define for n E N

(Dl)

D~l>(z) =

n

L

A:•-n

a~~l

4

(l)

_

n,-A: -

4

(l) n,A:•

= 1 is the so-called Pochham-

z•.

CHARACTERIZATIONS OF APPROXIMATE IDENTITIES

115 1

Note that for .\ = 1 we have a~~L = 1 for k = -n, ... , n. Thus D~ ) (z)

Dn(z) the Dirichlet kernel. For.\ = 2 we obtain a~~~

= 1- ( n!i)

2

fork

=

=

2

-n, ... , n. The family (D~ »nEN is usually called Bochner-Riesz kernel. We call {Di.\»neN generalized Dirichlet kernel with the parameter.\. The weight a~~l can be written as a quotient of Gamma functions. In fact we have (.\) r(.\ + n + k) r(.\ + n - k) (r(l + n)) 2 an,A: =

rc1 + ri + k) rc1 + n - k) (r(.\ + n)) 2.

The asymptotic properties of the Gamma function (see (1, St, 6.1.46)) give lim

a(.\)

n-oo n,A:

= lim (n + k).A (n - k).A n2 n-oo (n + k) (n - k) n2.\

= 1.

We just have proved that condition (4) of Corollary (9.2) holds. In order to derive (3) of Corollary (9.2) we apply that D!a.\)(e") is a Jacobi polynomial in x = cost. In fact, using an old formula of Gegenbauer (seel15]), one can show, that (02) where JliQ,IJ)(z) is then-th Jacobi polynomial to parameters 11>(1) = 1, and where a=.\ {J =-!,normalized such that

!,

.RiQ·

(nl) 2 (2.\)2n - ((.\)n) 2 (2n)I'

b(.\) _

"

The Jacobi polynomials JliQ·11>(x) are determined by the orthogonality relation

1 1

-1

J?!.Q,lf)(x) Jl!:·">(x)(l - z)Q(l + x) 11dx

= cSn,m h~Q,IJ) ·

With the representation (02) and asymptotic properties of Jacobi polynomials one can derive, that for each n E N

llDi.\>111 ~ M, provided we suppose that.\> 1. For the proofs we refer the reader to 115). According to Corollary (9.2} we have the following result.

CHAPTER IX

116

Corollary (9.9): Let B be a homogeneous Banach space on T . Let ~ > 1. Then the generalized Dirichlet kernel (DiA»nEN is an approximate identity for B. Figure 11 and Figure 12 show the approximating performance (n = 21) of the generalized Dirichlet kernel with the parameters ~ = 2 and ~ = 3.5 again for the Heaviside function /. It is remarkable, that (DiA»nEN is for ~ > 1 an approximate identity for any B, whereas for ~ = 1 this is not the case.

o.•

-]

-2

- 1

2

-o.

FIGURE

11: APPROXIMATION USING GENERALIZED DIRCHLET KERNEL WITH PARAMETER ~ = 2

CHARACTERIZATIONS OF APPROXIMATE IDENTITIF.S

117

o .s

-l

FIGURE

-·

•

-1

12: APPROXIMATION USING GENERALIZED DIRICHLET KERNEL WITH PARAMETER ~ = 3.5

Another family of functions forming an approximate identity is given by

(Jl)

u

3 (sin1f) Jn(e ) = n{2n 2 + 1) sin

i

4

The familly CJn)neN is called Jackson kernel. Now

(J2) is evident, where (Fn)neN is the

n

= 2,3, ....

Fej~r

kernel. Therefore, we write for

CHAPTER IX

118

with certain coefficients 0n,1c. The coefficient On,o can be calculated 88

3

On,o = 2n 2 :

=

1

(

n-1 (

k)2)

1 + 2 ~ 1 - ;i'

3n (l+(n-1)(2n-1})= 1 2n2 +1 3n ·

Hence CJn)neN is a family of positive functions with llJnll1 = 1. In order to use Corollary (9.7}, we have to determine On,1· By an elementary calculation we obtain

1 _k-l)(l-~)-2n2-2 °"· _ 2n26n+ 1 ~( ~ n n - 2n2 + 1 · 1

-

Therefore (Jnf(l} = On,1 -+ 1 88 n-+ oo. So we have proved:

Corollary (9.10):

Let B be a homogeneous Banach space Bon T. Let

CJn)neN denote the Jackson kernel on T. Then CJn)neN is an approximate

identity for B.

EXERCISES

{1} Let (k;);eI be a summation kernel. Show: If / E C(T) is n-times continuously differentiable, then lim ll(k; * f}(n) - /_l

(i.e. the (C, o)-kemel (Fn,D)neN of Chapter 2). It is easily shown that

and for

k = 0, ... , n - 1.

For a ~ 1 the sequences 1, 0n,1, . • . , On,n are convex. The asymptotic behaviour of the gamma function implies lim On• = 1 for every k EN (see n~oo

•

Exercise 1). Thus for a~ 1 the family (Fn,D)neN is an approximate identity. That holds also for 0 < a < 1 as we have already announced in Chapter 2. The following results allow to derive this fact.

CHAPTER X

124

Lemma {10.3): Let real numbers 4n,lc be given for n EN and k = 1, ... , n. Also let 4n,o = 1. Assume that for every n E N

where Mis a constant. Denoter= we get the following inequalities:

l!!.f!J, the integer part of np.

Then

L(k + l)IA 0n,1cl $ 2M, L(n - k)IA 0n,1cl $ 2M

r-1

n-1

2

2

lc=O and

(n + l)IA0n,rl $ 1 + 4M.

PROOF. It is evident that

~

L...J

J•n-lc

n-:3 k ?: (n _ k) k +n 1 >

k+l -2-

if 0$k ~ for 0 < t < 11'1 and I sin ti < t for

[o , n+f+2J the constant .;.(n + k + 2) (n - k)

majorizes the integrand, whereas on [ n+l+2 •n~l J the function

on [ n!l,

rr]

the function 11'2b are majorants.

f

n;lc and

This implies

2~ £l(k + l)F1c(z) -

1

(n + l)Fn(z)ldz

~

11'2 - k) In - ["'2 -(n - k) + -(n 211' 2 2

~

4 (n -

(n n-k + k + 2) + 2 (n - k 1)] 11'

-- -

2

-

,..

k) [ 2 +In n+k+2] n_ k .

11'

As n + k + 2 ~ 2n + 2, we have In ( n

+ k + 2) n-k

1 ~ ln(2) + In ( n-k n + ) ~ ln(2) + 1 + ~ ~, L..J J ;-n-lc

and we infer that

2~ l

T

l(k + l)F1c(z) - (n + l)Fn(z)I dz

~ C(n - k) (1 + J•n-lc3 t ~)

for an appropiate constant C. Combined with llFnlli {10.3), we conclude

= 1 and

Lemma

2~ hIAn{z)I dz~ L(k + 1) IA 0n,1:I + C L(n - k) IA 0n,1:I r-1

T

+c n-1 L( l:•r

2

k=O

L "

J-n-lc

n-:1

k) IA 0n,1cl + 2

and the desired estimate appears.

t

n-1

2

k=r

(n + l)IAan,rl

~ 6M + 3CM + 1,

127

TRIANGULAR SCHEMES

Theorem (10.5) : (Nikol'skii) Let real numbers On,A: for n E N be given and k = 1, ... , n and let On,o 1. Define n An(e") = 1 + 2 On,t coe(kt).

=

L

k=l

If the sequence 1, On i 1 • • • , On n is convex or concave, and if lim an 1c = '

t. ~·~ I

•

1 for all k E N and for all n E N lan,nl

~ Mi

n

and

ft-t()()

'

k :$; M2,

(M1 1 M2 constants), then (An)neN is an approximate identity for each homogeneous Banach space B on T . PROOF. We assume that 4 2 0n,ii: ~ 0 and show that the boundedness condition of Theorem {10.4) is satisfied. ( If 4 20n,1c :$; 0 one has only to change the sign in the following calculation.) In fact we can establish:

E (E;.,.-k DJ1) l4 0n,ii:I ii:...o 1

2

n-1 =

n

E o k=O

for each n

e N. Set for n e N and k = 0, 11 ••• ,n: =

n

n

k=O

k:::m

:E an,t(D.nm> = E

an,t

132

CHAPTER X

because of (DA:)lm) = 1, if lml

~

Now lim (Ln)lm)

n-+oo

holds for all m

k and (DA:)lm) = 0, if lml > k. n

= n-+oo lim ~ an A: = 1 ~ • A:=m

e N if and only if

Jim Bn ' A:= 0

n-+oo

and

n

lim ~BnA:

n-+oo~

'

A:=O

= 1.

Therefore Theorem (9.2) implies the following result.

Theorem (10.9) : (Lozinskii) Let Bn,A: for n e N and k = O, 1, •.. , n be real numbers. Consider

= L Bn,A:DA:(z). n

Ln(z)

k=O

If Bn,A: satisfy the properties Jim an A: = 0 n-+oo

for all

'

n

lim ~Bnk

£

and

n-+oo~ k=O

ILn(z)ldz

~M

'

k E No,

=1

for all

n E N,

then (Ln)neN is an approximate identity for each homogeneous Banach space B. The two limit conditions of Theorem (10.9) are the first and second of the so-called three 'lbeplitz-conditions of an infinite matrix (Bn,A:)neN, A:eNo:

(Tl)

Jim Bn,A:

n-oo

=0

for all k

eN

(T2) (T3)

00

E

A:=O

lan,A:I ~ C for all n e N (C is a constant) .

TRIANGULAR SCHEMES

133

Theorem (10.10) : Suppose that the 'lbeplitz..conditions {Tl), {T2) and {T3) are satisfied as well as the boundedness condition:

£

ILn(z)ldz :5 M

for all n E N.

If/ is an integrable function on T, and if/ is continuous in we have:

lim Ln •

n-oo

zo ET,

then

/(zo) = /(zo) .

PROOF. We can assume that zo = 1. Let 0 by the 'Ibeplitz-condition {T3) it follows :

< 6 < 11'. For 6 :5 t :5 211' -6

for all n e N. Since / is continuous in zo = 1, for given E > 0 there exists a number 6 > 0 and a function g e C{T) with /(1) for t E (0, 6) U [211' - 6, 211'] as well as 1

,2'r-f

11' J, 2

l/(e") - g(e")ldt

Then we have ILn • /(1) - Ln • g{l)I :5

;11'

1

£

01

.

l/(z) - g(z)l ILn (z- 1}1 dz g(e">l IL,.(e-")1 dt :5 2EM + e.

= C(T) we get in particular

lim L,. • g{l)

n-oo

showing


0 inequality (iv) gives

2~ L" w8(f, t) kJ(e") dt S w8 (1. ~) 2~ lo" (1 + ~t) 2 k1(e't) dt.

llk1•I - /"8 S

Since ~ S sin

~ fo"

! for 0 S t S 7r, one obtains

t 2 k1(e0 ) dt S 7r

L"

sin

2

(~)

k1(e") dt

=~

2

(1 - (k1)"(l)),

and then by the Cauchy-Schwarz inequality

; fo" tk1(e")dt S (; fo" t k1{e")dt) l (; fo" k1(e")dt) l S ~ (1 2

(kJ)"(l))l.

Therefore

llkj•f-fllB

(1. ~) [~ + ~ ~ (1- (k1)"(1))l + ~2 : (1- (k1)"(1))] = w8 (1. x) (~ + ~27r (kJ)"(t))l ) $

2

w8

2

(1-

ELEMENTS OF BEST APPROXIMATION

143

holds. Especially for..\= (1 - (ki)"(l))-i one gets

Ilk;•! - Ills = 0 ( w8

(1. Ji - (k1)"(1))). t

Considering in Theorem (11.4) and Theorem (11.5) the Fejer kernel (Fn)neN. we obtain by (Fn}"(l) = (Fn)"(-1) =

n:h:

For the Jackson kernel CJn)neN (see Chapter 9) by means of

(Jn)" (1)

= (Jn)" ( -1) = 1 -

2n23+ 1

we have:

Consequently for

f

E Lip 8 (cr) or

llJn•/ - /lls

IE Lip:S(cr), we obtain

= O(n-

0

)

as n - oo.

But note that Jn•/(z) is a trigonometric polynomial of degree 2n - 2 and not of degree n.

Definition (11.6) : Let B be a homogeneous Banach space on T. A family (Tn)neN of bounded linear operators acting on B is called Zygmundapproximation sequence if (i) (ii)

Tn(/) E Bn llTn(/) -

Ills

=0

(w8 (/, n- 1))

as

n-

oo .

A method similar to that of Theorem (7.6) yields an example of a Zygmundapproximation sequence.

144

CHAPTER XI

Lemma (11. 7) : For each n E N there exists an even nonnegative trigonometric polynomial kn(z) of degree n such that

(kn)"(l) =cos 2-2 .

n+

PROOF. Set On,•

= sin 0, we have = O(n-

0

)

as n - oo.

PROOF. Immediately we see that 1 - (FKn)"(l) = O(n- 2 ) as n and hence the statement follows. •

-+

oo,

Another family of nonnegative trigonometric polynomials forming a Zygmundapproximation sequence can be obtained from convex combination of generalized Dirichlet kernels D1A)(eit) as follows. Fix,\ 2: 1 and 0 ~ µ ~ ,\-1. For n e N, k O, ••• ,n set

=

and

n

olA,,i) _ ~ b(A,,i) Pia - L.,, n,A: • A:..o

We define

F.ce") n

= _1_ ~ ,q(A,,i) L.,, /Jn

lc=O

,,cA.11> Dce"). n,A: A:

146

CHAPTER XI

For ~ = 11 µ = 0, we see that F~A,p) is equal to Fn, the Fejer kernel. A profound result of Gasper (see 115)) implies that F~A,p) is nonnegative. We call (~A,p))neN generalized. Fej~r kernel with parameter " and µ. It is easy to show that lim (F~A·"»"(l) = 1, see (15]. According n-oo

to Corollary (9.7) the family (PnA,p»nEN Is an approximate identity for B. Moreover, if µ + ~ > 2, then 1 - (F~A·"»"(l) = O(n- 2) as n - oo, see (15]. Thus (PnA'"»nEN defines a Zygmund-approximation sequence provided ~ ~ 1, µ ~ 0 andµ+~> 2. The following figures 13 and 14 show approximations to the Heaviside function by means of the Fejer-Korovkin kernel and the generalized Fejer kernel of order n = 21 with parameters ~ = 2, µ = 1.

FIGURE 13: APPROXIMATION USING FEJER-KOROVKIN KERNEL

147

ELEMENTS OF BEST APPROXIMATION

FIGURE

14: APPROXIMATION USING GENERALIZED FEJER KERNEL WITHµ= 1, ..\ = 2

For the trigonometric polynomial of best approximation from B,. we evidently can derive from Corollary (11.8) or applying (F~~·"»raeN with

µ+..\>2:

Corollarv (11.9) : (Jackson) If B is a homogeneous Banach space on T, then for / E Lip8 (a) or Lip8(a) we have

I

E

So far we have studied the implications of the modulus of smoothness. For the converse problem, i.e., the conclusion of the convergence rate of 6s(/, B,.) to the modulus of smoothness of/, the inequality of Bernstein is essential (compare Exercise 4 of Chapter 2). In fact there is a fundamental theorem on the equivalence relating the rate of best approximation with

CHAPTER XI

148

the modulus of smoothness. For an overview on this topic we refer to (5, p. 153 ff).

Theorem (11.10) : (Bernstein) Let Pn(z) be a trigonometric polynomial of degree n. For each homogenous Bane.ch space B on T we have

PROOF. Denote Dn(z) then-th member of the Dirichlet kernel. By Theorem (1.5) follows for z = e't

Pn(z)

= Dn • Pn(z) = ;1r

£

Pn(u)Dn(z - u) du

t cos

= 21 111' Pn(e") (1 + 2 1r

-11'

A:=l

k(t -

s))

ds.

Differentiation of the integral with respect to t gives

The function

n-1

L ksin((2n A:sl

k)u) is a trigonometric polynomial consisting

of terms of degree greater than n. Since Pn has degree n, Theorem (1.5) yields

149

ELEMENTS OF BEST APPROXIMATION

Furthermore we have n-1

L k(sin(ku) + sin((2n - k)u)) + nsin(nu) n-1

= 2sin(nu) L

kcos((n - k)u)

k•l

+ nsin(nu)

n-1

= n sin(nu) ( 1+ 2 ~)1 - : )cos(mu)

)

m•l

= nsin(nu) Fn-1(e'"), where (Fn)neN denotes the Fej~r kernel. Hence

and estimating the B-valued integral, we obtain

Now we can show:

Theorem (11.11) : n-+ oo then

Let

I eB

and

Q

0(611 ) ws(/, 6) = 0(6l ln(6)1) { 0(6)

as 6-+

> 0. If 6s(/, Bn)

= O(n-

)

as

for 0 1

o+ .

PROOF. Denote P:(f) an element of best approximation The assumption implies a constant M such that

to/ from Bn.

for every n e N . Define Q2(z)

11

= P~(/)(z) and Qn(z) = P/!.(l)(z) -

Pf!- 1(/)(z)

CHAPTER XI

150

for n = 3,4, ... . For n = 3,4, ... we have

and

llQ2lls

Hence there exists a constant

11/lls

$

+ (r2a)M.

M such that

llQnllB $ r" 0 M for

n

= 2,3, ....

We have n

lim II/-~ n--too L..J Q11:lls A:=2

= n-....oo lim 11/-Pfn(f)lls = O,

and thus

L llQnllB < oo. 00

11/lls $

In a similar way we obtain for h E (0, 27r( and m

llLe-•"f - Ills

m

$

L

A:=2

= 2, 3, .•..

llLe-1i.Q11: - Q11:lls + 2

oo

L

llQ11:lls .

A:•m+l

Since

Le-•i.Q11:(z) - Q11:(z)

=

fo" Qn'(ze'•) ds

for almost every z e T ,

we get

where the Bernstein's inequality of Theorem (11.10) is used. Consequently we have

ws(/,6)

$

2M (6f 2"=+ 0

A:=2

f

A:-=m+l

r"=

0

)

= 2M6~2"= + 2M ra L..J

A:=2

1-2-0

•

ELEMENTS OF BEST APPROXIMATION

151

!.

We ma.y consider only 0 < 6 < Choose m e N such tha.t 2m-l S With Mi = ~~2°_1( the inequa.lity a.hove gives

l < 2m.

m

Wg(/,6) S 2if6L2l(t-o) +M16°. k=-2

Now we study three different cases: (i)

For 0

< a < 1 we ha.ve

j ,

a.nd thus by 2m+l S

For a = 1 we obtain

(ii)

ws(/,6) ~ 2M6(m -1) + M 16.

Since 2m-l S }, we ha.ve m -1 ~ lln(6)/ln(2)1 and then wg(J, cS) = 0(61 ln{6)1). For

(ill)

a> 1 immediately we get

ws(/,6) S

1

~~6_ 0 + M16° = 0(6)

88

6 - 0 +.

t

The corresponding result for the symmetric modulus of smoothness reads 88

follows:

CHAPTER XI

152

Theorem (11.12) : as n- oo then

w8(!, 6)

Let

=

f

E B and er

0(6°) {

0(6 l ln(6)1) 0(6 2 ) 2

> 0.

If 6s (/, Bn)

for

0

=0

(n- 0 )

L.J A:ml k < 6 < :n

t

2t

l~I for each y E A. Hence there does not exist an element of best approximation to x form A. (2) Let X be a normed space. If llxll = llYll = ll!(x + y)ll = 1 implies x = 11 for any x, 11 E X, then X is called strict convex. Show: If A is a closed subspace of a strict convex space X and x E X\A, then there exists at most one element of best approximation to x from A. Note that the spaces V(T), 1 < p < oo, are strict convex, whereas L 1 (T) and C(T) are not strict convex. Prove these statements.

(3) Denote B = L2 (T). Let IE Band n E No. Show: The element of best approximation to I from Bn is Dn•I =Sn(/) then-th Fourier sum. (Hint: Use Exercise 1, Chapter 4) (4) Let B be a homogeneous Banach space on T and IE B. Show: (a) If Pn is a trigonometric polynomial of Bn, then llSn(/) llSn(/) -

Ills Ills

+ llSnll 8 )111 - Pnlls, {1 + llSnllS) 6s(f, Bn)·

$ (1

~

(b) If B = L 1 (T) or B = C(T), then llSn(/) -

Ills

$ M(l + ln(n))

6s(/, Bn),

where M is a constant independent of I and n. If B = V(T) with 1 < p < oo , then

where M, is a constant depending only on p .

CHAPTER XI

158

(5) Let

I e B = V(T), 1 < p < oo . Show: For 0 < a < 2 we have

if and only if I is an element of LipB{a) . (6) Let B be a homogeneous Banach space on T and 0 1 we have

1"

t"'Fn(e")dt = O(n- 1 )

88

n-+ oo.

(Hint: Split the integral on [O, 1r] into two parts on [O,

[*, 1r]. Then use F11 (e") $

n

+ 1 and Fn(e")

$ (n + l)t 2 .)

(b) If IE Lip8(a), then one has with n-+ oo

llO"n(/) -

Ills

*] and

71'2

for 0 < a < 1 O(n-"') 1 = O(n- ln(n)) for a= 1 { O(n- 1 ) for 1 = 0 for all n E Z, then (q/f(n) = 0 for all n E Z, which means 7j/ = 0 almost everywhere, i.e. f = 0 almost everywhere. Now, on the one hand, enq, n E No, are orthogonal elements of A. On the other hand, all e_nq, n EN, are orthogonal to A, since q is orthogonal to e 1A. Therefore, A is equal to the closure of the linear hull of {enq : n E No}. This means exactly A= qH2(T). t Applying the Beurling-Helson theorem (12.6), we obtain some interesting consequences:

If A~ {O} is an invariant subspace of H2(T), then Corollary (12.7): A= qH2(T), where q is an inner function. PROOF. By A~ H2(T) there does not hold e1A =A. In fact, assuming e1A = A implies e_nA = A for all n E N. Select f E A ~ H 2(T), f ~ 0, and let n E No with lcn) ~ 0. Then e-n-1/ E A ~ H 2 (T) is valid in contradiction to (e_,._iff(-1) = f(n) ~ 0. Hence, case (2) of Theorem (12.6) applies. For the existing function q according to Theorem (12.6) we get qH2(T) =A~ H 2 (T). Consequently, q is an element of 112(T), i.e., q is an inner function. t We are also able to derive easily the result of Exercise 9, Chapter 4.

Corollary (12.8,): Let f E H2(T). Ifthe set {t E (0,211'(: /(e") = O} has a positive Lebesgue measure, then the function f is zero almost everywhere. PROOF. By/ e 112(T) we have A1 ~ H 2 (T). HI =-F 0 then A1 ~ {O} and by means of Corollary (12.7), we obtain A1 = qll2(T), where q is an inner function. Since f equals zero on {t E (0,211'(: /(e") = O}, each function of

POISSON INTEGRALS AND HARDY SPACES

169

A I is also equal to zero almost everywhere on this set, a contradiction to A1 =qH2 (T). t In the proofof Corollary {12.8) we have observed that for J E H 2 (T), J :f 0, it holds A1 = qH2(T), where q is an inner function. In particular, there is a function he H2(T) such that

f =qh. Obviously, h is an outer function, since we have

Theorem (12.9) : (Factorization theorem) 2 Let f e H (T), I :f 0. Then f can be written as a product

/=qh, where q is an inner function, h an outer function. This factorization is unique up to a constant of absolute value one. PROOF. The uniqueness of the factorization remains still to be shown. Let q1h1 = q-zh2 be almost everywhere, where qi. q2 are inner functions and hi, ~are outer functions. Then hi = -q1q-zh2 almost everywhere and

Hence, q 1q2 and q1q2 are elements of H2(T). Now functions g and g can be both in H 2 (T) only, if g is constant almost everywhere. Hence, q1q2 equals almost everywhere a constant of absolute value 1 showing the stated uniqueness.

+

The factorization theorem can be extended. For that we prove:

CHAPTER XII

170

vm

Lemmo {JS.JO} i For I E H 1 (T), I :/: 0, denote g = E L2 (T). 2 Then A1 = qH (T), where q is a function on T of absolute value 1 almost

everywhere.

PROOF. Set 'I'

Obviously, V' we have

if (e") -_ { /(e")/l/(ei')I 0 if

/(e") :/: O /(e") = 0.

e L00 (T) and / = g2VJ. Suppose now A, = e1A1 • For n

~ 1

Hence, gip is orthogonal to all functions gen, n E N. By A, = e1A, the element is orthogonal even to A,. Applying Theorem (12.6) we get 2 A,= XEL (T), E ~ T, Ea Borel set. By g e XEL 2 (T) we see g(e") = 0 for almost all e" E T\E. Furthermore, XE9TP E XEL 2 (T) = A1 implies

g-,,

i.e., g-,,(e") = O for almost all e0 EE. That means/= O, a contradiction. Thus e1A, :/:A, and by Theorem (12.6) the statement follows. t

Theorem (12.11) : Let IE H 1 (T), I:/: o. Then I can be written as product / = qh2 , where q is an inner function and h an outer function.

v'l7f

PROOF. According to Lemma {12.10) write g = as g = ijh, where h E H 2 (T) and q has absolute value 1 almost everywhere. We have Ah = H2(T), since Ah = qA1 = H2(T). Defining VJ as in the proof of Lemma (12.10), we obtain

If we prove that q = rp(q) 2 is an inner function the desired factorization off follows. Since h is an outer function, there exist trigonometric polynomials Pn e H 2 (T) such that Pnh tends with respect to the L 2 (T)-norm to the constant function 1 E H 2 (T) as n - oo. Then Pn/ E H 1(T) and Pn/ = cpq2(Pnh)h - cpq2h as n - oo with respect to the L 1 (T)-norm. Thus, cpq2h is an element of H 1 (T). Moreover, h e L2 (T) implies that v'rh even belongs to H2(T). Again multiplying by Pn we get Pncpq2h n - oo in the L 2 (T}-norm. Hence, e H 2 (T) and by Corollary (12.8)

w

was

171

POISSON INTEGRALS AND HARDY SPACES

the zero-set of tp(f has Lebesgue measure equal to zero. Consequently, r.p';f is an inner function.

+

From the preceding proof we know the zero-set of tp(f, and hence that off is a Lebesgue null set. Therefore, we have a stronger version of Corollary (12.8):

Corollary (12.12): lffor f E H 1 (T) the Lebesgue measure of {eit ET: /(e") = O} is positive, then f = 0 almost everywhere.

The factorization off E H 1 (T) in Theorem (12.11) contains also the following result.

Corollary (12.13): with I= Jih.

For given

f

E

H 1 (T) there exist Ji, h

E

H 2 (T)

PROOF. Take in Theorem (12.11) Ii = qh, h = h. Writing the Fourier coefficients of qh as a discrete convolution (compare Exercise 7(b)) we get

/i E H 2 (T). t

The decomposition off E H 1 (T) may be transferred to FE H 1(U). For f E H"(T) define F(re") = Pr • /(e"). Then F E H"(U) and the Taylor coefficients Cn of F(z) =

E enz" are exactly the Fourier coefficients /(n). ~

~

n•O

Then F is called the analytic extension of f. For the analytic extension F of an outer function f E H2(T) the following can be derived:

Theorem (12.14) : Let / E H2(T) be an outer function. Then the analytic extension FE H 2 (U) off has no zeros. PROOF. Suppose /(re") Then for each m E No

= Pr• /(e") = 0 for some r E IO, 11, ~

Pr• (em/)(e")

= Lr"f(n n•O

m)e'"'

t E

IO, 211'1-

172

CHAPTER XII

Therefore, Pr• g(e") = 0 for all functions g belonging to the linear hull of {em/ : m E No}. As g - Pr• g(e") is a continuous linear functional on L2 (T) we obtain Pr• g(e") 0 for all g e A1 H 2 (T), a contradiction. t

=

=

Theorem (12.16) : For F e H 1(U) there exist Q, S e H 2 (U) with F = QS2 and IQI 5 1. In particular, there exist Fi. F2 e H2(U) with F=F1F2. PROOF. We only consider F # 0. For all r e [O, l[ we decompose Fr according to Theorem {12.11) 88 Fr = qr(hr) 2 , where qr, hr e H 2 (T) and lqrl = 1 almost everywhere. We have llhrll~ = llFrll1 S M, and llqrll2 = 1. These boundedness conditions imply the existence of a sequence (rA:)A:eN. rk E [O, 1[, converging to 1 such that (hr.) tends weakly to some he H 2 (T) and (qr.) converges weakly to some q e H 2 (T) (see Appendix). Denote Qr I and Q, S the analytic extenSiODS Of qr 1 hr and Q 1 h On the open disc U. In particular, we get for 0 5 s < 1, 0 5 t < 211':

sr

and Um Qr•(ae") 11-00

= Q(se") 88 well.

Furthermore, IQr(z)l 5 1 for z

e U,

hence, IQ(z)I 5 1 and

F(rz)

= Qr(z)(sr(z)) 2

(compare also Exercise 7(c)). This yields for z

•

Finally, take Fi

= QS and F2 = S,

e U with k -

oo

proving the second statement

88

well .

POISSON INTEGRALS AND HARDY SPACES

173

Corollary Q2.16) : Let F e H 1 (U). For almost all t exists the radial limit /(e") := lim F(re"). Then f

e (0, 211'( there e H 1 (T) and

lim llFr -

r .....1-

(1)

r ..... 1-

/II 1 = 0 are valid. F\Jrthermore, the following holds:

11/111 = r .....1lim llFrll1 = sup{llFrllt:

0 $ r < l}.

(2) For all 0 $ r < 1 and t e (0, 211'( it holds Fr(e") (3) The mapping F -

I

= Pr • /(e").

is a bijection of H 1 (U) onto H 1 (T).

PROOF. Write F = F 1F 2 with Fi. F2 e H 2 (U). By Theorem (12.3) or Theorem (4.13) the radial limits of Fi and F2 exist almost everywhere. Hence,

exists for almost all t

/2 = r ..... Um1- F2 ' r

e (0, 211'(.

By the convergence

in 112(T) the element J =

Ii = r lim Fl,r ..... 1-

and

fih belongs to H (T). Also, we 1

have

hence,

llFr -

/111 $ llF2,rll2llF1,r - /ill2 + ll/ill2llF2,r - '2112 S ll/2ll2llF1,r - ftll2 + llftll2llF2,r - /112

yielding lim Fr = f in H 1 (T). Statement (2) holds, since the Taylor r-.1coeffi.cients of F coincide with the Fourier coefficients of/. Statements (1) and (3) can be demonstrated as in Theorem (12.3). t

Corollary 02.17) : (F. and M. Riesz) Let µ E M (T) with jl(n) = 0 for all n < O. Then µ is absolutely continuous with respect to the Lebesgue measure.

PROOF. The function F: U - C defined by

F(re")

= Pr• µ(e") = L 00

n•O

r"jl(n)e'"'

174

CHAPTER XII

for 0 ~ r < 1 and 0 ~ t < 211", belongs to H 1(U) since means of Theorem (12.16) (3) we get

llFrll1

~

111'11· By

Lr"J(n)e'"' 00

F(re") =

n=O

for some f E L 1 (T). Then J(n) = J'(n) holds for all n E Z implying /;/(e")dt = dµ(e") (see Exercise 11, Chapter 7). t

EXERCISES (1) Prove that harmonic functions F, u: U - Care defined for f E L 1(T) andµ e M(T) by F(re") =Pr• /(e") and

u(re")

= Pr • µ(e"), r E (0, 1( and t E (0, 27r(.

(2) Prove: If F: U - C is a harmonic function then there exist complex numbers Cn, n E Z, such that

F(re")

=

L

00

n--oo

enr'"'e'"'

for r E (0, 1(, t E (0, 27r[. Fixing r, the series converges absolutely and uniformly. (3) Let F: U - C be a harmonic function. Let zo e U and 0 < r < 1 such that {z E C : lzo - zl ~ r} ~ U. Prove the following mean value property of F:

F(zo)

1 ("

= 211" Jo

F(zo + re")dt.

(4) Let F: U - C be a harmonic function. Prove: (a) If F(z) ~ 0 for all z E U, then there exists a positive measure µ E M(T) satisfying F(re") = Pr• µ(e") for all r E [O, 1[, t E [0,27r[. (b) If there exists

f

E

L 1 (T) with lim

r-t-

llFr - /111 = 0, then

F(re") =Pr• /(e") for all r E [O, 1(, t E [O, 27r[.

POISSON INTEGRALS AND HARDY SPACES

(5) Let

f

E C(T). Define F : 0

-

C on the closed unit disc by

F(re•c) = {Pr• /(e")

/(e")

175

for r E [O, 1[ for r = 1.

Prove that F is a continuous function on

0.

(6) Let E ~ T be a Borel set. Prove that the closure of the linear hull of {xsen: n E Z} in L 2 (T) equals XEL 2 (T). (7) Let /, g E H2(T). Prove: (a) / • g (b)

e H 00 (T) and (/ • g)ln) = /(n)g(n).

Jg E H 1 (T), and for n E No it holds (/g)ln)

(c) Pr• (Jg)

=L n

k=O

f(k)g(n - k).

= (Pr• /)(Pr• g) for all r E [O, 1[.

(8) Let/ E H2(T). Demonstrate the equivalence of the following two statements: (a)

J is an outer function.

(b) If g E H2(T) and if g/f E L2(T), then g/f E H2(T). (9) Let

f e L 1 (T). Show:

If there existsµ E M(T) such that

#l(n) = (-i)sign(n)/(n), then µ is absolutely continuous with respect to the Lebesgue measure.

(.?\ Taylor & Francis ~-

Taylor & Francis Group http://taylora n dfra ncis.co m

13

CONJUGATION OF APPROXIMATE IDENTITIES Applying the results of Chapter 8 we obtain for 1 < p

llFn • I

< oo

- illp = llH(Fn). I - H(f)llp -

0

as n - oo for every/ E V(T). AB before H(/) is the Hilbert transform of/ and

Fn(z)

=

t

A:•-n

(-i)sign(k)

(1- nl~ 1)z = H(Fn)(z) 11

the conjugate Fejer kernel. Similar results hold for other conjugate kernels. At first we prove a weak type {1,1) inequality, a result of Kolmogorov.

Theorem (13.1) : Let / E C(T) and

~

(Kolmogorov)

> 0. Then we have

c m({z ET: IH(f)(z)I >~})$xii/Iii,

where c is a constant independent of / and ized Lebesgue measure on T.

~.

Here m denotes the normal-

PROOF. We start with

/(z) = Q(z) =

L n

A:•-n

a1czA:

a t~~nometric polynomial such that Q(e") ~ 0 fo!:_ each t E [O, 211'[ and /;; f0 Q(eit)dt = 1. We define h(z) = Q(z) + iQ(z). Then we have (compare Chapter 8): h(z) n

_

n

·-· n

= ao + 2 E a1cz•,

E 2Re(a•z"), Q(z) = E 21m(a•z•),

11-1

Q(O)

= 1.

lc•l

177

Reh(z)

a1c

= G:i,

Q(z)

= ao +

= Q(z) and h(O) = ao =

CHAPTER XIII

178

z = reit E Uthe identity Q(z) =Pr• q(e"). Consequently we have Reh(z) ~ 0 for every z EU. Hence for A > 0 there is a neighborhood V of U = { z E C : lzl S 1} such that ( ) h z -..\ ip(z) = h(z) + ..\ + 1 , z E V

If we write for the moment q = QIT, we get for

:+i

is a holomorphic function on V. Now observe that w + 1 maps the set {w EC: Rew~ O} onto the disc {w EC: lw -11$1}, and moreover {w EC: Rew~ 0, lwl ~..\}onto {w EC: lw-11 $1, Rew~ 1}. Set

E>.

= {e" ET: IQ(e")I > ..\}.

Since lh(eit)I ~..\fore" EE>., we have Reip(eit) ~ 1 on E>. and Reip(e'') ~ 0 for every t E [O, 2w(. By Cauchy's theorem it follows that ip(O)

= 2~ fo2fr ip(e")dt .

!x and

But ip(O) = ~+i + 1 = 1

ip(O) = Reip(O) =

~ That means m(E>.) $

2

2~ fo • Reip(e") dt

1 f Recp(eit) dt 21r }E,.

~

m(E>.).

f!x $ i·

Now assume f E C(T), f ~ 0. Replace f by / • Fn, where Fn is an element of the Fejer kernel. The trigonometric polynomials / • Fn are nonnegative and by the results of Chapter 8 we have llH(f. Fn) - H(f)llt $ llH(f. Fn - /)112 $

II/• Fn -

/112 - 0

as n - oo. Hence we obtain m{{z ET: IH(f • Fn)- H(f)I > ..\})- 0 n - oo. Therefore the assertion follows from the result derived for nonnegative trigonometric polynomials. Finally write f E C(T) 88 f = ft-h+if3-if4, where fj ~ 0and11/Jlh $ 11/llt for;= 1,2,3,4. Then H(/) = H(/1)-H(h) +iH(/3)-iH(f,) and

88

.

{e" ET: IH(f)(eit)I > ..\} give the statement. t

'

~ U {eit ET: IH(f;)(e't)I > ~} ;-1

CONJUGATION OF APPROXIMATE IDENTITIES

179

Let g e C(T) and f E L 1(T). Then H(g • /) and H(g • /)(z) = H(g) • /(z) for almost every z e T. Lemma (13.!) :

PROOF. Since g E C{T) ~ L 1 (T) we can choose for P: 0 =to (/(ze-") - /(ze")} cot(~) dt (P

= 211' Jo 1

1'

(/(ze-") - /(ze")) H(k;)(e")dt

(/(ze-") - /(ze")) X;(e")dt - -1 211' tp(J) -

2~ /," (/(ze-") - /(ze")) X;(e")dt = 11(i) + 12(;) + /3(j).

Given E > 0 choose /J > 0 such that llLeu/ - Ills < E for any 0 $ ltl and then select j near to io such that 0 < tp{j) < 6, then llI1(i)lls $

and also

=.11')o/tp(J) IH(k;)(e")ldt $

hence

lim

J-+Jo

ll/3(j)lls

MiE, 11'

0 is a constant independent of n and/.

Note that Lemma (8.1) contains a corresponding result with respect to pointwise convergence. Next we consider the Poisson kernel (Pr)re(o,1(· For the Hilbert transform H(Pr) =Pr we derive: H(Pr)(z)

=

L 00

00

(-i)sign(k)rlklz"

hi

~-00

= 21m

= 2L:r"sin(kt)

2 (~(rz)") =Im~= rsint L..J 1-rz l-2rcost+r2 k•l

for z = e". According to the notation of Theorem (13.5) we set cp(r) Since IH(Pr)(z)I $ 2

2 L rk = _ r 1- r 00

A:=t

= 1-r.

CONJUGATION OF APPROXIMATE IDENTITIES

183

for every z E T we have

Ll-r

jH(Pr)(e")I dt 5 2r 5 2.

Furthermore it holds

r(e") =cot(!)2rsint X 2 1 - 2r cost + r 2

=

2

(1- r) cot(0 1 - 2r cost + r 2

_ (1-r) 2 oot(j) _ (1-r) 2 oot(j) - (1 - r) 2 + 2r(l - cost) - (1 - r)2 + 4r (sin(~)) 2 ' and hence we obtain

r lxr(e")ldt $ 11-r

(l - r) 4r

r . Ji_,.

2

1 , 3dt (sm 2)

(1 - r) 2 w3 /.• dt 11'3 < - O. Then m({z ET: IH(f)(z)I

> ~})

where c is a constant independent of / e.nd

c s xll/1111

~.

PROOF. As in the proof of Theorem (13.1) we can assume that f E L 1 (T) satisfies/~ 0 almost everywhere. Moreover by considering/• Fn, we can even reduce the problem to nonnegative trigonometric polynomials.

CHAPTER XIII

186

In fact / • Fn converges in L 1-norm to/ on the one side. On the other hand Corollary {13.10) and a result in the Appendix A imp~

m({z ET: IH(f • Fn)- H(l)I > ~}) - 0 as n - oo. Now the statement follows by the first part of the proof of Theorem (13.1). t Having shown Theorem (13.11) we can improve Lemma (13.2):

Let/ E L 1(T) and let g be another element of L 1(T) such that H(g) Then H(g • /) E L 1(T) and H(g • /)(z) = H(g) • /(z) for almost every z e T. Lemma {19.Jf) :

E L 1(T).

PROOF. We proceed as in Lemma (13.2), but apply Theorem (13.11) instead of Theorem (13.1). t

Theorem (13.13) : Let B be a homogeneous Banach space on T. Let / e B such that H(/) e B. Then the following holds: (1) n-ooo Jim llH(Fn) •I - H(/)lls = O. (2) lim llH(Pr) • / - H(f)lls r-ol-

= 0.

(3) lim llHc(/) - H(f)lls = O. c-oO+

= j almost everywhere, and for every k E Z.

(4) H(/)

(H(/)f(k)

= (-i)sign(k)lck)

(5) H(H(/))(z) = - /(z) + 2~ fT /(u)du for almost all z ET. PROOF. Interchanging in Lemma (13.12) the roles of Fn and/, we get

H(Fn) • I = H(Fn • /) = Fn • H(I) almost everywhere. Hence lim llH(Fn) •

n-+oo

f - H(f)lls

= n-+oo lim llFn • H(/) - H(f)lls = 0,

CONJUGATION OF APPROXIMATE IDENTITIES

187

and (1) is shown. In the same manner (2) can be proved. For (3) note that for E = 1-r

llH.(J) - H(J)lls ~ llH1-,.(/) - H(P,.) •Ills+ llH(P,.) • / - H(J)llB· Now use Corollary (13.7)(1) and statement (2) above to get assertion (3). Further (3) implies

(Hff'(k) = lim (un(/)f'(k). n-oo

Since for

lkl

~ n

we have

un(Jf'(k) = (-i) sign(k) ( 1 we obtain

nl~ 1 )

lck),

= (-i)sign(k)f(k) for every k e Z, which also gives H(J) = j almost everywhere. Finally to prove (5) we observe that H(H(Fn)) = -F" + 1. By Corollary (H(/)f'(k)

(13.10)(1) we have

Um H(Fn) • H(J)(z)

n-oo

= H(H(J))(z)

for almost all z E T. By Lemma (13.12) we also have

H(Fn) • H(J)(z) = H(H(Fn)) • /(z) = -Fn • /(z) + for almost all

1

7r

2

£

/(u)du

z e T . Applying Corollary (3.3), statement (5) follows. •

We illustrate the results of this chapter by the conjugate function f, see Chapter 2. One can check that

is the conjugate of /. In fact function. Moreover

1of the Heaviside

t ..... In (jcot (UI) , [-•, 1"]

-+ R is an even

CHAPTER XIII

188

Hence, it is sufficient to prove

L-1' ln (loot (;)I) dt < oo, to get

j

e L 1 (T). But this follows by

Furthermore, one can derive that

- = {o-inf unn>

for n even for n odd,

i.e., clnn) = (-i) sign(n)f(n) and ") S2n+1(/)(e

~ 2k1 1 cos((2k + l)t). - " ) = -4 L.J = S2n+1(/)(e 11' k=O

The following figures show

FIGURE

Si1 (/)

and

+

tT) ds

_!_ ('ht In (l/(e'-)1) Pr(ei(t-•>) ds. 211"

Jo

+

Lemma {1.4.!) : Let I E V(T) with 1 $ p $ oo, such that ln(l/I) E L 1 (T). Then there exists h E H"(T) with lhl =I/I almost everywhere. PROOF. For n E Z let On = (ln(l/l)t(n) and for z E U set

ao + 2 L 0nz" 00

G(z) =

and Q(z) = exp(G(z)) .

n•l

G and Q are holomorphic functions satisfying IQ(re")I = exp(ReG(re"))

SZEG0-KOLMOROGOV THEOREM

for t E (0, 27r[, r E (0, 1(. Since G-n =

193

a;, we have

ReG(re") =Pr• (ln(lfl))(e") and Theorem (3.5) implies

lim IQ(reit)I = exp(ln(lfl)(eit))

r-1-

= l/l(e")

for almost every t E (0, 2w(. Next we derive that Q E H"(U). By Jensen's inequality (see Exercise 1) we have IQ(re")I =exp (Pr• (ln(lfl)) (e"})

= exp (

2~ fo • ln(l/l)(e")Pr(e'(t-•>) ds) 2

~ _!_ /h l/l(e")Pr(e'Ct-•>) ds = Pr• l/l(e") . 2w lo Consequently sup llQrllp ~ sup llPr •

OSrI) ds

> -oo

0

already yields ln(l/I) E L 1 (T). Conversely, suppose ln(l/I) E L 1 (T). By Lemma (14.2) there exists h E H 2 (T) such that lhl = I/I almost everywhere. In particular f = qh, where q is a function with modulus 1 almost everywhere. Assume that e1A1 = A1. Then we get

Ara

= qA1 = qe1A1 = e1Ara .

That would imply Ara = enAra for every n E Z. Since Ara ~ H 2 (T), this is only possible when h 0 almost everywhere, but this contradicts In(lhl) E L 1{T). t

=

Theorem (14.4) : Let f E L 1{T), f ~ 0 and f following conditions are equivalent: (i)

f

(ii)

There exists an element h E H 2 (T) such that where.

satisfies the Kolmogorov-SzegO-property.

:F

0. Then the

f = lhl 2 almost every-

PROOF. The square root g = ..fl is an element of L 2 {T). Obviously In(/) E L 1(T) is equivalent to ln(g) E L 1(T). Suppose at first that In(/) E L 1 (T). By Lemma (14.2) exists a function h E H 2 (T) such that f = g2 = lhl 2 almost everywhere. Conversely if J = lhl 2 , h E H 2 (T), by the factorization Theorem (12.9) we can assume that his an outer function. Now by Exercise (7)(b) of Chapter 12 the given function I is an element of H 1 (T). Moreover Theorem (12.14) and Exercise (7)(c) of Chapter 12 imply that the analytic extension F of f has no zeros. Hence Theorem (14.1) gives ln(J) e L 1 (T). t We insert a short section to demonstrate how Theorem (14.4) can be applied to solve a linear prediction problem of time series analysis. Let (Xn)nez be a weakly stationary sequence of random variables with finite variances, see Chapter 7, Exercise 8. These random variables are elements of a Hilbert space L 2 (P), where Pis a probability measure on a

SZEG6-KOLMOROGOV THEOREM

195

certain measure space. We assume that E(Xn) = 0 for every n e Zand Var(Xn) > 0. A sequence (Zn)nEZ of random variables is called "white noise" if E(Zn) = 0 and Cov(Zn, Zm) = 6n,m 1 i.e., the random variables Zn have zero mean value and are uncorrelated. The spectral measure of (Zn)nez is the (normalized) Lebesgue measure on T. Remind the notion of the spectral measure µ introduced in Exercise 8 of Chapter 7 by means of Herglotz' Theorem (7.10): Cov(Xn,Xo) =

£

z-"dµ(z).

Let Hn(X) be the closed linear subspace of L 2 (P), spanned by {XA:: k :Sn}, and H(X) the Hilbert space spanned by {XA:: k e Z}. Obviously Hn-1(X) ~ Hn(X) ~ H(X). Let H-oo(X)

=

n

nEZ

Hn(X) .

On the Hilbert space H(X) an unitary linear operator U: H(X)- H(X) = Xn+l· In fact the weak stationarity implies that the linear extension of the shift-mapping is well-defined and unitary. Thus U can be extended to H(X), see Exercise 2. Evidently U(Hn-1(X)) = Hn(X) is valid. A weakly stationary sequence (Xn)nez is called regular if is given that extends the shift operator, i.e., U(Xn)

H_ 00 (X) = {O} . Regular weakly stationary sequences are not predictable in the sense that its values are not determined by its "past". Given a weakly stationary sequence (Xn)nez a random sequence (Zn)nez is called innovation sequence for (Xn)nez if (Zn)nez is white noise and if Hn(X) = Hn(Z) for every n e Z. The following characterization of regular weakly stationary sequences can be proved merely by tools of Hilbert space theory.

Theorem (14.5) : A weakly stationary sequence (Xn)nez is regular if and only if there exists an innovation sequence (Zn)nez for (Xn)nez and

L l0nl 00

a sequence (On)nENo of complex numbers with

n•O

allneZ

L BA:Zn-A: . 00

Xn =

k=O

2

< oo such that for

CHAPTER XIV

196

The series converges in L 2 (P). A random sequence with such a representation is called one-sided moving average process. PROOF. Assume the regularity of (Xn)neZ· Represent Hn(X) in the form Hn(X)

= Hn-1(X) $Cn,

Cn

= (Hn-1(X))J.,

compare Exercise (3)(c) of Chapter 4. The dimension of Cn is one or zero, since Hn(X) is spanned by elements of Hn-1(X) and elementsaXn, a EC. But if Cn is trivial for one n E Z, then Cn is trivial for every n E Z by the weak stationarity of (Xn)nez, and then H_ 00 (X) = H(X) holds, contradicting the regularity of (Xn)nEZ· Therefore dim Cn = 1 for each n E Z. Choose Zo E Co with llZolh1 = 1, where the norm is ta.ken in the Hilbert space L2(P). Put Zn = U" Zo E Cn for each n E Z. For given n

e Z and k E No we have

•-1

= E a1Zn-I + Pn-Ai(Xn)1 where Pn-Ai(Xn) J-0 OJ = Cov(Xn, Zn-J) = Cov(Xo, Z-1>· We have Then Xn

00

L 1011

2

E Hn-A!(X) and

~ llXnll~ 1

1~0

and so

E a1Zn-I converges in L 2 (P). 00

l•O

It remains to prove that

Again this assertion is independent of n. We consider the case n = 0. Since II

P-•(Xo)

= Po(Xo) + L

1-1

(P-1(Xo) - P-1+1(Xo)), (Po(Xo)

= Xo),

and the terms in this sum a.re orthogonal, we have

t

1-1

llP-1(Xo) - P-1+1(Xo)ll:

= 11t(P-1(Xo) - P-1+1(Xo)),, 1-1

= llP-11(Xo) -

2 2

Po(Xo)ll= ~ 4 llXoll=.

SZEG0-KOLMOROGOV THEOREM

197

•-oo

Therefore Um P-•(Xo) exists in L2(P). Now P_.(Xo) E H-1c(X) and hence which means

=L 00

a;Zn-J· J•O The proof that a one-sided moving average process is regular is left as an exercise. t Xn

Corollary 04.6) : (Kolmogorov) Let (Xn)nez be a regular weakly stationary sequence with zero mean value. Then the corresponding spectral measureµ has the form µ(z) = 2~/(z)dz, where f satisfies the Kolmogorov·Szeg~property.

PROOF. Given the one-sided moving average representation of (Xn)nez define h(z)

=

H 2 {T). Since

00

Ea•~ for all z E T. The function h is an element of

•-o

= L Blc~-n = 00

e_nh(z)

1i:-o

L BHnz•, 00

lc•-n

we have by Corollary (4.10)(2)

Thus the spectral measureµ bas the form µ(z) = f.f(z)dz with /(z) lh(z)j 2 , and Theorem (14.4) gives In(/) E L 1 {T). t

=

&mark : Even the converse implication in Corollary (14.6) is valid. However for the proof one needs the concept of stochastic measures, and so we refer to (21). We point out that for the proof of Theorem (14.5) and Corollary {14.6) we have only used methods of Hilbert space theory and Fourier aeries. A direct proof yields an isometric isomorphism between the Hilbert space H(X) and the Hilbert space L 2 (µ), where µ is the spectral measure of

CHAPTER XIV

198

L

(Xra)nez, and L2 (µ) consists of all Borel measurable functions/ on T with

lf(z)l 2dµ(z) < oo. This isomorphism maps Xn to e_n, see Exercise 4.

The best linear prediction for the future point of time n > 0 given the complete past k $ O, is the random variable Xn E L2 {P) that satisfies

x,Ct

Applying the one-sided moving average representation of a regular weakly stationary sequence (Xn)nez, we immediately obtain

=L 00

Xn

k=n

BA;Zn-kt

and for the prediction error

see Exercise 5. The following result gives the error for prediction one step ahead.

Theorem (14.T} :

(Szego)

Let/ E L 1 (T), / ~ 0 , and let PA be the set of all polynomials Q with Q(O) = 1. Then the following identity is valid

inf {

2~ fo " IQ(e")l2/{e")dt: Q EPA}= exp (;7r 2

Lh

In {/(e"))

dt)

Here the right side of the identity equals 0 when In{/) '/. L 1{T). PROOF. Assume that In{/) E L 1{T). As in the proof of Theorem {14.4) we can write/= lhl 2 , where his an outer function. Since ln{lhl) E L 1{T), the analytic extension of ln(lhl) is ln(ISI), where S is the analytic extension of h, compare Corollary {12.16). In particular we have

SZEG0-KOLMOROGOV THEOREM

199

Since h is an outer function we can approximate in L 2 (T) the constant function with value h(O) = S(O) by functions Qh, where Q are analytic polynomials. We can suppose that Q(O) = 1. Hence we get inf {

2~ fo " IQ(e")l 2/(e")dt : Q e PA} 2

2~ Lh 1Qh(e")l dt : Q E PA} :5 18(0)1 =exp ( ~ Lh In (/(e")) dt) 2 2

=inf {

But for every Q

e PA

3

holds

E l{Qhr 0 almost everywhere, and define the measure #J/ by 1J1(z) = 2~/(z)dz . Then the set of all analytic polynomials (i.e. the linear span of {en : n E No}) is dense in L2 (µI) if and only if In(/) ¢ L1 (T). PROOF. Suppose that In(/) E L 1 (T). As in the proofs before we have / = lhl2, where h is an outer function. Evidently the set of all analytic polynomials is not dense in L 2 (µ/) exactly when A" is a proper subspace of L2 (T). But we have A1a = H2(T) and hence the set of all analytic polynomials are not dense in L 2 (µ1 ). For the converse statement put g = '1J E L2 (T). The assumption now is that A, is a proper subspace of L2 (T). We show that A1 1' e1 A1 • If A 1 = e 1 A11 , then A, = XEL 2 (T), where E is a Borel subset of T, compare Theorem (12.6). But we have g > 0 almost everywhere, hence A1 = L2 (T) in contradiction to the assumption. Therefore A, 1' e1A, holds, which implies ln(g) e L 1 (T). But then In(/) E L 1 (T) is valid, too. t

EXERCISES (1) (Jensen's inequality.) Show: If~: R-+ Risa convex function, / : (0, 211'(-+ R is Borel measurable, and 1:• /(t)dt and / 02• ~(J(t))dt exist, then

(2) Let (Xn)nez be a weakly stationary sequence, and let H(X) be the Hilbert space spanned by {XA: : k E Z}. Show that the linear extension of the mapping U(XA:) = Xt+l is well-defined and unitary. Prove that it can be extended to H(X). (3) Show that a one-sided moving average process is regular. (4) Let (Xn)nez be a weakly stationary sequence with corresponding spectral measureµ. As usual let e,.(.z) = z" for all n E Z. Show that the linear extension of the mapping ~(en) = X-n, n E Z, is well-defined and unitary. Prove that it can be extended to an isometric isomorphism of L2 (µ) onto H(X). (5) Prove the 888ertions concerning the best linear prediction stated before Theorem (14.7).

201

SZEG0-KOLMOROGOV THEOREM

(6) Let 'I' E L 00 (T) and a for NE N

= (On)nENot b = (bn)neNo N

•N(o,b) = Prove

(7) Let

I

L

n,m•O

l•N(o,b)I $ E

E l 2 (No). Define

iP(n + m)o,,bm.

ll'l'lloo llall2 llbll:a ·

H 1 (T).

(a) Given rp E L 00 (T) with fP(n) ~ 0 for n E No, show that

L iP(n) lf(n)I $ ll'l'llooll/lh· 00

n-0

(Hint: Use the factorization of Theorem (12.11) in order to get I= gh with 11911~ = llhll~ = II/Iii. Then apply Exercise 6.) (b) Prove Hardy's inequality:

E~~~I

n•O

(Hint: Consider rp(e")

$

ll/111 .

= ie-"(w -

t) and use (a).)

(.;:\ Taylor & Francis ~

Taylor & Francis Group http://tayl o ra ndfra ncis.com

ABSOLUTE CONVERGENCE OF FOURIER SERIES For a particular subspace of C(T) the sup-norm convergence of (Sn(/))nEN to / E C(T) follows immediately. AB in Chapter 1 let

A(T)

= {! E C(T): JE l1 (Z)}

with norm

11/llA(T) =

L 00

n•-oo

lf(n)j.

We have already noted that (A(T), 11 llA(T)) is a homogeneous Banach space on T. We point out that the Bane.ch space A(T) is isometric isomorphic to the Banach space l 1 (Z). The isomorphism A(T) onto l 1 (Z) is given by and its inverse mapping is

/ -1

L 00

(On)nEZ -

n•-oo

anzn.

We start with a result of M. Riesz, which induces the existence of certain functions f E A(T) with prescribed values.

Theorem (16.1) : (M. Riesz) Let f E C(T). Then f E A(T) if and only if there exist g, h E L 2 (T) such that / g • h. Moreover 11/llA(T) ~ 119112 llhll2·

=

PROO~ Let g, h E L 2 (Tl, and consider / = g • h. Then J(n) = g(n)h(n) and g, he l 2 (Z). Hence/ E l 1 (Z), i.e. / e A(T). (Compare also Exercise

7 of Chapter 4.) Conversely if f E A(T) set On = IJ(n)l 112 and bn = ane".. , where t,. e [O, 211'( such that J(n) = IJ(n)le"". Evidently (On)nez. (bn)raEZ E 12 (Z). Corollary (4.10) yields elements g, h E L 2 (T) with g(n) =an, h(n) = b,.. Since J(n) = (g • h)ln) for every n E Z we get I = g • h. t

203

CHAPTER XV

204

Corollary (15.2): Let E ~ T be a closed set, and let U be an open subset with E ~ U ~ T. Then there exists a function f E A(T) such that

/IE = 1 and JIT\U =o.

PROOF. An elementary consideration shows that there exists an open neighbourhood V of 1 e T such that EVV ~ U. By Theorem (15.1) the function 1

I=

m(V)xv • XEV

is an element of A(T). As before let m be the normalized Lebesgue measure on T. Then we have

/(z)

and thus /(z)

1 (

1

= m(V) 211" lv XEv(zii)du,

= 1 for z e E and /(z) = 0 for z ¢ EVV. t

The Banach space A(T) is (with pointwise multiplication) even a Banach algebra with unit-element. Applying the Gelfand theory of commutative Banach algebras, we get important results in a very elegant way.

Theorem (15.3) : The Banach space A(T) is (with ordinary pointwise multiplication) a commutative Banach algebra with unit eo = 1. PROOF. We have to show that for /, g element of A(T) and

E

A(T) the product Jg is an

ll/9llA(T) $ 11/llA(T) ll9llA(T)·

We define for n E Z On

=

ex>

L

f(k)g(n - k).

A:•-CX>

Then (an)nez E l 1 (Z). In fact ex>

L

n•-ex>

lanl $

=

L E ex>

ex>

n=-ex> A:=-ex> ex>

L

A:•-ex>

llck)I

lf(k)l li(n -

L ex>

n•-ex>

li(n -

k)I k)I = lllllA(T)ll9llA(T)•

ABSOLUTE CONVERGENCE OF FOURIER SERIES

205

where the interchanging of the series summation is possible by means of the absolute summability of the sequences (lck))keZ and (g(k))kEZ· FUrthermore we also have for z e T:

L

00

n•-oo

a,.z"

= =

L L 00

00

n--oo k•-oo

L

f(k)g(n - k)z"

00

•--oo

L 00

f(k)zk

m--oo

g(m)zm

= /(z)g(z).

Thus we have shown fg e A(T) and (/gf(n) =a,.. Evidently ll/911.A(T) ~ ll/llA(T)ll9llACT) is already proved above, too. Finally eo e A(T) and lleoll.A(T) 1. t

=

&mark: In Theorem (15.3) we actually have shown that l 1 (Z) is a Banach algebra with unit 60. The algebra operation is given by the convolution on Z, i.e., given a= (On)nez, b = (bn)nez define the convolution by a• b(n)

=

L 00

a(n - k)b(k).

Jr•-oo

Theorem (15.3) contains in fact that a• b = (a• b(n))nez E l 1(Z) and Ila• blli ~ llall1 llbll1· The reader should compare the convolution on Z with the convolution on T in Theorem (1.3). We mention that such a commutative operation may be defined on any locally compact Abelian group G, see e.g. (13, Ch. VII). Compare also with Exercise 1. The fact that A(T) is a Banach algebra with unit we shall utilize now to derive important results of Levy and Wiener. Let A be a commutative Banach algebra with unit. Let &(A)= {h: A - C: h :f. 0, h linear and multiplicative} be the set of all nonzero homomorphisms of the algebra A. Thus each h e &(A) satisfies h(az + /311) oh(x) + {Jh(y) and h(:.i:71) h(x)h(11) for x, 'II e A and o, /3 e c.

=

=

The set &(A) can be identified with the set of all maximal ideals of A by h - kerh, where kerh = {x e A: h(x) = O}. A proof of this result can be found in the Appendix C. As a consequence we obtain the following characterization of invertible elements of A.

CHAPTER XV

206

Theorem (15.4) : Let A be a. commutative Bane.ch e.lgebre. with unit. Then x E A is invertible if e.nd only if h(x) #: 0 for every h E d(A). PROOF. Denote e the unit-element of A. Obviously h(e) = 1 is valid and hence 1 = h(xx- 1) = h(x)h(x- 1) if xis invertible. Therefore h(x) #: 0 for e.ny h E d(A). Now assume h(x) #: 0 for every he d(A}, but x not invertible. Then Ax is an idea.I in A. Let M be a. maxima.I idea.I in A such the.t Ax ~ M. By the result mentioned above there exists a. homomorphism h E d(A) such that Ax~ M ker h. But then 0 h(ex} h(x}, a. contra.diction. t

=

=

=

Next we determine the set of a.II homomorphims of A(T}.

Lemma {15.5) : We have d(A(T)) = {ha : z E T}, where the linear functione.ls ha a.re defined by ha(/) = /(z} for a.II f E A(T). The mapping z - ha of Tonto d(A(T)) is a. bijection.

PROOF. It is easy to check the.tea.ch h11 , z ET, is a. homomorphism on A(T). Denote e 1(z) = z and e-1(z} = z- 1. Ee.ch homomorphism is a continuous linear functione.1 with norm less or equal to 1, see Appendix C. Hence for h E d(A(T)) we get lh(e1}l ~ 1 e.nd l/lh(e1}I = lh(e-11 ~ 1. That means h(e1} ET. Let z h(e1) ET. Then h(eo) 1, h{e-1) z- 1 and finally h(en} = z" for every n E Z. Therefore h(/} = /(z} holds for a.II f E A(T), and we have shown h =ha. Since for z1, z2 ET, z1 #: z2, we have h111 (e1) = z1 ./: z2 = h112 (e1} the injectivity of z - ha follows, and the lemma is proved completely. t

=

=

=

Theorem (15.6) : (Wiener) If f E A(T) satisfies /(z) ./: 0 for every z E T, then 1/f E A(T).

PROOF. By Lemma (15.5) and Theorem (15.4) the function f is an invertible element of the Bane.ch e.lgebre. A(T). Hence there exists g E A(T) such the.t Jg= eo. That means 1// = g. • Now we ce.n prove Wiener's Te.uberie.n theorem, which states the denseness of the linear span of 11 (Z)-transle.tes of one single function f E A(T) in A(T) provided f has no zeros. The l 1 (Z)-translate off E A(T) is defined by

(e1r/)(z) fork E Z.

=

L 00

n=-oo

f(n - k)z"

ABSOLUTE CONVERGENCE OF FOURIER SERIES

207

Wiener's Tauberian theorem is an important result in spectral synthesis, compare (4).

Theorem (15. 7) : Let / E A(T). Then for every g E A(T) and there exist n E N and C-n• .•• , en E C such that

Ilg -

L n

A:•-n

E

>0

c1:(e1:/)llA(T) < e

if and only if /(z) :/: 0 for every z E T. PROOF. First assume /(zo) = 0 for zo E T. Then (e1:/)(zo) = 0 for each k E Z. If the linear span of {e1:/: k E Z} would be dense in A(T), then ker h- = A(T), a contradiction to ker h- being a maximal ideal in A(T), compare Lemma (15.5). Now suppose /(z) :/: 0 for every z E T. Given g E A(T) consider g// according to Theorem (15.6). Since (g/ ff E l 1 (Z) fore> O there exists an element h E A(T) and n E N such that h(k) = 0 for lkl > n and Ilg// - hllA(T) < e/11/llA(T)· Then

Ilg and since hf=

E

hfllA(T) ~ 11/llA(T} Ilg/f

ft

A:•-n

.....

-

hllA(T) < E,

h(k}(e1:/), the statement is proved.

t

In the original proof Wiener has used a technique of collecting local information to make a global statement. In fact the following result can be shown.

Theorem (15.8) : Let / E C(T). If for every z E T there exists a function g, E A(T) such that g, = / on a neighbourhood of z, then IE A(T). PROOF. Given z E T let g, E A(T) and v.. an open subset of T such that g.. IV.. = /IV... There exists a function 'P.s on T, where rp.. is at least twice continuously differentiable such that rp.. IV.. > 0 and rp.. IT\V, = 0. By Corollary (1.7) tp1 is an element of A(T).

208

CHAPTER XV

Since T is compact, there exists a finite set {zi. ... , Zn}

u n

n

~

T such that

v.r The function~= E 'Pa; is an element of A(T) and ~(z) > 0 Jal J=l for all z ET. Set t/J., ='Pa,/~. The functions t/J.1 are elements of A(T) and

T =

E" t/J., (z) = 1 for all z e T as well as /t/J., = g.1 t/Ja;, j = 1, ... , n. J•l Therefore we obtain

satisfy

I

=

L J.P., = Lg., .P., E A(T). • n

n

J=l

J•l

Another interesting consequence of Wiener's theorem (15.6) concerns ideals in A(T). Given a closed subset E ~ T denote k(E) := {/ E A(T) : /IE

= O}

and

j(E) := {/ E A(T) : /IUE = O, UE a neighbourhood of E}. The linear subspaces k(E) and j(E) are ideals in A(T), and j(E) Further k(E) is a closed ideal.

~

k(E).

Theorem (15.9) :

Let I be a closed ideal in A(T), I -:/: A(T). Then there exists a nonvoid closed subset E ~ T such that

j(E)

~

I

~

k(E).

PROOF. Set E = {z ET: /(z) = 0 for every f e I}. We show that Eis nonvoid. Assuming E = 0, we can find for each z E T an element /z E I and a neighbourhood u. ~ T of z such that /.(w) -:/: 0 for every w E u•. The compactness of T yields z1, ... , Zn E T such that T = En /z;/a 1 is an element of I

U u.,. n

J•l

Then

and /(z) > 0 for every z e T . Theorem J•l (15.6) implies that 1// E A(T), and hence 1 =///EI. Thus I= A(T), a contradiction. So we have E -:/: 0.

/ =

ABSOLUTE CONVERGENCE OF FOURIER SERIES

209

It is obvious that E is closed, and that I ~ k(E). Next let g E j(E), i.e., glUs = 0 where UE is a neighbourhood of E. Denote suppg := clos {z E T: g(z) ::/: O}. (Here clos stands for the closure in T.) By the definition of E for each z E supp g exists an element 'Pa E I such that rp,,(z) = 1. As before we can derive finitely many z1, ... , Zn E supp g such that

'{) = L cp,,, E I n

J•l

is positive on suppg. Now we set B = {z ET: 1.p(z) = O} ~ T\suppg. (ff B = 0 we can skip the next step of the proof.) By Corollary (15.2) there exists a function h E A(T) such that hlB = 1 and hi supp g = 0. Then 1.p+h E A(T) and (1.p+h)(z) > 0 for every z ET. Hence 1/(cp+h) E A(T) and VI = cp/(1.p + h) E I. If z E T with g(z) '/. 0 (i.e. z E suppg and hence h(z) = 0), we have cp(z) = 1. Therefore g =?/Jg E I. That was to be shown. t

Remark : A closed subset E of Tis called S-set (or Wiener set) if j(E) = k(E). That means (applying Theorem (15.9)) there is exactly one closed proper ideal I in A(T) such that h(I) = {z ET: /(z) = 0 for all/ E I} equals E. The set h(l) is called hull or cospectrum of the ideal I. Riesz' Theorem (15.1) is a characterization of A(T). The next result gives a sufficient condition for f E C(T) to be a member of A(T).

Theorem (15.10) ; / e A{T).

Let

I

E C{T).

If f(n) ~ 0 for every n E Z, then

PROOF. By Theorem {2.10) we get lim O'n(/)(1)

n-+oo

= n-+oo lim ~ L...,,

1:•-n

....

(1 - ~) + 1 f(k) = n

By /(k) 2:: 0 the partial sums 5n(/)(1)

n

= E

/(e'°)

= /(1).

....

/(k) will converge, if

1:•-n (5n(/){l))neN is bounded. Assuming the unboundedness of (5n(/)(l))neN for each M > 0, there exists no E N such that 5n(/)(1) 2:: M for every n 2:: no. But for n > no we have O'n{/)(1) = n

1

no

1

+ 1L81:(/)(l) + n + 1 1:-0

L n

•=no+1

81:(/)(l)

CHAPTER XV

210

and hence limsup un(/)(1) ~ M, contradicting the convergence of the n-oo

Fejer sum un(/)(1) to /(1). Therefore we have proved / e A(T). An important example are the functions A,. e C(T), where 0 for for Setting 'P>./2(e")

./2J(t) a direct computation shows

1"'

2

1 -A/2 X(-A/2,>./2J(t - s)ds 'P>./2. 'P>./2(e") = 211'

~ = 211'

(

!ti) 1 -1"

Hence

and therefore A>. e A(T). Note that

L

00

llA>.llA(T) =

na-oo

(A,.)ln}

= A>.(l} = 1 .

Evidently there is a strong relationship to the Fejer kernel in L1(R), eee Chapter 17. Next consider for 0

< ~ ~ i the function V,.(e")

= 2A2>.(e") -

A>.(e") .

Forte (-~.~)one has V>.(e•') = 1 and for 2~ < ltl ~ 11' we get V,.(e") Obviously llV>.llA(T) ~ 3. Furthermore the following statement holds:

Lemma {15.11) :

.\o > 0 such that

lle1: V>. - V>.llA(T)

Let n E N be fixed. For every

0 there exists

and all k = -n ... , n.

ABSOLUTE CONVERGENCE OF FOURIER SERIES

211

PROOF. Since

it suffices to prove lle1:A2>. - A2>.llA(T) small enough. From above we know

lle1:An - A2>.llA(T)


.. "'") - "'" * IP>.llA(T) 11' = xll(e1:1P>.) * (e1:1P>.) - IP>.* IP>.llA(T) 11' 11' $ xll(e1:

.) * {e1:

. -

.)llA(T) + xll

. * (e1:

. 211'

s T""'"ll2 ll(e1: -

.)llA(T)

1)cp,.ll2 •

where the last inequality follows from Theorem (15.1). Now choose a neighbourhood U of 1 in T such that

le1:(z}- ll

= lzt -11 < E/2

for all k = -n ... ,n and

z EU.

and then choose .\o > 0 such that e'" E U for every l.\I $ .\o. Since ll

.11~ = .\/1r and as ll(et - l}r,o>.11~ < £ 2.\/(411') for 0 < .\ $ .\o and k = -n ... , n, we finally obtain

Theorem (15.12) ; (Wiener-Ditkin) Let f E A{T) with /(1) = 0. We then have lim

,......o+

PROOF. Since

00

E

k•-oo

.....

/(le)

11/ -

/{l - V>.)llA(T)

= 0.

= /(1) = 0 we have for n E Z

i. v,.(n) = E

t•-00

E 00

00

i(le)V,.(n - le) -

f(le)V,.(n),

212

CHAPTER XV

and then

11/VAllA(T) =

llJ• VAll11(z) ~ =

E 00

1:... -00

E 00

IJ(k)I

n=-oo

IVA(n -

k) - VA(n)I

E IJ(k)I lle1: V.\ - V.\llACT>1:--00 00

Given e > 0 choose n E Z such that

E

l/(k)I < e/12. Further by Lemma

IA:l>n (15.11) there exist~> 0 such that lle1:V.\ - VAllA(T) all k = -n, ... , n and 0 < ~ ~ ~. Now it is obvious that

< e{211/llAI

= 111 +

oo

IA:I ....

E E lf~k>I .

a:--oo n-1 I I A:"O

Also formally 00

IA:I lf{k)I

lf(k)I

00

EE-=EEa:--oo n•l lkl n•l fA:l>n lkl A:,AO

-

by summing along the columns of the triangular scheme. Applying the inequality of Cauchy-Schwarz

E 11~>1 ~ ( E

IA:l~n I I and

IA:l~n

:2) 112 (

E

IA:l~n

lf12) 112

001 1001 1 -tb:=L.J k2 z2 n A:an n ~-
../(').z) for >.. E R, >.. #: 0, we have

(f>.f(a) = J(l).

225

CHAPTER XVI

226

Theorem (16.3) : Let f e L 1 (R). The function and uniformly continuous.

1: R -

C is bounded

PROOF. By Theorem (16.2)(4) only the uniform continuity has to be proved. Since

J(a + b) - f(a)

=

J.

e-'- f(x)(e- • - 1) dx for all a, b ER,

we obtain

lf(a + b) - f(a)I $ J.1/(x)l ·le-• - lldx . As l/(x)l · le-fk - ll - 0 when b - 0 for all x e R , and l/(x).l · le-ibz -11$21/(x)I, Lebesgue's theorP.m of dominated convergence yields Ilea+ b) - /(a)I - 0 as b - 0, independent of a e R. Similar as in L 1 (T) a convolution can be defined in L 1(R).

+

Let /, g

Theorem (16.4) : function

e L 1(R).

For almost all x E R the

Y - f(x - y)g(y)

is integrable. Denoting we have h E L 1(R) and

h(x) = J.1 0 it holds lim A-oo

1

lsl>I

(Ma constant).

lkA(x)ldx = 0.

A simple way to construct summation kernels on R is the following: If f is a continuous Lebesgue-integrable function with set for~ e }0,oo( and x e R

(1) Obviously we have

as well as

J. J.

kA(x)dx

lkA(x)ldx

=

J.

!(x)dx

=1

= J.1/(x)ldx = 11/111

fa f(x)dx

= 1, then

CHAPTER XVI

230

and for 6 > 0

f

Jl•l>I

lk.\(x)ldx =

f

Jl•l>l.I

if~-+ oo.

lf(x)ldx-+ 0

Thus the family (klhe)O,oo( defined by (1) depending only on C(R) is a summation kernel on R.

f

E

L 1(R) n

As in the foregoing chapters the Fejer kernel is very important. Set

m;a

1 ( . ") F(x) := 271'

(2)

2

In order to apply the above mentioned construction we only have to show that fa F(x)dx = 1. The family (FlheJO,oo( with Fl(x) = .\F(..U) is called the Fej.Sr kernel on R. 1b prove fa F(x)dx = 1 we apply the results for the Fejer kernel on T. With Lemma (2.7) we have for all n e N

1•

=ft) l dt=l · 2

-1- (sin 271' -• n + 1 sin

-1

lim -

1

n-oo 271'

/.2tr-I -1- (sin -=j!t) n +1

1

sin

!

2

dt=O

(this is the condition (S3) of Definition (2.2)), it holds lim - 1

1'

=j!t) ! dt 2

-1- (sin n-oo 271' _1 n + 1 sin

(*)

For 0

for all

= 1.

< E < 1 there exists 6 > 0 with

ltl < 6.

Consequently we get for this 6 > 0: _1

_.!._1' _ 1 (sin=jlt)2 dt < _.!._

1 + E 211' _ 6 n + 1

sin

l

-

r

_1

21" }_, n + 1

-r

(~)2 dt

FOURIER TRANSFORM ON R


0 by Theorem {16.2)(5):

where

(1+e)ei(•cte = ;__..!..re-Ml-lJ, 14 a 2

cos a = _!_ (sin j)

2

21"

2

= F(a).

= Fis proved completely.

FA(a)

= ~(.Xa) = .XA(.Xa) = (A.\na),

A.\(y)

= { -j;{l - tp) 0

'

for for

1111 :5 .x 1111

> .x.

Thus

CHAPTER XVI

232

Now one can show as in the preceding chapters that summation kernels on R are approximate identities in L1 (R), where a family (A:>.he)O,oo( of continuous functions on R is called an approximate identity for L1 (R), if lim Ilk>. • I - /Iii = o A-+oo holds for all / E L 1 (R). Indeed for summation kernels (k>.heJO,oo( on R one can directly transfer the results corresponding to Lemma (2.3) and Theorem (2.4). To show the analogue of Lemma (2.5) one has to note that the space of continuous functions with compact support (this subspace we will denote by Coo(R)) is dense in L 1 (R). Thus one obtains:

Theorem (16.12) : kernel on R. Then

f

Let lim

A-+oo

E

L 1(R) and (k>.heJO,oo( be a summation

II/ - A:,. • /II 1 = o.

In particular considering the Fejer kernel it follows:

Let

Corollary (16.13):

f

'1>.(/)(:i:) :::::

E

2~

l:

L 1 (R). Denote for~> 0 the function

(1- 1~1 )f(a)e'-do.

Then with respect to the L 1 (R)-norm we have

I= A-+oo lim ""(/). PROOF. By the foregoing we have F,.(x)

l:(

Hence by Theorem (16.6) F>. • f(x)

=

2~

1-

= 2~

l: (1-

l~I) e'°sdo.

l~I) f(a)ewda = '1>.(/)(x). •

Important consequences of this result are:

233

FOURIER TRANSFORM ON R

Corollary Q6.14) : (Uniqueness theorem) Let/ E L 1 (R) such that /(a) = 0 holds for all a ER. Then f everywhere.

Corollary (16.15) :

Let/ E

L1(R)

= 0 almost

(Inversion formula) such that E L 1 (R). Then

1

/(x)

= 2111'

!. . . 8

/(a)e'-do

for almost every x E R. Moreover that equality holds in every point where is continuous.

I

PROOF. For fixed x

e R the functions a-

X[-.\,A)

(1-

l~I) /(a)e'-•

converge pointwise (as.\ - oo) to the function a - /(a)ew. The absolute value of these functions is majorized by By Lebesgue's dominated convergence theorem it follows that

1/i.

lim FA• /(x)

.\-oo

1 = .\-oo llm 211'

j" (1 -A

1 1~1) j(a)ewdo = 2 "

11'

J. ica)e'-do. a

By Theorem (16.18} (which will be shown at the end of this chapter) FA•f(x) converges to /(x) for almost every x E R as .\ - oo and the

assertion follows. The same theorem gives the additional statement concerning the points of continuity of/. t

R.emark: Defining the ~called inverse Fourier transform of g e L 1(R)

by

g(x) :=

2~ J. g(a)e'-do

for all x E R, the inversion formula of Corollary {16.15) can be written as

I = (ft almost everywhere. However, observe for the application of Corollary (16.15) we have to 888Ume that j E L1 (R).

CHAPTER XVI

234

We have already shown that FA(x) the inversion formula follows

for all a

eR

d-'(a)

= (d-'f(x) for all x e R is valid.

By

. Corollary (16.15) gives namely

= 21~

J.

•

FA(x)etudx =

Corollary (16.16): in L 1 (R).

J.

1 FA(x)e-•sdx 2~ •

The space of all functions with

PROOF. It holds (FA• /f(a)

= _!_pA(a). 2~

j e C00 (R) is dense

= FA(a)f(a) = X(-A,AJ(l -

fore the Fourier transform of FA•/ vanishes outside (16.13) the statement follows.

+

lal)/(a). There-

of(-..\,~. By Corollary

Corollary (16.17): (Riemann-Lebesgue lemma) For each/ E L 1(R) we have

lim /(a)= 0. lal-oo PROOF. Let E > 0. By Corollary (16.16) there exists g E L 1 (R) with g(a) = 0 for lal ~ ..\ (..\ > 0 is a suitable constant) and II/ - glh < e. Consequently for lal ~ ..\ :

Ilea)!

= lf(a) -

g(a)I ~

II/ -

9111

< E.

+

Hence we have shown that the Fourier transform is injective by Corollary (16.14) and the image of L 1 (R) is contained in Co{R)

= {g e C{R): lal-oo lim g(a) = O}

according to Corollary {16.17). Exercise 13 shows however that the Fourier transform is not onto Co(R).

235

FOURIER TRANSFORM ON R

In addition to the Fejer kernel {FAheJO,oo( we introduce the following summation kernels: (1) Set VA(x) := 2F2A(x) - Fl(x). Then {VAheJo,oo( is a summation kernel on R and is called de la Vallee-Pouuln kernel. A straightforward calculation shows: VA(o) =

{

2-

1 0

¥

for lal ~ .\ for .\ ~ lol ~ 2.\ for 2.\ ~ lal.

{2) Let P(x) := ~ 1}z2. Since fa r};rdx = arctanxl~oo ='II" we see that (Plhe)o,oo{ with Pl(x) = .\P(.\x) is a summation kernel. (PA)Ae)o,oo( is called Cauchy-Poisson kernel. For the Fourier transform we have: PA(o) =exp(-lxD for all a ER. To show this, consider /(x) = X(O,oo(e-s and g(x) = XJ-oo,o(es. The corresponding Fourier transforms are f(a) = (1 + ia)- 1 and g(o) = (1 - U:a>- 1• Setting h{x) = /(x) + g(x) = e-lsl, we obtain h{a) = ~· Ash and h elements of L1 (R) the inversion formula gives PA(o)

=

J.

=~ 21r = 2~

PA(x)e-'-dx

J.

J. •

h{.\x)e-'-dx h{x}exp(-io x/.\ )dx = h(-a./.\). 1

(3) Set Gl(x) = .\G{.\x), where G(x) := -j;e-s

•

The summation

kernel {GAheJO,oo( is called Gaussian kernel and it holds .....

a 2

GA(a.) =exp(-(-) ) 2.\ (see Exercise 9}. Finally we prove a result similar to Corollary (3.3).

CHAPTER XVI

236

Theorem (16.18) : Let/ be an integrable function on R. Denoting u,.{/)(x) = /; f~>. ( 1 fco)e'-da, we have U>.(/)(x) - /(x) as ~ - 00 for almost all x e R. Moreover in points x, where I is continuous, u,.(J)(x) converges to /(x) 88 ~ - oo.

-1X1)

PROOF. We proceed 88 in Theorem (3.1). First one obtains as in Lemma (3.2) that for almost all x e R

r"

c·>

lim 11cx +JI)+ /(x - JI) -1cx>I dJI = 0 1.-0+)0 2

holds. Indeed the prooffor / e L 1{R) is similar to that one for f e L 1 (T). In particular (*) holds for the points of continuity of /. Now let x e R be given such that (*) holds. Then one obtains

=

lu,.{/)(x) - /(x)I

If.

F>.(Jl)(/(x - JI) - /(x))dJll

:5 2 lfoao F>.(JI) ( /(x +JI); /(x- JI) - /(x)) dJll :5 2 Lao F>.(JI) I/(x +JI); f(x - JI) -

/(x)I dJI.

For arbitrary 6 > 0 it holds

f

< .!. -

I

F>.(JI) /(x +JI); /(x - JI) - /(x)I dy

'II'~

[I •

f(x + y) + /(x - JI) - /(

using the inequality F>.(Y) :5

f

•

I

?

1rAJI2

2

2

>I

dy y2'

for y :/: 0. Hence we have

F ( ) /(x + 11) + /(x - fl) - /( ). y

x

x

>Idy -< _!!___ 11'~62'

where M is a constant . Chooeing 6 = 6,. = ~ -i the latter integral converges to 0 aa ~ - oo. Assuming ~ > 1 without any restriction, we have < 6>,. Now by llF>.llao :5 ~/27r we obtain

i

11/>. F,.(y) I/(x + y); f(x - y) - /(x)I dy :5 2: fol/>. I/(x + y); f(x - y) - /(x)I dy.

FOURIER TRANSFORM ON R

237

As (*) holds, this integral tends to 0 as ~ - oo. Finally by integration by parts (as in the proof of Theorem (3.1)) the integral

I

('" F~(fl) /(x +fl)+ /(z - fl) .11/A 2

/(x)I

dy

gets arbitrarily small, when A is ch06en sufficiently large. The details shall be elaborated in the exercises. t

EXERCISES (1) Show that L3 (R) is not a subset of L1(R) and vice versa.

(2) Let/ E L 1 (R). Show: (a) The function /is real-valued if and only if

lea) =/(-a) for all

a

e R.

(b) The function / is even (respectively odd) if and only if is even (respectively odd) .

1

(3) Given/ E L 1 ()0, ool) denote for a E R

/c(a) := 2

fo

00

/(z) cos(ax)dx

the so-called Fourier-cosine transform and

/.(a) := 2

fo

00

/(z) sin(ax)dx

the so-called Fourier-sine transform of /. Show:

(a) If/ E L1 (R) is an even function, then /(a) = /c(a) for every a e R. If/ e L1(R) is an odd function, then /(a) = (-i)/.(a) for every a E R. (b) For IE L 1{]0, oo[} it holds lhn /c(a) = 0 and lim f.(a) = 0. Jaf-oo Jal-oo (c)

If/ E

L1

00, oo[) such that Tc E L1(R) respectively f. E L1(R),

CHAPTER XVI

238

then for almost all x

/(x)

e JO, oo( we have

11

00

= -11'

0

.fc(a)cos(ax)da

respectively

/(x)

= -11'1 Loo /.(a) sin(ax)da. 0

(Hint: In (b) and (c) extend odd function on R.)

f

E L 1 (JO, oo[)

to an even respectively

(4) Show for a e R\{O}: (a) For /(x) = X(a,~(x), x ER, we have /(a) = i(e-'~-e-'-)0- 1 •

= X)o,111(x) for x > 0, then ic(a) = 2sin(Jja)a- 1 and /.(a) = 2(1 - coe(Pa))a-•.

(b) If /(x)

Are these Fourier transforms elements of L 1 (R)7 (5) The Bessel function J,, of order v expansion J,,(x) = for x e R. For v > Show:

/(a)

>

-1 is defined by the series

oo (-l)" (J)"+2n nl r(v + n + 1)

L

n=O

-i denote /(x) = (1 - x 2 )"-i XJ-1.11(x).

= 2"f

(v + ~) ./i J:~a).

Derive the following integral representation (Poisson) for v

> -l:

(6) Let /(x) = e-IJzX(o,oo((x) with P > 0. Determine Re/(a), Im/(a), lf(a)l and arg(/(a)) for all a e R. (7) Let u be a solution of the differential equation

u" - u = - I

(/ e L 1 (R)).

Determine u under the assumption that u, u'

e L 1 (R)

holds.

FOURIER TRANSFORM ON R

239

(8) Define / E L 1 (R) by

/(x) = {

0

~ ~! :~~ 1:1 ~ ~

Ix!> 1.

for

For abbreviation we denote by rCx) = I• ... • /(x) the n-fold convolution of /. Show: 1 (a) r(x) = [(x + n)/(n- 1>(:1: + 1) - (z - n)J'"-l)(x - 1))

2(n -1)

for n = 2,3,4, ... and x ER. {Hint: Use the Fourier transform and Theorem (16.8).)

1 (b) /"(x) = 2"{n _ l)I for n

"to(-1)• (~) (x + n - 2k)~- 1

= 2,3,4, . . . , where (x)+ = max{O,x} .

= -j;e-•' and GA(x) = AG(.U). Show that (GAhe)o,oo( is a summation kernel and determine G.\. Illustrate the limit functions

(9) Let G(x)

of G,. and {j,_ as~ - oo.

{10) In the proof of Theorem (16.18) carry out the missing integral estimate.

(11) Denote (PAheJO,oo( the Cauchy-Poisson kernel and (G>.heJO,oo( the Gaussian kernel. Show:

(a) Given / E L 1(R) we have for all x E R

p,_ • /(x) G>. • /(x)

!. = 2~ J.

= 2~

exp

(-lxl) /(a)e'-da

exp (-C

2~) ) /(a)e'-da. 2

(b) Further {P>.) • /(x) - /(x) and (G,.) • /(x) - /(x) as~ - oo for alm08t all x E R. In the points of continuity of / both convergence statements are holding.

(12) Suppose that an integrable function / is continuous at x = 0. Furthermore assume that /(a) ~ 0 for all a E R. Show that E L 1 (R). In particular the inversion formula of Corollary (16.15) can be used and it holds

!. . .

/(0) = 111" R /(a)da. 2

1

240

CHAPTER XVI

(Hint: Use Theorem (16.18) and the theorem of the monotone convergence.) (13) (a) Let I e L 1(R) be odd. Show that: for arbitrary numbers 0 < a < f3 < oo is valid:

11:¥da,~M for a constant M which is independent of a and {J. (Hint: Use Lemma (6.6).) (b) Set for a 2:: 0 O~a~e

a> e.

and g(-a) = -g(a), where e is the Euler constant. Show that: e C0 (R) holds, but there does not exist/ e L 1 (R) with j = g.

g

17

PLANCHEREL TRANSFORM ON R In Chapter 4 we have shown that the Fourier transform is an isometric isomorphism of L 2 (T) onto l2(Z). A corresponding result for L 2 (R) does not exist without any modification, since the Fourier transform is generally not defined for f e L 2 (R). We know that L 2 (R) is a Hilbert space with the inner product

f

< /, g > = J.1(x)g(:t}d:e, Denote

88

/, g e L2 (R).

before C00 (R) the linear space of all continuous functions with

compact support. Obviously, C00 (R) is a subset of L 1 (R) n L2 (R), which is dense with respect to the L 1-norm 88 well as the £ 2-norm in L 1 (R) respectively L 2 (R) .

Lemma {17.1):

For

I e Coo(R) we have 1e L 2 (R)

and

2~ J.1f(a)l da = J.1/(x)l d:e . 2

2

PROOF. First we 888Ume that /(:t) = 0 for all l:tl 2: 7r. Then we can consider fas an element of L 2 (T) . By Corollary (4.10)(1) it follows 2

J.1/(x)l u =

2~ n~oo lf(n)l

2

•

Replacing /(:t) by e-'-/(:t) with a ER, one obtains by the same result

!. •

L 00

1

l/(x)l2d:e = 27r

n•-oo

lf(n + a)l 2 •

The theorem of B. Levi (see Appendix A) yields:

r1

r1

lo n~oo lf(n + o)l 2do = n~oo lo lf(n + o)l 2do = J.1f(a)l2da. 00

00

241

CHAPTER XVII

242

Therefore it follows

J.1t(x)l2dx = 2~ J.1f(a)l2da.

For arbitrary f e Coo(R) choose ~ > 0 such that g(x) = 0 for where g(x) = ./>./(~). Using that g(a) = */(i) we obtain

J.1/(x)l 2dx = J.1g(x)l 2dx =

2

lxl

~

""

2~ J.19(a)l da = 2~ J.1f(a)l da. t 2

Since Coo(R) is dense in L 2 {R) we can extend the Fourier transform of C00 (R) to L2 (R) by applying Lemma (17.1). In fact if f e L2 (R), then there exists a sequence (/ri)neN1 /n E Coo(R) with II/ - /nll2 - 0 as n - oo. By Lemma (17.1) the Fourier transforms are in L 2 (R) and llfm - inll2 = v'21fll/m - /nll2· L 2 (R) being complete, P(/) = n-oo lim 2 exists in L (R). As a function P(/) is defined almost everywhere. The element P(/) e L 2 {R) is independent of the choice of the approximating sequence (/n)neN· Indeed if {gn)neN is another sequence with 9n E Coo(R) and lim ll9n -./112 = O, then II/n - 9nll2 ~ 11/n - /112 +II/ - 9nll2 - 0. By

In

n-oo

Lemma (17.1) it follows L 2 (R).

in

llin - 9nll2 - 0, that means nlim in= nlim 9n in ......oo ......oo

Hence we can define:

Defnition (17.2): For J e L 2 {R) the Pl~cherel transform P(/) e L 2(R) is defined as the limit in the L 2-norm of the sequence Cin)neN where (/n)neN is an arbitrary sequence of functions of Coo(R) con\rerging to/ in the L 2 -norm . The Plancherel transform P and the Fourier transform .... are defined on L 1(R) n L 2 (R). Both transforms in fact coincide. If namely f e L 1 (R) n L 2 (R), then choose a sequence (/n)neN1 In E Coo(R) with 11/n - /112 - 0 and 11/n - /Iii - 0 as n - oo. Then Hin - P(/)112 - 0 and fn(a.) - /(a) for all a e R. The convergence of to P(/) with respect to the L 2-norm implies a subsequence (nt)A:eN of N such that /n,,(a) converges to P(/}(a.) for almost all a e R (compare Exercise 9 of Chapter 3). Hence = P(J) holds almost everywhere. By Lemma (17.1) and Definition (17.2) we get:

in

1

PLANCHEREL TRANSFORM ON R

e

Lemma (17.9.) :

For / everywhere. Furthermore

L 1 (R)

n L 2 (R)

we have

243

1=

P(/) almost

2~ J.1f 0 set b. = X(-A,Alf· Then fA e L 1 (R) n L 2 (R) and J;. e L 2 (R). Moreover the functions J;. converge to P(/) with respect to the L 2 -norm as ~ - oo. PROOF. By the Holder inequality it follows b. e L 1(R)nL2 (R) and therefore J;. E L 2 (R). If (gn)neN is a sequence of functions 9n E Coo(R) with ll9n - /112 - O, then

1

....

Jt,:llin - /All2 = ll9n - /All2 ~ ll9n - /112 +II/ - /All2 - 0 as n - oo and ~ - oo. That means lim

n-oo

J;. = P(f) in the L2-norm. t

So far we have shown that the Plancherel transform P is a linear bounded operator of L 2 (R) into L 2 (R). Moreover we know that 1

v'21f llP(!)ll:i = 11/112 for all/ E L 2 (R). In particular, Pis injective. Now we prove an inversion formula implying the surjectivity of P.

Lemma (17.5):

For /, g

e L2 (R) it holds

< P(f),g > = < P(g),/ >. PROOF. For/, g

e Coo(R)

the theorem of Fubini gives:

< f.9 ::> = J. f(a)g(a)da = =

!./.

/(x)e-'-sdzg(a)da

J.1(x)g(x)dz = < g, / > .

244

CHAPTER XVII

Now for /, g e L2 (R) choose /,., g,. E Coo(R) with /,. - /, g,. - g in the L 2-norm. By the continuity of the inner product one obtains

< P(f),g >

= n,m.....oo lim < Jn,9m > = lim < im.ln > = < P(g),j >. + n,m-+oo

Theorem {17.6) : Let / e L2 (R). Then

(Inversion formula)

I = ~ P(P(f))

2

almost everywhere.

PROOF. For abbreviation set g = P(/). By the linearity of the inner product we get

Moreover one obtains by Lemma (17.5):

< /, -/;P(g) >

= < -j;P(g),j > = /; < P(/),g > = /;llPC/)11~ = II/II~

and

< 2~P(g),/ > = 2~< P(g),/ >=II/II~. Finally

resulting in

II/ - 2~P(g)ll~

= O, which is our assertion. +

We sum up the main properties of the Plancherel transform P:

PLANCHEREL TRANSFORM ON R

245

Theorem (17.7) :

(Plancherel) The linear operator P: L 2 (R) -+ L 2 (R) is an isomorphism of L 2 (R) onto L 2 (R), satisfying (1) -/,; < P(/),P(g) > = < /,g > for all /,g in particular V'\;llP(f)ll2 = 11/112·

(2) P(/)

=f

(3) P(/)(a) (4) /(x)

for

/

e L2 (R),

e L 1 (R) n L 2 (R).

= A-+oo Um JA /(x)e-'°"dz with respect to the L2-norm. -A

= -/,; A-+oo lim JA P(/)(a)ewda with respect to the L2-norm. -A

PROOF. (1) Applying Lemma (17.5) and Theorem (17.6) one obtains

-Ii< P(f),P(g) >=-Ii< P(P(g)),j > = < g, I>=< J,g >.

The other assertions are contained in the foregoing results of Lemma (17.3), Theorem (17.4) and Theorem (17.6). • To analyze the pointwise convergence of tT>..(/) to/ for square integrable functions /, we define the convolution of a function g E L1 (R) with a function / E V'(R), 1 < p < oo. We set g•/(x) =fa /(x-y)g(y)dy and show that g•/(x) exists for almost all x e R such that g •I e V'(R) and 119 •Ill,$ 119111 11/11,. The function (x, 11) -+ l/(x - 11)l'lu(11)I is measurable. As in the proof of Theorem (1.3) we obtain that

x- J.11cx - y)I' lg(y)ldy

exists for almost all x E R and that it is an integrable function for which

holds

J. J.1/(x - Y)l'lg(y)jdydz = 11111: 119111·

Further let q be given by~+~= 1. By the Holder inequality it follows

lg• /(x)I :S J.11 = 0 for every n E No it follows that = 0 almost everywhere, that is f = 0

By Corollary (16.14) we get g almost everywhere. t

Corollary (17.12):

For / E L 2 (R) we have 00

(1)

f=LWn · n=O

Moreover 00

II/II~= LI< f,Wn > 12 ,

(2)

n• O

00

(3)

P(/)

= ../2WL:(-i)" < f,Wn > Wn· n ..

PROOF. (4.7). •

o

The assertions follow immediately by Theorem (17.11) and Lemma

Remark: It holds fa Hn(z)Hrn(z)e-s' dx = nl 2"./i6nrn (see Exercise 8), hence

(4)

llhnll2 = (~Ji) i

Wn(x)

and therefore

1

= ( v'i2"nl

)

1/2

I

Hn(z)e-s

11

•

The identity (3) of Corollary (17.12) can also be used for defining the Plancherel transform.

PLANCHEREL TRANSFORM ON R

251

EXERCISES (1) Give an example of a function f E L 2 (R), which is not in L 1(R) and whose Plancherel transform P(/) is integrable.

(2) Determine the Plancherel transform P(/) of /(:r:)

1 e'°ir - e•flz

= 211' i --x--

(x E R\{O}, a< {J).

(3) Show

(4) Let 1 Sp< oo and IE V(R). Show: The mapping y - L'tl/ of R into V(R) is uniformly continuous. (5) Let 1 Sp < oo and : + ~ g • f(x) =

= 1.

For / E V(R) and g E L 9 (R) set

J.1 0, we see that GA(n) = exp(-.m2 ). For fixed s > 0 the partial sums

FN(e")

=

(7r/ 1)1/ 2

N

L

exp(-(t + 211'n) 2 /4s)

n--N

converge uniformly on (0, 211'[. Obviously (GA(n))nez E l 1 (Z). Hence we can use Theorem (18.3) and get for every t E (0, 211'[ (11'/s) 112

L

00

exp(-(t + 211'n) 2 /4s) =

n•-oo

L

00

n=-oo

exp(-.m2 )e'"t.

This identity can be used to derive the postponed proof of the fact, that the Gaussian kernel (G.)ae)o,oo( on T (see Chapter 2) consists of positive functions. In fact, on the right of the above equality we find G.(e"). Therefore it follows that G• (eit) > 0 for all t E (0, 211'[. This identity is moreover very Important in number theory. The function 1'(s)

=

L 00

n=-oo

exp(-7rm2 ),

is called theta function. With t = 0 and replacing s by 11'8, we obtain from above the so-called Jacobi identity 1'(s) = 8 -1/21'(s-1).

257

POISSON SUMMATION FORMULA

Lemma {18 ..4) :

Let

partial sums FN(e"}

f

E L 1 (R) be of bounded variation. N

= 271" E

n•-N

Then the

/(t + 2wn) converge uniformly on

The limit function F has bounded variation on

IO, 2wl.

IO, 2wl.

PROOF. Let Vn be the total variation off restricted to l2mr, (2n + 2)w],

IO, 2w( such that the series E 00

n E Z. Further choose to E

convergent. Given E > 0 choose no E N, such that and

E

lnlf?no

L

lnl2:no

< e/2. Then for every t

Vn

l/(t + 211'n)I S

S

E

E

lnl2:no

lf(to+21rn)I is

lf(to+2wn)I < e/2

IO, 2w( we have

L

l/(to + 211'n)I +

L

lf(to + 2wn)I +

lnl2:no

n-=-oo

~2:no

L

l/(t + 2wn) - /(to+ 211'n)I

L

n
0. Then we have for every x e R /(x) =

E I ( ~)

1:--00

lal > 11'W,

sinc(wx - k).

This series converges uniformly and absolutely on R.

Remarks: (1) It is easily checked that Theorem {18.8) and Theorem (18.10) also hold, if f e L 1 (R) n C(R) and j e L 1 (R). Also in Corollary (18.9), respectively Corollary (18.11) we may assume that f E L 1 (R) n C(R) with /(a) = 0 for lal > 11' respectively lal > 11'W. In applications of signal processing f is in general assumed to have finite energy, i.e., f E L 2 (T). If one extends the Fourier transforma.tion to V(R), 1 < p < 2, (which is possible, see e.g. (5]), then the corresponding results from above are also valid for V(R)-functions.

263

POISSON SUMMATION FORMULA

(2) Functions/ e L 2 (R) with P(/) vanishing almost everywhere outside of the interval (-cS, cS) are called band-limited. Choosing the smallest a such that P(f)IR\[-a, a) = 0 almost everywhere and writing a = 1f'W the sampling rate 1/w is called Nyquist rate. According to Corollary (18.11) the Nyquist rate is the greatest sampling rate allowing exact reconstruction of f by the values f(k/w), k E Z. {3) The band-limited functions f e L 2 {R) n C(R) can be characterized as follows by the Paley-Wiener theorem. The function/ e L2 (R) satisfies P(f)IR\(-a, a) = 0 almost everywhere if and only if f has an extension to the whole complex plane C (also denoted by/) such that / is an entire function and for every z E C it holds

(b)

l/(z)I ~ M exp(alzl),

where Mis a positive constant independent of z. To prove that a band-limited function f e L 2 (R) n C(R) has an entire extension ofexponential type (i.e., the boundedness condition {b) is satisfied) is rather easy, see Exercise 5. For the more involved converse implication we refer to [18, Chapter 19). Many further results on this topic can be found in (2).

EXERCISES (1) Show that the Plancherel transform of the sinc-fucntion is X[-w,w)· {2) Let w > O. Define for k .sA:(x)

e Z and x e R

= (w/21r) 112

sinc{wx - k).

Show that {sA:}A:ez is an orthonormal basis of L2 (R). Apply this to derive Shannon's sampling theorem {18.11) (with equality in the Hilbert space L 2 (R)). (3) Let x E R and let N E N. For a given set {aA:}f.-N of complex numbers define a distance by N

dN({aA:)) = sup{l/(x) -

L

A:•-N

/(k)aA:I : IE B~,

11/112 ~ 1},

CHAPTER XVIII

264

where B~ = {/ e L 2 (R) : P(f)IR\(-1", 11'] = 0 almost everywhere}. Prove that dN((aA:)) is minimized if we choose OA: = slnc(x - k) for

k=-N, ... ,N.

(Hint: Reformulate the problem to see that the Fourier coefficients of es minimize dN.)

(4) Applying Theorem (18.3) prove the identity 00

2 ~ L.J 112 n=-oo

II

+ (t + 2mr)

= 2

00

~ e-lnlr e•nt ~ n=-oo

for each t e [O, 21"( and y > 0. (Hint: Use the Cauchy-Poisson kernel of Chapter 16.) (5) Let J e L 1{R), J ~ 0 almost everywhere. Show: If /(n) = 0 for every n e Z, then J = 0 almost everywhere. (6) Let J e L 2 {R) such that P(f)IR\[-u, u) = 0 almost everywhere. Show that J has an entire extension to C such that

l/(z)I ~ M exp(ulzl) is true for every z e C. (Hint: Consider J~,, /(e")e1tsdt.)

(7) Let /, g

e L 2 (R).

[/,g)(e")

Define for t E (0, 2w[

L 00

= 21"

f(t

+ 21rn) g(t + 21rn).

n~-oo

Prove that [/,g](e") is defined for almost all t [/, g) E L 1{T). F\irther show: (a) (b)

I(/, gJl 2

~ [/, /]

[g, g) almost everywhere.

[/,gnk) = (/ ·J)lk) for all k e /. J:•[/,g)(e")dt

and hence

f0,.[f, g)(e")dt

e [0,211'[, and

=

z.

In particular

= < f,g > = ii< P(/),P(g) >,

ii f0,.[P(/), P(g)){e")dt.

APPENDICES

(.?\ Taylor & Francis ~-

Taylor & Francis Group http://taylora n dfra ncis.co m

A

MEASURE THEORY Our interest will be confined to convergence theorems. For an introduction to u-algebras, positive measures, complex measures, measurable functions, integration of positive functions, integration of complex functions we refer to [18, Chapter 1). Given a measure space (X, M, µ), where M is a 0 holds lim µ({x EX: l/n(:r:) - /(x)I ~ E}) = 0.

n-oo

MEASURE THEORY

269

Theorem (A.6) : Let f and /n, n e N, be complex measurable functions on X. (1) Suppose (/n)neN converges in measure to /. Then there exists a subsequence Un)neN such that Jim /,.. (x) = /(x) for µ-almost n-+oo every x EX. (2) Assume that µ(X)

< oo and n-+oo Jim /n(X)

= f(x)

for µ-almost every

x EX. Then (/n)raeN converges in measure to/.

PROOF. {l) For each k EN there exists n1:

e N with n1: > nt-l such that

~}) 0 now set Bn = {x E X : l/1:(x) - /(x)I < E for every k 2: n}. Obviously Bn ~ Bn+l· Now let A EM be a µ-zero set, (i.e., µ(A) = 0), such that lim /n(x) = /(x) for all x e X\A. That means X\A ~

00

LJ

n•l

n-+oo

B,. or equivalently

"l~~ µ(X\B,.) = µ (

n

n•l

n X\Bn ~ A. Hence 00

n•l

X\Bn) S µ(A)

where (18, Theorem 1.19] is used.

= O,

APPENDIX A

270

Since {x E X : l/n(x) - /(x)I ~ e} ~ X\Bn, the convergence in me88ure follows.

+

The measure space, which we usually consider, is X = T = { z E C : lzl = I}. There is an obvious identification of T with (0, 21r[ by the mapping t - e". By that mapping the Lebesgue u-algebra and the Lebesgue measure is transferred from (0, 21r( to T. In this way the Lebesgue measure dz on T is the restriction of the Lebesgue measure dt on R to (0, 21r[. Thus a function is Lebesgue integrable on T if the corresponding function on (0, 211"[ is Lebesgue-integrable. As topological space we consider T as subset of the complex plane C. The reader should note that T and (0, 21r[ are not homeomorphic. The most important feature of the Lebesgue measure dz on T is the translation invariance, i.e.,

£

/(zzo )dz = 1

£

/(z)dz

for all Lebesgue-integrable functions on T. If the measure µ on T is the Lebesgue measure we write V(T) instead of V(µ). Further we point out that T is a group, which can be identified with R/27rZ. Moreover T is the most prominent example of a compact commutative topological group, see e.g. (19

II).

B

BANACH SPACES We use two important results of functional analysis: The principle of uni· form boundedness {which often is also called Banach.Steinhaus theorem) and the conditionally sequentially compactness of norm·bounded subsets in LP-spaces, 1 < p < oo, (which is part of the theorem of Eberlein·Smulian). Since our problems of Fourier series can be attached with greater ease when they are placed within the framework of Hilbert spaces and Banach spaces, we give their bBSic notions.

Definition (B. ll : A linear Banach space B over the complex numbers C is said to be a normed linear space, if there exists a mapping II II : B !O, oo! called norm, if the following rules hold:

llxll = 0 if and only if x = 0. llaxll = lal llxll for a e C, x e B. llx + Yll $ llxll + llYll for x, y E B . By d(x, y) = llx - Yll a metric is defined on B. (1) (2) (3)

A normed linear space B is called Banach space if B is complete in the metric defined by its norm. Consider V(µ), 1 $ p < oo, as defined in Appendix A. The collection V(µ) is a linear space. The only property of a norm not satisfied by

11/11,, =

(fx 1/(x)I" dµ(x))

l/p

is that 11/11., may be zero without J being the zero-function. Since 11/11,. = 0 precisely when /(x) = 0 for µ·almost all x, we introduce an equivalence relation in V(µ) by writing f""" g if and only if II/ - gll., = O. The set of all equivalence classes is also denoted by V(µ), and it is straightforward to check now V(µ) with II II,. is a normed linear space. But note that when investigating V{µ), not a space whose elements are functions, but a space whose elements are equivalent classes of functions is considered. Nevertheless for the sake of simplicity we will speak of V(µ) as a space of

functions.

271

APPENDIX B

272

It is very important that V(µ), 1 ~ p < oo, is a Banach space. The proof of the completeness uses the lemma from Fatou, see [18, Theorem 3.11). To deal with p = oo, the essential supremum is introduced. For a complex measurable function / define

11/lloo = ess sup l/(x)I = inf{M 2: 0: µ{x EX: l/(x)I 2: M} = O}, :r:EX

and let L 00 (µ) be the linear space of all complex measurable functions with 11/11 00 < oo. Identifying all functions, which differ only on a µ-zero set, one obtains, that L 00 (µ) with norm II Hoo is also a Banach space. (The careful reader will note that the essential supremum is sometimes introduced by means of locally µ-zero sets, resulting in another L00 (µ)-space, see e.g. [UJ. But in our text we always consider a-finite meMures, and in this case both definitions coincide.) An important subcl888 of Banach spaces are Hilbert spaces.

Definition (B.2) : A linear space H over the complex numbers C is said to be an inner product space, if there exists a mapping < , > : H x H - C called scalar product, if the following properties are satisfied:

< x, x > 2: 0 for all x E H and < x,x > = 0 if and only if x=O. (2) < ax + {3y, z > = a < x, z > + {3 < y, z > for x, y, z E H, a,{3 EC. (3) < x, y > = < y, x > for x,y EH. (1)

By Hxll = < x, y > 112 a norm is defined on H, such that H becomes a normed linear space, and H is called Hilbert space, when H is complete. The space L 2 (µ) is a Hilbert space, where the scalar product is defined by

< /, g > =

l

f(x) g(x) dµ(x).

We will often use the notion linear operators or linear functionals when analyzing the behaviour of Fourier series.

BANACH SPACES

273

Definition (B.3) : For a linear operator S from a normed linear space B into a normed linear space Y define the operator-norm

llSll

= sup{llSxll:

xe

X,

llxll

$1}.

If llSll < oo, the operator Sis called bounded linear operator. When we take the complex field C for Y we talk about bounded linear functionals. Further B* is the collection of all bounded linear functions on B. The bounded linear opeartors are exactly the continuous linear operators. The linear space B* is with the norm above a Banach space, often called dual space of B. We leave the verification of these properties as exercise.

Theorem (B.4) : (Uniform boundedness principle) Let B be a Banach space and Y a normed linear space. Suppose S is a collection of bounded linear operators of B into Y, such that sup {llSxll

: S e S} < oo

for every x e B. Then sup {llSll : S e S}

< oo.

PROOF. Assume that sup{llSll : Se S} = oo. We construct inductively Xn e B and Sn e S, such that (a) llxnll = 4-n (b) llSnxll

?! n

N,

for all

n e

for all

n e N,

E Xn· To achieve this suppose we have found x1 1 ••• ,Xn-1 EB n•l with property (a), and 81 1 ••• ,Sn-l e S such that

where x = (c)

00

llSaixaill > 2(M11-1 + k)

where Mo

for

k

= 1, ... , n -

1,

= 1, Mai = sup {llS(x1 +···+ z11)ll : S e S} < oo, and

(d) llS11ztll > Jl!Saill llxaill for k = 1, ... , n - 1 are true. By the assumption sup{llSll : S e S} = oo, there exist Sn e S with

APPENDIX B

274

and an element Zn E B with 11%nll = 1 and llSnXnll Xn = 4-n Zn· Then llxnll = 4-n and

Thus (c) and (d) also hold for k

x

00

=E

n•l

= n.

>

~llSnll· Now set

But then (b) is true. In fact, for

Xn we have

llSnxll

=

~

Sn

(E k=l

Xt +xn +

f

k•n+l

Xt)

us.z.11 - 11s.(z1 + ... + Zn-1)11 - Is. (~. Z•) II·

Further

and hence

Therefore we obtain by (c) and (d): llSnxll ~ llSnXnll - Mn-1 ~ llSnXnll - Mn-1 ~

1

3 llSnll

llxnll

1

2 llSnXnll

1

2 llSnxnll -Mn-l > n.

But evidently (b) is in contradiction to the 888umption sup{llSxll : S E S} < oo, and hence sup{llSll : S E S} < oo follows.

t

Remark : This proof of the Banach-Steinhaus theorem is due to the Hausdorff [8). For another proof (using Baire's category theorem) see [20).

275

BANACH SPACES

Theorem (B.5) :

Let B be a separable Banach space, (i.e., there ex-

ists a dense countable subset in B). Suppose that (Fn)nEN is a norm-

bounded sequence in B•. Then there exist a subsequence (Fn.)teN and an element F E B• such that

for all x

e B.

PROOF. Let {xJ};eN be a dense subset of B. The sequence (Fn(x1))neN is a bounded sequence in C. Hence there exists a convergent subsequence, which we write as (Fn1(x1))n1eN· By the same reason this subsequence contains a further subsequence such that (Fn1(x2))neN is convergent. It is easily seen that the diagonal sequence (Fnn)neN has the property that each sequence (Fnn(xJ))neN is convergent. Since {xJ}JeN is dense in B, we even obtain the convergence of (Fnn(X))nEN for every x EB. Now consider the linear functional Since {IFnn(x)I : n E N} is bounded for each x E X, the principle of uniform boundedness yields sup{llFnnll : n E N} = M and we obtain for each x

< oo

eB

Thus we have found a subsequence (the diagonal sequence) and F with the derived property.

+

e

B•

We will apply Theorem (B.5) in case of B = L 11 (T), 1 < q < oo. The Banach spaces L"(T), 1 :s; q < oo are separable. The dual space L"(T)• can be identified with V(T), where p is determined by ~ + = 1, see 118, Theorem 6.16) . In fact, a representation theorem of Rlesz says, that for every bounded linear functional F e (L"(µW, 1 :s; q < oo there exists a unique g e V(µ), ~ + ~ = 1, such that

!

F(f)

=

l

/(x) g(x) dµ(x)

and llFll = llgll,,. In the case of q = 1, we have to assume that µ is u-finite. For B = L"(T) the above theorem gives:

APPENDIX B

276

Corollary (B.6): A norm-bounded sequence (gn)neN of L•(T), 1 < q < oo contains a weakly convergent subsequence (gn.)lleN, i.e., there exists an element g E L"(T) such that lim

f

11-oo)T for every

f(z)

g"• (z) dz = f

)T

f(z) g(z) dz

f e V(T), where~+!= 1.

Remark: Theorem (B.5) is part of a far more general result, the theorem of Eberlein-Smulian, see e.g. (23).

c BANACH ALGEBRAS In Chapter 15 we applied some powerful results of Gelfand's theory on commutative Banach algebras.

Definition (C.1) : A Banach space A (over C) is called Banach algebra with unit, if there exists a multiplication in A such that for x, y, ze A (1) (xy)z = x(yz), (2) (x + y)z = xz + yz and x(y + z) = xy + xz, (3) a(xy) = (ax)y = x(ay) (a E C),

(4) llx11ll S llxll 111111·

(5) There exists a unit element e EA which satisfeis :re= ex= x and llell = 1. A Banach algebra A is called commutative if xy = yx holds for all x, y E A. Examples of Banach algebras with unit are C(T), L 00 (T), M(T) and l 1 (Z). Also on L 1 (T) is defined a multiplication (the convolution). However, there does not exist a unit element in L 1 (T). Note that as throughout this book we consider complex Banach algebras. The spectrum of an element x E A is the set of all complex numbers .\ such that .\e - xis not invertible. We denote the spectrum of x by cr(:r). It is a very important fact that u(z) is a not empty set of C. We give a sketch of the proof of this result due to Beurling and Gelfand.

Lemma (C.!J: Let A be a Banach algebra with unit. For every x EA the spectrum of xis not empty.

PROOF. Assume cr(x) = 0. For every bounded linear functional FE A* define RF : C - C, RF(.\) = F((.\e - x)- 1 ). Now we can give a Taylor expansion of RF in each point .\o E C. 277

278

APPENDIX C

In fact, for

~

e C and ..\ e C with 1

I.\ - .\ol < ll~e - xii we have 1

00

(.Xe - x)- = ~)~ - ..\)" [(~e n=O

zr•r+l.

It is straightforward to check, that this series is the inverse of (.Xe - x). Hence we obtain

L 00

RF(..\)=

n=O

F ( ((..Xoe

-xr )"+1) 1

(..\o - ..\)".

So far we have shown that RF : C - C is an entire function. Applying the continuity of the inversion, we get

Therefore RF is the zero-function. However there exists F E A• with F((..\e - x)- 1 ) IO, a contradiction. t

Corollary (C.3) : (Gelfand-Mazur) If A is a Banach algebra with unit in which every nonzero element is invertible, then A is isometrically isomorphic to C. PROOF. Since O'(x) I 0, there exists ..\ E C such that .Xe - x is not invertible, and must be 0 by the assumption. Moreover there can be at most one..\ with .Xe= x. Thus we have shown that there exists exactly one ..\ = ..\(x) such that .Xe = x. The mapping x - ..\(x) is the stated isometric isomorphism. t We consider commutative Banach algebras A with unit. Associate with A the set ~(A)=

{h: A - C : h I 0, h linear and multiplicative}

and the set

M(A)

= {M ~ A:

M

is a maximal ideal of A}.

BANACH ALGEBRAS

279

Maximal ideals are proper ideals which are not contained in any larger proper ideals. We shall show that A(A) and M(A) can be identified. But first we prove that each h E A(A) is a bounded linear functional.

Theorem (C.4) : Let A be a commutative Banach algebra with unit e. Then A(A) ~ A•, and moreover llhll = 1 = h(e) is true for every he A(A). PROOF. Let h E A(A). Assume that lh(xo)I > Uxoll for some xo e A. Put x = xo/h(xo). Then llxll < 1 and h(x) = 1. Since llx"ll ~ llxll" the elements

Yn

= -x -

x2 -

form a Cauchy sequence in A. Let y Xfln-li

we obtain x

••• -

x"

e A with y = n-oo Jim Yn.

+ y = xy, which implies h(x)

Since x + Yn

=

+ h(y) = h(x) h(y).

But this contra.diets h(x) = 1. Thus h h(e) = 1, we also have llhll = 1. t

e

A• and llhll ~ 1 follows. Since

Theorem (C.5) : Let A be a commutative Banach algebra with unit. For h E A(A) let ker h = {x e A : h(x) = O}. Then ker h is a maximal ideal of A, and the mapping

h _. ker h,

A(A) _. M(A)

is bijective. PROOF. Evidently ker h is a proper ideal of A, and A/ ker h is isomorphic to C. Hence ker his a maximal ideal of A. If hi, h2 e A(A) and x EA with hi(x) :I= h2(x), then x - h2(x)e e kerh2 and x - h:r(x)e fj kerh1, proving that h _. ker his one-to-one. Finally consider Me M(A). Observing that M is closed, it is easy to see that A/M is a Banach algebra with unit. The quotient space A/Mis even a field. To see this, let x e A\M, and put

I = {ax + y : a e A, y e M}.

280

APPENDIX C

Since I is an ideal, which contains M properly, we get I= M . Hence there exists an element ao e A with aox + y = e. That means ao + M is the inverse of x+M in A/M. Now by Corollary (C.3) there is an isomorphism ~of A/M onto the complex field C. If h = ~ o 'Ir, where 1r: A-+ A/Mis the canonical projection, then h e AA and ker h = M. t Concludingly we note, that the set A(A) can be given a compact Hausdorff topology such that A can be represented es an algebra of continuous functions contained in C(A(A)). This representation is the most important tool in the investigation of commutative Banach algebras. AP. general references for Banach algebras we mention [20j.

REFERENCES 1. Abramowitz, M.; Stegun, I.A., Handbook of Mathematical l'Unctiona, Dover Publ., New York, 1972. 2. Achieser, N.I., Theory of Approximation, Dover Publ., New York, 1992. 3. Bari, N.K., A '.lreatUe on '.IHgonometric Seriea, I, II, Pergamon Press, Oxford, 1964. 4. Benedetto, J.J., Sputrol Synthem, Teubner, Stuttgart, 1975. 5. Butzer, P.L.; Jansche, S.; Stens, R.L., l'Unctional Analytic Methoda in the Solution of the l'Undamental Theorema and Beat-Weighted Algebraic Appnnimation, Approximation Theory (G.A. Anastassiou, ed.), Marcel Dekker, New York, 1992, pp. 151 - 205. 6. Butzer, P.L.; Nessel, R.J., Fourier Analym and Approximation, Birkhiueer, Basel, 1971. 7. Edwards, R.E., Fourier Seriea, I, II, Springer, New York, 1968. 8. Hausdorff, F., Zur Theorie der linearen metmchen Rtiume, J. Reine Angew. Math 167 (1932), 294 - 311. 9. Helson, ff., Harmonic Analym, Addison-Wesley, Reading, 1983. 10. Hewitt, E.; Hewitt, R.E., The Gibba-Wilbrnham Phenomenon: An E,,Uode in Fourier Analym, Archive for History of Exact Sciences 21, 1979, pp. 129 - 160. 11. Hewitt, E.; Stromberg, K., Real and Abatract Analym, Springer, New York, 1965. 12. Jorsboe, O.G.; Mejlboro, L., The Carleaon-Hunt Theorem on Fourier aeriea, Lecture Notes Math. 911, Springer, Berlin, 1982. 13. Katznelson, Y., Introduction to Harmonic Analym, Wiley, New York, 1968. 14. Komer, T.W., Fourier Analym, Cambridge University Press, Cambridge, 1988. 15. Lasser, R., Fourier Summation with Kernels defined by Jacobi Polynomials, Proc. Amer. Math. Soc. 114 (1992), 677 - 682. 16. Lozinski, S., On ConvtfJence and Summability of Fourier Seriea and Interpolation Procu1e1, Math. Sbornik 14 (1944), 175 - 268. 17. Rudin, W., Some Theorema on Fourier Coefficienta, Proc. Amer. Math.

Soc. 10 (1959), 855 - 859.

281

282

REFERENCES

18. _ _ , Real and Complex Analysia, Mc Graw-Hill, New York, 1987. 19. _ _ , Fourier Analy3ia on Groupa, Wiley, New York, 1962.

20. _ _ , FUnctional Analysis, Mc Graw-Hill, New York, 1973.

21. Shlryayev, A.N., Probability, Springer, New York, 1984. 22. 'lbrchinsky, A., Real-Variable Methods in Harmonic Analysia, Academic Press, Orlando, 1986. 23. Whitley, R., An Elementary Proof of the Eberlein-Smulian Theorem, Math. Ann. 172 (1967), 116 - 118. 24. Zygmund, A., '.lHgonometric Seriea, I, II, Cambridge University Press, Cambridge, 1959.

INDEX

Abel lemma 59 summable 29 absolute convergence of Fourier series 203 analytic extension 171, 191 approximate identity 11, 107 for L 1 R 232 approximate sampling theorem 262 Banach algebra 3, 205, 277 band-limited function 263 Bernstein 217 inequality 23, 148 Bessel function 238 inequality 39 best approximation 137 Beurling-Helson theorem 167 cardinal series 259 Cesaro summable 29 conjugate Fourier series 53 function 53, 98, 102 kernel 93, 177 Dirichlet kernel 189 Fej~r kernel 177, 182, 186 Poisson kernel 182, 184, 186 conjugation admits 53, 103 continuity theorem of P. Levy 84 continuous measure 85 convolution 3, 85, 205, 225 operator 107 covariance 91, 105 Denjoy-Lusin theorem 224

Dini criterion 65 discrete measure 85 factorization theorem 169 Fast Fourier transform 36 Fourier coefficient 1, 76 cosine transform 237 series 1 sine transform 237 Stieltjes transform 77 transform on B• 76 transform on R 225 transformation 4 Gelfand theory 204 Gibbs phenomenon 34 Hardy 31 inequality 201 space 43, 159, 191 Hausdorff-Young theorem 56 Herglotz theorem 83 Hermite polynomial 247, 252 function 247, 252 Hilbert formulae 104 transform 95, 177 homogeneous Banach space 5, 155, 213 ideal 205, 208 inner function 166, 169, 170 innovation sequence 195 invariant subspace 166 inverse Fourier transform 233 inversion formula 233, 244 283

284

INDEX

involution 8 isoperimetric problem 47

Lusin-Privalov theorem 184 modulus of smoothness 138

Jacobi identity 256 polynomial 115 Jensen inequality 193, 200 Jordan criterion 66 kernel Bochner-Riesz 115 (C, a)- 18, 123, 128 Dirichlet 14 Fejer 14 Fejer-Korovkin 145 Gaussian 18 generalized Dirichlet 115 generalized Fejer 145 Jackson 117 moving-average 119 Poisson 17, 161 Rogosinski 112 summation 11 de la VallOO-Poussin 17 kernel on R Cauchy-Poisson 235 Fejer 230 Gaussian 235, 256 summation 229 de la VallOO-Poussin 235 Kolmogorov 177, 195 theorem 185 -Szego property 193, 197 Korovkin 131 Lebesgue 25, 27 constant 50 property in zo 25 test 60 linear prediction 194 Lipschitz condition 139 local 65 symmetric 139 Lozinskii 131, 132 Lusin 98, 184

moment 228 Nikol'skii 127 Nyquist rate 263 one-sided moving average process 196 operator-norm 50, 107 outer function 166, 169, 170, 171 Paley-Wiener theorem 263 Parseval 39, 78 Plancherel isomorphism 42, 245 transform on R 242 Poisson integral 159, 191 summation formula 254 Poisson-Stieltjes integral 159 Polya criterion 88 positive definite 83 Privalov 98, 184 projection 53 projection admits 54 pseudomeasure 223 regular process 195 Riemann-Lebesgue lemma 16, 234 Riesz, M. 203 theorem 101, 103 Riesz, F. and M. theorem 173 Riesz-Thorin theorem 57 Rudin-Shapiro polynomials 217, 224 Shannon-Whittaker-Kotel'nikov theorem 262 sine-function 259 spectral measure 91, 195 synthesis 207

Jackson theorem 147

symmetric 138

S-set 209

Stechkin 219, 220 Szasz 216 Szego 196 SzegO-Kolmogorov theorem 191 Sz.-Nagy 125

INDEX

Toeplitz condition 132 matrix 83 trigonometric polynomial 1 uniqueness theorem 16 1 233 de la VallOO-Poussin criterion 68 weakly stationary 91 1 195, 200 weak type (1 1 l) inequality 177 Weyl's theorem 24 Wiener 88 1 248 theorem 206 Wiener-Ditkin set 212 theorem 211 Wiener-Levy theorem 213 Wiener Tauberian theorem 206 Young inequality 61 Zygmund-approximation sequence 143

285