Introduction to Fourier Analysis and Wavelets (Brooks Cole Series in Advanced Mathematics)

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Introduction to Fourier Analysis and Wavelets

Mark A. Pinsky

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Introduction to Fourier Analysis and Wavelets

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Mark A. Pinsky: Introduction to Fourier An.tiysis and Wavelets (ISBN: 0-534-37660-6). Original edition copyright CD 2002 Wadsworth Group. Brooks/Cole is an imprint of the Wadsworth Group, a division of Thomson Learning, Inc. First published by Brooks/Cole, an imprint of Thomson Learning, United States of America. All rights reserved. Reprinted for the People's Republic of China by Thomson Asia Pte Ltd and China Machine Press under the authorization of Thomson Learning. No part of this book may be reproduced in any form without the express written permission of Thomson Learning Asia and China Machine Press . .:$:~~)C~~P!l&mrmm~'+JJ ili!l&U "3fJL~I~ili!l!HtiftFWJt!L

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CONTENTS

1

FOURIER SERIES ON THE CIRCLE 1.1

1.2

1.3

Motivation and Heuristics 1.1.1 Motivation from Physics 1.1.1.1 The Vibrating String 1.1. 1.2 Heat Flow in Solids 1.1.2 Absolutely Convergent Trigonometric Series 1.1.3 *Examples of Factorial and B~ssel Functions 1.1.4 Poisson Kernel Example 1.1.5 *Proof of Laplace's Method 1.1.6 *Nonabsolutely Convergent Trigonometric Series Formulation of Fourier Series 1.2.1 Fourier Coefficients and Their Basic Properties 1.2.2 Fourier Series of Finite Measures 1.2.3 *Rates of Decay of Fourier Coefficients 1.2.3.1 Piecewise Smooth Functions 1.2.3.2 Fourier Characterization of Analytic Functions 1.2.4 Sine Integral 1.2.4.1 Other Proofs That Si (<X>) = 1 1.2.5 Pointwise Convergence Criteria 1.2.6 *Integration of Fourier Series 1.2.6.1 Convergence of Fourier Series of Measures 1.2.7 Riemann Localization Principle 1.2.8 Gibbs-Wilbraham Phenomenon 1.2.8.1 The General Case Fourier Series in L 2 1.3.1 Mean Square Approximation-Parseval's Theorem 1.3.2 *Application to the Isoperimetric Inequality

1 I

2 3 6 7 9 l1 13 13 19 20 21 22 24 24 25 29 30 31 31 34 35 35 38

IV

CONTI:Nl s

*Rates of Convergence in L 2 1.3.3.1 Application to Absolutely-Conv ergent Fourier Series Norm Convergence and Summability 1 .4. 1 Approximate Identities 1.4.1.1 Almost-Everywh ere Convergence of the Abel Means 1.4.2 Summability Matrices Fejer Mean~ of a Fourier Serie~ 1.4 . .3 1.4.3.1 Wiener's Closure Theorem on the Circle *Equidistributio n Modulo One I .4.4 I .4.5 *Hardy's Tauberian Theorem Improved Trigonometric Approximation 1.5. I Rate.., of Convergence in C ( Jf) 1.5.2 Approximation with Fejer Means 1.5.3 *Jackson's Theorem 1.5.4 *Higher-Order Approximation 1.5.5 *Converse Theorems of Bernstein Divergence of Fourier Series I .6.1 The Example of du Bois-Reymond 1 .6.2 Analysis via Lebesgue Constants 1.6.3 Divergence in the Space L 1 *Appendix: Complements on Laplace's Method I. 7.0.1 Fir~t Variation on the Theme-Gaussian Approximation 1.7.0.2 Second Variation on the Theme-Improve d Error Estimate 1.7.1 *Application to Bessel Functions 1.7.2 *The Local Limit Theorem of DeMoivre-Lapla ce Appendix: Proof of the Uniform Roundedness Theorem *Appendix: Higher-Order Bessel functions Appendix: Cantor's Uniqueness Theorem 1.3.3

1.4

1.5

1.6

1.7

1.8 1.9 1.10

2

39 43

45 45 49 51 54 57 57 59 61 61 62 65 66 70 73 74 75 78

80 80 80 81 82 84 85 86

FOURIER TRANSFORMS ON THE LINE AND SPACE

89

2.1

89

2.2

Motivation and Heuristics Basic Properties of the Fourier Transform 2.2.1 Riemann-Lebesg ue Lemma 2.2.2 Approximate Identities and Gaussian Summability 2.2.2. I Improved Approximate Identities for Pointwise Convergence 2.2.2.2 Application to the Fourier Transform 2.2.2.3 Then-Dimensio nal Poisson Kernel

91

94 97 100 102 106

V

CONTENTS

2.3

2.4

2.5

2.6

2.7

2.2.3 Fourier Transforms of Tempered Distributions 2.2.4 *Characterizatio n of the Gaussian Density 2.2.5 *Wiener's Density Theorem Fourier Inversion in One Dimension 2.3.1 Dirichlet Kernel and Symmetric Partial Sums 2.3.2 Example of the Indicator Function 2.3.3 Gibbs-Wilbraha m Phenomenon 2.3.4 Dini Convergence Theorem 2.3.4.1 Extension to Fourier's Single Integral 2.3.5 Smoothing Operations in lR 1-Averaging and Summability 2.3.6 Averaging and Weak Convergence 2.3.7 Cesaro Summability 2.3.7.1 Approximation Properties of the Fejer Kernel 2.3.8 Bernstein's Inequality 2.3.9 *One-Sided Fourier Integral Representation 2.3.9.1 Fourier Cosine Transform 2.3.9.2 Fourier Sine Transform 2.3.9.3 Generalized h-Transform 2 L Theory in IR" 2.4.1 Plancherel's Theorem 2.4.2 *Bernstein's Theorem for Fourier Transforms 2.4.3 The Uncertainty Principle 2.4.3.1 Uncertainty Principle on the Circle 2.4.4 Spectral Analysis of the Fourier Transform 2.4.4. I Hermite Polynomials 2.4.4.2 Eigenfunction of the Fourier Transform 2.4.4.3 Orthogonality Properties 2.4.4.4 Completeness Spherical Fourier Inversion in IR" 2.5.1 Bochner's Approach 2.5.2 Piecewise Smooth Viewpoint 2.5.3 Relations with the Wave Equation 2.5.3.1 The Method of Brandolini and Colzani 2.5.4 Bochner-Riesz Summability 2.5.4.1 A General Theorem on Almost-Everywh ere Summability Bessel Functions 2.6.1 Fourier Transforms of Radial Functions 2.6.2 L 2 -Restriction Theorems for the Fourier Transform 2.6.2.1 An Improved Result 2.6.2.2 Limitations on the Range of p The Method of Stationary Phase 2.7.1 Statement of the Result 2.7.2 Application to Bessel Functions 2.7.3 Proof of the Method of Stationary Phase 2.7.4 Abel's Lemma

108 109 110

112 112 114 115 115 117 117 118 119 121 122 124 124 125 125 128 128 129 131 133 134 134 136

137 138 139 139 145 146 149 152 153 154 157 158 159 161

162 163

164 165 167

VI

3

CONTFNl

FOURIER ANALYSIS IN U' SPACES

169

3.1 3.2

169 169 174 174 175 176 177 179 183 184 185 186 187 188

3.3

3.4

3.5

3.6 3.7 3.8 3.9

4

s

Motivation and Heuristics TheM. Riesz-Thorin Interpolation Theorem 3.2.0. I Generalized Young's Inequality 3.2.0.2 The Hausdorff-Youn g Inequality 3.2.1 Stein's Complex Interpolation Theorem The Conjugate Function or Discrete Hilbert Transform 3.3 .I L'' Theory of the Conjugate Function 3.3.2 L 1 Theory of the Conjugate Function 3.3.2.1 Identification as a Singular Integral The Hilbert Transform on JR. 3. 4. 1 L 2 Theory of the Hi Ibert Transform 3.4.2 L'' Theory of the Hilbert Transform, I < p < oo 3.4.2.1 Applications to Convergence of Fourier Integrals 3.4.3 L 1 Theory of the Hilbert Transform and Extensions 3.4.3.1 Kolmogorov's Inequality for the Hilbert Transform 3.4.4 Application to Singular Integrals with Odd Kernels Hardy-Littlewoo d Maximal Function 3.5. I Application to the Lebesgue Differentiation Theorem 3.5.2 Application to Radial Convolution Operators 3.5.3 Maximal Inequalities for Spherical Averages The Marcinkiewicz Interpolation Theorem Calder6n-Zygmu nd Decomposition A Class of Singular Integrals Properties of Harmonic Functions 3.9.1 General Properties 3.9.2 Representation Theorems in the Disk 3.9.3 Representation Theorems in the Upper Half-Plane 3.9.4 Herglotz/Bochne r Theorems and Positive Definite Functions

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES 4.1 4.2

4.3

Motivation and Heuristics The Poisson Summation Formula in JR 1 4.2.1 Periodization of a Function 4.2.2 Statement and Proof 4.2.3 Shannon Sampling Multiple Fourier Series 4.3.1 Basic L 1 Theory 4.3 .I .I Pointwise Convergence for Smooth Functions 4.3.1.2 Representation of Spherical Partial Sums

192 194 197 200 202 203 206 209 210 212 212 214 216 219

222 222 223 223 225 228 230 231 233 233

CONTENTS

4.4 4.5

4.6

4. 7 5

APPLICATI ONS TO PROBABIL ITY THEORY 5.1 5.2

5.3 5.4

5.5 5.6 5.7 6

4.3.2 Basic L 2 Theory 4.3.3 Restriction Theorems for Fourier Coefficients Poisson Summation Formula in IRJ 4.4.1 *Simultaneo us Nonlocalizat ion Application to Lattice Points 4.5.1 Kendall's Mean Square Error 4.5.2 Landau's Asymptotic Formula 4.5.3 Application to Multiple Fourier Series 4.5.3.1 Three-Dime nsional Case 4.5.3.2 Higher-Dime nsional Case Schrodinger Equation and Gauss Sums 4.6.1 Distributions on the Circle 4.6.2 The Schrodinger Equation on the Circle Recurrence of Random Walk

Motivation and Heuristics Basic Definitions 5.2.1 The Central Limit Theorem 5.2.1.1 Restatement in Terms of Independent Random Variables Extension to Gap Series 5.3.1 Extension to Abel Sums Weak Convergence of Measures 5.4.1 An Improved Continuity Theorem 5.4.1.1 Another Proof of Bochner's Theorem Convolution Semigroups The Berry-Essee n Theorem 5 .6.1 Extension to Different Distributions The Law of the Iterated Logarithm

INTRODUC TION TO WAVELETS 6.1 6.2 6.3

Motivation and Heuristics 6.1.1 Heuristic Treatment of the Wavelet Transform Wavelet Transform 6.2.0.1 Wavelet Characteriza tion of Smoothness Haar Wavelet Expansion 6.3.1 Haar Functions and Haar Series 6.3.2 Haar Sums and Dyadic Projections 6.3.3 Completenes s of the Haar Functions 6.3.3.1 Haar Series in Co and Lp Spaces 6.3.3.2 Pointwise Convergence of Haar Series

VII 235 236 238 239 241 241 243 244 245 247 247 248 250 252 256 256 256 260 261 262 266 268 269 270 272 276 279 280 284 284 285 286 290 291 291 292 295 296 298

VIII

CONTENl "

299

6.3.4 6.3.5 6.3.6 6.3.7 6.4

*Con-..truction of Standard Brownian Motion *Haar Function Representation of Brownian Motion *Proof of Contmuity *Levy'-. Modulus of Continuity Multire~olution Analy-;is 6.4.1 Orthonormal Systems and Riesz Sy~tems 6.4.2 Scaling Equations and Structure Constants 6.4.3 From Scaling Function to MRA 6.4.3. I Additional Remark-. 6.4.4 Meyer Wavelet-; 6.4.5 From Scaling Function to Orthonorn1"l Wavelet 6.4 . ."i.l Direct Proof that VI 8 v() I-. Spanned hy

301 301 302

303 304 310 313 315 318 319 324

{\.IJ(f- k)}, . .

6.4 . ."i.2 6.5

6.6

6.7

Null Integrahility of Wavelets Without Scaling Function-.

325 326 327 330 331 333

Waveleh with Compact Support 6.5.1 From Scaling Filter to Scaling Function 6.5.2 Explicit Con-.truction of Compact Wavelets 6 .."i.2.1 Dauhechie~ Recipe 6.5.2.2 Hernande:t-\Vei~~ Recipe 6.5.3 Smoothness of Wavelet-> 6.5.3. I A Negative Re~ult 6.5.4 Cohen's Exten~ion of Theorem 6.5.1 Convergence Properties of Wavelet Expansion~ 6.6.1 Wavelet Series in U' Space~ 6.6. I. I Large Scale Analysis 6.6. I .2 Almost-Everywhere Convergence 6.6. I .3 Convergence at a Preas~igned Point 6.6.2 Jack-.on and !~ern~tein Approximation Theorems Waveleb in Several Variables 6.7.1 Two Important Examples 6. 7. I . I Tensor Product of Wavelets 6.7.2 General Formulation of MRA and Wavelets in IR;" 6.7.2.1 Notations for Subgroups and Coscts 6.7.2.2 Riesz Systems and Orthonormal Systems in 6.7.2.3 Scaling Equation and Structure Constants 6.7.2.4 Existence of the Wavelet Set 6.7.2.5 Proof That the Wavelet Set Spans V 1 8 V0 6.7.2.6 Cohen's Theorem in IR;" 6.7.3 Examples of Wavelets in JRd

334 336

338 341 341

345 346 347 347

352 352

354 354 355

rn;c1

356 357 358

361 362 362

References

365

Notations

369

Index

373

PREFACE

To my parents, Harry A. Pinsky and Helen M. Pinsky, who led me to the path of learning

This book provides a ~elf-contained treatment of classical Fourier analysis at the upper undergraduate or begining graduate level. I assume that the reader is familiar with the rudiments of Lebesgue measure and integral on the real line. My viewpoint is mostly classical and concrete, preferring explicit calculations to existential arguments. In some cases, several different proofs are offered for a given proposition to compare different methods. The book contains more than 17 5 exercises that are an integral part of the text. It can be expected that a careful reader will be able to complete all of these exercises. Starred sections contain material that may be considered supplementary to the main themes of Fourier analysis. In this connection, it is fitting to comment on the role of Fourier analysis, which plays the dual role of queen and servant of mathematics. Fourier-analytic ideas have an inner harmony and beauty quite apart from any applications to number theory, approximation theory, partial differential equations, or probability theory. In writing this book it has been difficult to resist the temptation to develop some of these applications as a testimonial of the power and flexibility of the subject. The following list of "extra topics" are included in the starred sections: Stirling's formula, Laplace asymptotic method, the isoperimetric inequality, equidistribution modulo one, Jackson/Bernstein theorems, Wiener's density theorem, one-sided heat equation with Robin boundary condition, the uncertainty principle, Landau's asymptotic lattice point formula, Gaussian sums and the Schrodinger equation. the central limit theorem, the Berry-Esseen theorem and the law of the iterated logarithm. While none of these topics is "mainstream Fourier anaysis," each of them has a definite relation to some part of the subject. A word about the organization of the first two chapters, which are essentially independent of one another. Readers with some sophistication but little previous knowledge of Fourier series can begin with Chapter 2 and anticipate a self-contained treatment of the n-dimensional Fourier transform and many of its applications. By contrast, readers who wish an introductory treatment of Fourier series should begin with Chapter I, which provides a reasonably complete introduction to Fourier analysis on the circle. In both cases I emphasize the Riesz-Fischer and Plancherel theorems, which demonstrate the

X

FI<Ef-1\( 'I

natural harmony of Fourier analysis with the Hilbert ~pace" L 2 Clf) and L 2 (lR"). However much of modern harmonic analysis is carried out in the U' spaces for p -=/= 2, which is the "ubject of Chapter 3. Here we find the interpolation theorems of Riesz- Thorin and Marcinkiewicz, which are applied to discuss the boundedness of the Hilbert transform and its application to the U' convergence of Fourier series and integrals. In Chapter 4 I merge the subjects of Fourier series and Fourier transforms by means of the Poisson summation formula in one and several dimensions. Thi~ also has applications to number theory and multiple Fourier series, a~ noted above. Chapter 5 explores the application of Fourier methods to probability theory. Limit theorems for sums of independent random variables are equivalent to the study of iterated convolutions of a probability measure on the line, leading to the central limit theorem for convergence and the Berry-Esseen theorems for error estimates. These are then applied to prove the law of the iterated logarithm. The final Chapter 6 deals with wavelets, which form a class of orthogonal expansions that can be studied by means of Fourier analysis-specifi cally the Plancherel theorem from Chapter 2. In contrast to Fourier series and integral expansions, which require one parameter (the frequency), wavelet expansions involve two indices-the scale and the location parameter. This allows additional freedom and leads to improved convergence properties of wavelet expansions in contrast with Fourier expansions. I include a brief application to Brownian motion, where the wavelet approach furnishes an easy access to the precise modulus of continuity of the standard Brownian motion. Many of the topics in this book have been "class-tested" to a group of graduate students and faculty members at Northwestern University during the academic years 1998-2000. I am grateful to this audience for the opportunity to develop and improve my original efforts. I owe a debt of gratitude to Paul Sally, Jr., who encouraged this project from the beginning. Gary Ostedt gave me full editorial support at the initial stages followed by Bob Pirtle and his efficient staff. Further thanks are due to Robert Fetterman, whose lectures provided much of the inspiration for the basic parts of the book. Further assistance and feedback was provided by Marshall Ash, William Beckner, Miron Bekker, Leonardo Colzani, Galia Dafni, George Gasper, Umberto Neri. Cora Sadosky, Aurel Stan, and Michael Taylor. Needless to say, the writing of Chapter I was strongly influenced by the classical treatise of Zygmund and the elegant text of Katznelson. The latter chapters were influenced in many ways by the books of Stein and Stein/Weiss. The final chapter on wavelets owes much to the texts of Hernandez/Weis s and Wojtaszczyk.

Mark A. Pinsky

CHAPTER

1 FOURIER SERIES ON THE CIRCLE

1.1

MOTIVATION AND HEURISTICS

1 . 1 .1

Motivation from Physics

Two major sources of Fourier series are the mathematical models for (i) the vibrating string and (ii) heat ftpw in solids.

1. 1. 1. 1

The vibrating string

The first systematic use of trigonometric series can be found in the work of Daniel Bernoulli ( 1753) on the vibrating string. A simple harmonic motion of a string of length rc is defined by the formula (1.1.1)

f(x, t) =A sin nx cos(nt- a)

for suitable constants A, a, and n = 1, 2, .... A is the amplitude, n is the angular frequency, and a is the phase shift. The simple harmonic motion is a solution of the differential equation fu = fxx, which is supposed to describe the small transverse displacement f(x, t) of a tightly stretched string whose ends are fixed at x = 0 and x = rc . More complex, multiple harmonic motions are obtained by linear superposition N

(1.1.2)

f(x, t) = LAn sinnxcos(nt- an). n=l

Functions of this form can be used to satisfy a variety of initial conditions. if we are given the values off and the partial derivative aj ;at when t = 0. This is possible whenever f(x, 0) and 8f j8t(x, 0) are expressed as finite linear combinations L~=l an sin nx. This may be less obvious in other cases; for example

1

2

INTRODUCTION TO FOURILR ANALYSI~ AND WAVELETS

f(x, 0) = sin 3 x = (3 sinx - sin 3x)/4, whereas sin 2 x cannot be so expressed. In order to work with these trigonometric sums, we note the property of orthogonality, expressed as

(1.1.3)

frr

Jo

sinmxsinnxdx = 0,

If a function has the form f(x) = L~=l forr f(x) sin nx dx = 0 for n > N. Exercise 1 .1 .1. Show that form L~= 1 ak sin kx.

if N is odd,

m

a,

# n.

sin kx,

then we must have

sinN x can be written as a finite sum of the

Exercise 1.1.2. Suppose that we have a convergent series expansion of sin 2 x = 'L:~ 1 ak sin kx on the interval 0 :S x :=:: rr. Prove that ak is nonzero for infinitely many values of k. Hint: Assume a finite expansion and use the orthogonality relation ( 1.1.3) to obtain a contradiction.

Exercise 1.1.3. Generalize Exercise 1.1.2 to any even power of sin x, showing the impossibility of an expansion sin" x = L~= 1 ak sin kx for 0 < x :S 7T where n = 4,6, ....

Any multiple harmonic motion (1.1.2) is a 2rr-periodic function of time: f(x, t + 2rr) = f(x, t) for all -oo < x, oo, -oo < t < oo. It also is a 2rr-periodic function of x and is odd with respect to x = 0 and x = rr, meaning thatf(-x) = -f(x) and f(rr + x) = -f(rr - x) for all x. Exercise 1 .1 .4. Suppose that f (x), - oo < x < oo is given. Show that any two of the following properties imply the third: (i)f(x + 2rr) =f(x), Yx; (ii)f(-x) = -f(x), Yx; (iii)f(rr - x) = -f(rr + x), Yx. 1. 1. 1.2 Heat flow in solids The vibrating string suggests the use of sine series, since the ends of the string are fixed. More general trigonometric series are suggested by the study of heat flow in a circular ring, assumed to have circumference 2rr. In this model it is natural to assume that the temperature u(x, t) is a 2rr-periodic function of x (but not periodic in time).. Fourier (1822) formulated the heat equation u, = u:tx to describe the time evolution of the temperature. It is satisfied by any function of the form (A, cos nx + B, sin nx)e-" 2 ' where n = 0, 1, 2, ... , t ::: 0 and -rr < x < rr. Taking linear combinations of these, we arrive at a "general solution" N

(1.1.4)

u(x, t) =

L(A, cos nx + B, sin nx)e-" n=O

2 '.

FOL'R!FR 'i!-R!I,S ON THE CIRCLE

3

This will fit an initial temperatu re profile j(x) if and only iff is expressed as a finite trigonome tric sum N

( 1.1.5)

j(x) =

L(A

11

cosnx+B 11 sinnx).

11=0

The coefficien ts A 11 , 8 11 can be found by using the orthogona lity relations

f:

(1.1.6)

sin mx sin nx dx = 0

(1.1.7)

m =/= n,

J:

( 1.1.8)

together with the norms:

sin mx cos nx dx = 0

f':rr sin 2 nx dx

=

JT

=

(1.1.9)

Ao = - I /,. f(x) dx, 2n -rr

(1.1.10)

A 11 = -I /"" j(x) cos nx dx JT

(1.1.11)

m =!= n.

B 11 = -I JT

all m, n,

f'_'_rr cos 2 nx dx, n =

I, 2, ... Thus

n =I, 2, ... ,

-,7

!"

j(x) sin mx dx

n =I, 2, ....

-rr

Fourier's thesis is that ( 1.1.5) will also remain true for N = CXJ, if the coefficients A 11 , B, are defined by these formulas. This is most easily done in case the series L~ 0 ()A,j + )B,)) converges .

1.1.2

Absolute ly Converg ent Trigonom etric Series

We begin the mathemat ics by considerin g functions defined by (1.1.12)

=

f(f3) =

L(A, cosnB +

B, sinnB)

n=O

where :L~ 0 ()A, I+ IB~~I) < CXJ. The values off are determine d on any interval of length 2n. A standard choice is the interval 1r = ( -JT, n], where we identify 2n-period ic

functions on lR with functions on 'IT'. A function on 1r is considere d continuou s (resp. differentia ble) if the correspon ding periodic function on lR is continuou s (resp. differentiable). In concrete terms this means that/ is continuou s (resp. differentia ble) on ( -JT, JT] withf(n - 0) =f(-JT + 0) (resp. f'(n- 0) =f'(-JT + 0)).

4

INTRODUCTION TO FOURIER ANALYSIS AND WAVFLET<;

In order to simplify the notations throughout, we recall the Euler formula for the complex exponential function e' 8 = cos() + i sin() and its consequence s · n s1n u =

(1.1.13)

l ( tH i

2

e

- e -ie) .

This allows us to rewrite the trigonometri c series ( 1.1.12) in the complex form CXJ

(1.1.14)

f(fJ)

L

C,.eme

n=~oc

Complex notation is especially efficient when we multiply two such functions, the formula e' 8 e"" = ei(R+t/J) being a streamlined expression of the addition formulas for both the sine and cosine functions. Similar efficiency is realized in the integration formulas J eux dx = a- 1e= when a is a nonzero complex number. The passage from real functions to complex functions also suggests the natural definition of convergence of the series (1.1.14), namely as the limit of the symmetric partial sums L."!..w We use throughout the notation LnEZ for this limiting process. If we try to consider more general definitions of convergence , difficulties will arise. Throughout the text we will systematical ly use the Lebesgue integral and its many properties. In some cases a more elementary definition of integration will suffice, but we prefer to systematical ly employ the Lebesgue theory-both for its increased generality and its ease with respect to passage to the limit. Theorem 1.1.5. Suppose that LnEZ IC,. I < CXJ. Then f defined by ( 1.1.14) is a continuous function on 'Jr. The coefficients are obtained as (1.1.15)

n

E

Z.

If g is any other L 1 function on 'Jr, we have the Fourie~ reciprocity formula (1.1.16)

where D,. is the Fourier coefficient of g, defined by ( 1.1.15) withf replaced by g. In particular we have Parseval's identity ( 1.1.17)

Proof. The uniform convergence of (1.1.14) follows from the Weierstrass M test, since \C,e'" 11 \ = \C, \,the general term of a convergent numerical series, so that the limit function is continuous. This uniform convergence also holds for the series defined by e-'N 11f(B),

5

FOI 'RIER <;FRIE<; ON THF CIRCLE

which we may therefore integrate term-by-ter m. In the process we encounter the integral

J:

(1.1.18)

e -tNO emil d(J =

0

(n

=/= N.

n E Z. N E Z)

while the integral is 2n for n = N. Equation ( 1.1.18) is known as the complex orthogonal ity relation. This shows that (1.1.19)

(NEZ)

which was to be proved. To prove ( 1.1.16 ). we multiply the series ( 1. L 14) by g( 0). The partial sums are bounded by a multiple of the integrable function lg(e) 1. hence we can apply Lebesgue's dominated convergenc e and integrate term-by-ter m to obtain (1.1.20)

-I

27T

!"

_n

f((})g((J) de

=

C f' e"' g(e) de = - 1 L L -" 27T _,., 27T 11

'" ·:.

C,D "

nEZ

which was to be proved. Taking g =] gives the Parseval identity ( 1.1.17).

•

Exercise 1.1.6. Suppose that LnEZ InC" I < oo. Prove that the series ( 1.1.14) defines a differentia ble function withf' ((}) = LnEZ inCnemo a continuou s function. Hint: Use the inequality le'h - 11 :5 lhl to justify passage to the limit.

Exercise 1 .1 . 7. Suppose that LnEZ In" Cn I < oo for some k = 2, 3, .... Prove that the series ( 1. 1.14) defines a k-times differentia ble function with f (8) = g(¢ - (}), we compute the Fourier coefficien t by writing

which, when substitute d into ( 1.1.16), yields the following.

Corollary 1.1.8. The convolutio n ofan absolutely integrable trigonome tric series f with an arbitrary L 1 function g has the representa tion

(1.1.21)

A large source of examples of absolutely convergen t trigonome tric senes is obtained from power series. Consider a general Laurent series :X:

(1.1.22)

~ f( Z ) --L-Jan ....~" n=O

+

"X.

~ h n .~-11 L-J .. , n=l

6

INTRODUCTION TO H)l IRIFR ANALYSIS AND WAVEU-:1 ~

assumed to be absolutely convergent in an annular region n < lzl < r2. Then 'X.

f(re'li) =

(1.1.23)

L

a,r"e"' 11

+

n=O

="-

L

b,r "c- 11111 ,

n=l

is an absolutely convergent trigonometric series. In particular we will apply this to e~ = L:::o z" jn! in the next Section 1.1.3.

1 .1 .3

*Examples of Factorial and Bessel Functions

We can generate many useful examples beginning with the power series of the exponential function.

Example 1.1.9. Let C, = 0 for n < 0 and C 11 = r"jn! where r > 0 and n = 1, 2, .... Then F(fJ) =ere'" and the Fourier coefficient formula ( 1.1.15) and Pars eva/ identit_v ( 1.1. 1 7) speciali;:_e to 7

1 / -r" = e"''"e-"'HdfJ, n! 2n _,.

(1.1.24)

(1.1.25)

fo(2r) : =

L""

(r"I ):.o n.

n=O

= -1 2Jr

/::r

r > 0, n = 0, 1, 2, ...

e2rn"H dfJ,

r

:=:: 0.

_,.

Equation (1. 1.25) gives an integral formula for the modified Bessel function / 0 (2r) defined by the power series on the left side. In particular we will determine the asymptotic behavior of / 0 (2r) when r -+ oo by analysis of the integral on the right side of ( 1.1.25). Meanwhile ( 1.1.24) gives a useful representation of the factorial function. We will use this to present a self-contained treatment of Stirling's formula in the form lim n!/n"d e-" = ~-

(1.1.26)

n---x..,

To obtain this result we take r = n in ( 1.1.24 ), to obtain (1.1.27) where F(fJ) = e' 11 - 1 - ifJ. This is set up to apply the Laplace asymptotic method, whose proof will be given at the end of the section, and whose statement follows.

Proposition 1.1.10. Suppose that F(fJ),

-Jr :Sf) :S Jr is a continuous complexvaluedfunction so that Re F(fJ) has a unique maximum at f) = 0 with F(O) = 0, limH _. 0 F(fJ)jfJ 2 = -k with Re k > 0. Then

(1.1.28)

f

rr enflli)df) -7

=

(-1 ). v["if+ -;;k Jn 0

n-+ oo.

In the present case F((J) = e''1 - 1 - ifJ, so the conditions are satisfied with ReF(fJ) = cos(} -1 < Oforn > lfJI > Oandk -F"(0)/2 1/2.

=

=

FOURIER ->ERIES ON THE CIRCLE

7

Exercise 1 .1 .11. Prove that the factorial function satisfies the two-sided system of inequalities C 1 nne-n < n! < C2(n + l)n+Ie- for positive constants C1, C2 andn = 1, 2, .... Hint: Compare

L;= 1 log k

with the integral of log x and then exponentiate.

As a second application of Laplace's method, we take F(()) = cos()- 1 to deduce the asymptotic formula for the modified Bessel function defined by ( 1.1.25): lo(2r) = e 2 r

(1.1.29)

fl(l + o(l)), '/4;;;

r

~

CXJ.

Returning to the theory, one can discuss further general properties of the class of absolutely convergent trigonometric series. They form an algebra of functions, meaning that the sum and product of two is again in the same class. In detail

L

(1.1.30)

Ane'"li

nEZ

+

L

Bneinli =

nEZ

L

(An

+ Bn)einO,

nEZ

(1.1.31) We will prove later that the class of absolutely convergent trigonometric series contains the class of HOlder continuous functions of exponents greater than

!.

1 .1 .4

Poisson Kernel Example

A second useful example of an absolutely convergent trigonometric series is generated by the functionf(z) = 1/(1 - z), defined in the unit disc lzl < 1. Thus we obtain the absolutely convergent series (1.1.32)

o:::r
When we take the real and imaginary parts, we obtain the real series 1 - r cos () = 1 1 -------= 1 + ~ r" cos n() = - + - ~ rlnleinli 1 + r2 - 2r cos() ~ 2 2 L...., n=l

=

r~n()

----=-----= ~ rn sin nO 2 1+ r

-

2rcos()

L....,

nEZ

'

·

= - - ~ rlnl sgn(n)einO, 1

2 L...., nEZ

n=l

where the signum function sgn (n) is defined by setting sgn (0) = 0, sgn (n) = 1 if n > 1 and sgn (n) = -1 if n :::: -1. These can be rewritten as (1.1.33)

(1.1.34)

Pr(()) :=

Qr(()) .-

1+

1+

1- r 2 = r 2 - 2rcos()

2r sin() = -i r 2 - 2r cos ()

L rlnleinO, nEZ

L nEZ

sgn (n)rlnleinB.

8

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Pr is the Poisson kernel and Qr is the conjugate Poisson kernel. See Figures 1.1.1 and 1.1.2. It is easily checked that both Pr and Qr are solutions of Laplace's equation. Equations (1.1.15) and (1.1.17) yield n E Z,

(1.1.35) (1.1.36)

-1 2 7r

1" .

e-ml} Qr(B) dB

=

- i sgn (n)rlnl,

0 < r < 1, n E

Z,

0 < r < 1,

-rr

(1.1.37)

0::;: r < 1,

(1.1.38)

O
Setting n = 0 in ( 1.1. 35) yields (1.1.39)

1 1 =2rr

lrr -rr

1-

1+

r2 -

r2

2r cos(}

d(}

'

o::::r
These integration formulas will be used frequently in dealing with the summability of Fourier series.

y

10

FIGURE 1.1.1 Poisson kernel Pr(O) with r = 0.8

FOURIER SERIES ON THE CIRCLE

9

y

FIGURE 1.1.2 Conjugate Poisson kernel Q,(B) with r

= 0.8

Exercise 1 . 1 . 1 2. Prove that P r and Qr are both solutions of Laplace's equation + Uyy = 0 where x + iy = reiB.

Uxx

Hint: Using (x + iyY = r"einfJ shows that each term in the series is a solution of Laplace's equation. Another approach is to recall the form of Laplace's equation in polar coordinates, namely u,, + (1/r)u, + (ljr 2 )u99 = 0.

1 .1 .5

*Proof of Laplace's Method

We formulate this in somewhat greater generality as follows: Let A(x), B(x) be continuous functions defined on an interval a < x < b so that ReB(x) < ReB(xo) for x =I= xo E (a, b). B satisfies the asymptotic relation (B(x) - B(xo))f(x - xo) 2 - - -k when x -- xo E (a, b) with Re(k) > 0. We will prove that when t -- oo

(1.1.40)

C(t) :=

1b

A(x)etB(x) dx = erB(xo) ( A(xo)f!£

Proof. Without loss of generality we can assume that Xo

+ o ( ~))

.

= 0 and B(O) = 0.

10

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Step 1. Localizing the range of integration: For any 8 > 0 the maximum of Re B(x) when lxl 2: 8 is negative so that we can write

=

C(f)

{

A(x)e'B(r)

dx

+

O(e-l>r)

}lri<8

for some D > 0. Step 2. Replacing B by its quadratic approximation : Now we write the expansion of B(x) around x = 0:

For any two complex numbers z 1

= x 1 + iy 1 , z 2 = x 2 + iy 2

we have the inequality

( 1.1.41)

Applying this with z 1 = B(x), z 2 = -kx 2 , we can take z 3 enough. With this choice of 8 we can write

-kRx 2 /2 by taking 8 small

::;:

lxl ::0 8.

The error in replacing the exponential by the purely quadratic exponential term is bounded by tClE

1 ' • z,, lxl-e-'•R-'

1
by making the substitution

I-

dx < tC3E

foe lxl

2

k

e-r Rx

'I'

-

-oc

dx =

const

E --

../i

v = x../i.

Step 3. Replacing A(x) by A(O): Since A is continuous, we can rechoose 8 > 0 so that E when lxl < 8. The error made in replacing A(x) by A(O) is bounded by

IA(x) - A(O)I <

E

1 · lxl<~

also by making the substitution C(t)

1

00

e-'•R< 2 1 2 dx <

E

" ' 1 2 dx = e-'Rx

const E --

-oo

v = x ../i. From Steps = A(O)

1:

e-rkr

2

dx

+

../i

1, 2 and 3 we have

o( ~) _

Step 4. Integral over the real line: Finally we replace the integral on -8 < x < 8 by the integral over -oo < x < oo with an exponential error, since

11.

e-rkx'

ltl>o

dxl

<

.!_ { lx!e-'~R•' dx 8 Jlxl>~

=

_1_e-rkRo 2 8kRt

But the integral over -oo < x < oo is (1.1.42)

f

oo e--rkx'

-oc

dx =

fir. Vkf

If k is complex, we must take the square root with positive real part, since the integral clearly has this property. Combining these steps completes the proof. •

Exercise 1.1.13. /fRe k > 0, show that the real part of f~oo e-k.x' dx is positive.

FOURIER SERIES ON THE CIRCLE

11

the change of variable x 2 = y to reduce consideration to which is an infinite sum of integrals over (knjb, (k + l)njb). Show that the first of these is positive and the remaining terms altemate in sign and decrease to zero. Altematively, one mat prove this by using complex analysis, considering the analytic function k ~ rj>(k) = forxo e-Ax dx, defined for Re k > 0. On the real axis rj>(k) agrees with the branch of the function k ~ (rr 1k) 112 , which is positive and real on the positive real axis. Hence these two analytic functions agree on the entire halfplane Re k > 0. Hint:

First

make

J0rxo(e-aYjy 112)cos(by)dy,

Exercise 1.1.14. Suppose, in addition, that A(x) is Lipschitz continuous: IA(x)- A(y)i _::: Klx- yl and that the second derivative B"(x) exists and is also Lipschitz continuous. By going through the steps of the above proof, show that in this case the error term o(lj ../i) in Laplace's method can be replaced by 0(1/ t). Computation of (1. 1.42). This is obtained 2 I := fiR e-kx dx in polar coordinates. Thus

=

LL

=

111'

by considering the square of

e-k(x2+y2) dx dy

-rr

{oo e-kr2 r dr dB

lo 1

= 2JT-. 2k

1.1.6

*Nonabsolutely Convergent Trigonometric Series

It is possible to deal with trigonometric series with monotonically decreasing coefficients by the method of summation by parts to produce convergent series. Given any sequence of complex numbers an, n = 0, 1, 2, ... , define (.6.a)n = an -an-t for n > 1. The basic identity is that for any two sequences an, bn N

( 1.1.43)

aNbN- aMbM =

L

N

ak(.6.b)k

k=M+l

+

L

bk_,(.6.ah,

M
k=M+l

The proof is left as an exercise. Exercise 1.1.15. Prove ( 1.1.43 ). Hint: Write the left side as a telescoping sum and show that akbk- aA-lbk-l = ak(D.b)k

+

bk_,(D.a}k.

As a first application of summation-by-parts, we can deduce the convergence of certain trigonometric series. Proposition 1.1 .1 6. Suppose that An > 0 and An > An+ 1 with lim An = 0. Then the trigonometric series L~o Anemx is convergent for x =1= 0.

12

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

Proof. We apply ( 1.1.43) with an = An, bn = Bn (x) = sum with 1-

(1.1.44)

IBn(x)l =

1

ei(n-t-l)x I

1- e'•

1

L:Z=o e'kx. This is a finite geometric

:5 sin (x/2)

Applying ( 1.1.43) shows that N

L

(1.1.45)

N

ake'kx

L

= aNBN(x )- aMBM(x) -

k=M+l

Bk-t (x)(Aa)k.

k=M+l

The first two terms tend to zero when M, N --+ oo. The sum is estimated by

I L Bk-l(x)( Aah :5 k=M+l

I . (

N

(1.1.46)

I

SlD X

12

)

L IAaki =(aM -aN) k=M+l N

.

1

(

SID X

12)

•

which tends to zero when M, N--+ oo.

Example 1.1.17. The trigonom etric series Ln> einx jlog n is converge nt for 2 x =1- 0. By taking the real and imagina ry parts, we see that the series Ln> 2 cos nxjlog n is converg ent for x =I= 0 and the series Ln> 2 sin nxjlog n is converge nt for all x. Exercise 1.1.18. Prove that we have uniform converge nce of Ln::: einx /log n on 2 any closed interval not containi ng x = 0.

In an appendix to this chapter we prove the basic Cantor uniquene ss theorem , which allows one to identify the coefficie nts An from the sum of the (conditio nally converge nt) trigonom etric series. Summat ion by parts can also be used to estimate the modulus of continui ty of an absolute ly converge nt trigonom etric series in terms of the tail sum, defined as 00

L

E,. :=

(IAki

+

IA-ki).

k=n+I

To do this, let h > 0 and write f(x +h)- f(x) = LAnein x(einh- 1), nEZ

if(x +h)- f(x)l <

L

!nhAnl

L

+2

lni~N

!Ani.

lni>N

The second sum is 2EN. The first sum is rewritten as N

L lni~N

N-1

lnhAnl = -h Ln(dE )n = -hNEN

+ h LEn,

n=O

n=O

so that 1 lf(x +h)- f(x)l .::5 hN EN+ N (

?; En

N-1

)

+

2EN.

FOURIER SERIES ON THE CIRCLE

13

The first and third terms are balanced by taking hN = 1. The middle term is an average of the last term and can be estimated therefrom. Specific forms of the tail sum will lead to various concrete estimates. Exercise 1.1.19. Suppose that the tail sum satisfies En < en-a for some 0 1. Exercise 1.1.22. Illustrate the results of the previous exercise in the following three cases:

L

CXl

00

L

2-n cos(3nx),

n=O

1.2

L

CXl

3-n cos(2nx),

n=O

2-n cos(2nx).

n=O

FORMULATIO N OF FOURIER SERIES

Armed with some motivation, we now begin the formal study of Fourier series.

1.2.1

Fourier Coefficients and Their Basic Properties

We begin with an integrable function/ on 1r = ( -1r, rr]. Any such function can naturally be identified with a 2rr -periodic function on the entire real line. This extension is helpful in many ways, especially in computing integrals since we may write J..lf f(x + ()) d() =

jT

j((})d(}.

The Fourier coefficients, or discrete Fourier transform, off by the formula (1.2.1)

1

A

f(n) =

2

7r

JT{

E

L 1 (1r) are defined

0

f(8)e-m d().

This is a linear transformation from the space L 1 (1r) to the space bilateral sequences. We formalize this as a proposition.

fCXl (Z)

of bounded

Proposition 1.2.1. The space of discrete Fourier transforms is an algebra, as expressed by (1.2.2) (1.2.3)

14

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

where the convolution of two L 1 functions is defined by

Ut *fz)(e) =

2~ l

ft (
and where the integral is convergent for almost every e. The convolution product is commutative and associative: ft *fz = fz *ft. (ft *fz) *h = ft * (fz */3). Furthermore the mapping f ~ j is a contraction, meaning that if(n)i < - 1

(1.2.4)

A

2n

i

1r

lf(e)l de.

Proof. The first formula is immediate from the linearity of the integral. To prove the second formula, we note that the convolution is well defined by the Fubini theorem, since 2rrlf, */2!(8) :=:: j.., l/1 (c/>)11/2(8- c/>)1 de/>. The latter integral has a finite integral over 1!', since by Fubini and a change of variable we find that

l de llf, (c/>)llhW- c/>)1 de/>=

li

If, (t9)1!/2(c/>)l de/> dt9 =

i

If, (8)1 dt9

i

l/2(8)! de.

Therefore the integral defining the convolution of 1/1 1 * lhl is finite almost everywhere, and dominates the convolution / 1 * h. Having made these preparations, we can multiply the Fourier coefficients and transform by Fubini:

4rr 2f,(n)

xf2(n)

=

f/,(t9)e-"' 0 lf2(c/>)e-m¢ =

l

e-"'"' (if,(0)/2(1/r- t9)d8) dl/r

which was to be proved. The commutative and associative properties are most easily deduced from these properties of ordinary multiplication, once we have used Fubini to identify the convolution as the unique L 1 function F with the property that for any bounded function h, f-n: f-n: h(x + y)f, (x)f2(y) dx dy = 2rr f-n: h(z)F(z) dz. The contraction property is immediate from (1.2.1). •

Exercise 1.2.2. Let f 1 ,fz

E

L 1 (1f').

Show that for any bounded measurable

function h we have

Ll

h(x+y)ft(x)fz(y )dxdy = 2n

l

h(z)(f1 *h)(z)dz.

One may be tempted to conclude that the convolutionf1 * h is represented by the Fourier series LnEZJ, (n)]2 (n)e'" 8 . This cannot be literally true in general, because the latter series does not converge pointwise or in L 1 , for a general L 1 function. Nevertheless the following special case of Fourier reciprocity is quite useful.

Proposition 1.2.3. Suppose that LnEZ IAnl < CXl and we have an absolutely convergent Fourier series K(e) = LnEzAneino. Then for any f E L 1 (1f') we have ( 1.2.5)

(f

* K)(e)

= LAn](n)ein8. nEZ

Proof. Begin with the identity

/(8- c/>)K(c/>) = LAne"'¢/(8- c/>). nEZ

FOURIER SERIES ON THE CJRCLS

15

By hypothesis, this is a series of L 1 functions that is convergent and bounded pointwise by the integrable function 1/1 x Lnez IAnl. so that by Lebesgue's dominated convergence theorem we can integrate term-by-term to conclude { j(O- )K() d¢

}y

=

LAn { emj((JnEZ

=

) d¢

JT

2rr LA,j(n)eme nEZ

•

which was to be proved.

This can be immediately applied to the Poisson kernel K(8) = Pr(B) = LneZ rlnieinO. If Lnezf(n)einO is the Fourier series of some L 1 function f, we may now assert that (f

* Pr)(B)

= Lf(n)rlnieinO. nEZ

Another example is obtained from the Dirichlet kernel, defined by the finite trigonometric sum N

DN(f)) =

(1.2.6)

L einO. n=-N

The Fourier partial sum operator N

SNJ = Lf(n)einO = L -N

lr-N.NJ(n)f(n)einO

neZ

is equivalently obtained as the convolution (1.2.7)

I SN/(8) =

(DN *f)(8).

Here lA is the indicator function, defined by lA (n) = 1 if n E A and equals zero otherwise. From (1.2.6), it is clear that DN is an even function and that 1-rr: DN = 21r. See Figure 1.2.1 for the Dirichlet kernel and Figure 1.2.2 for the Fourier partial sums of j(x) =X, -JT < X < 1T.

Exercise 1.2.4. Show that the Dirichlet kernel can be equivalently expressed as (1.2.8)

DN(f)) =

sin(N +~)f) sin f) /2

f) =I= 0, ±21T, ...

by summing a finite geometric series.

Exercise 1.2.5. Suppose that K

E L 1 ('Ir)

is an evenfunction: K(-8) = -K(8). Prove that the convolution operator f -+ K * f is self-adjoint, meaning that for any f E L 1('lf') and any bounded function g we have 1-rr:(K * f)g = 1-rr:f(K *g).

16

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

21t

FIGURE 1.2.1

The Dirichlet kernel DN(O) for N = 5. FromM. Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications. Reprinted by permission of The McGraw-Hill Companies. y

·..\ .i t.~'.h

fz ,'.

.·

/I

.

"·

'·!\

\

.

'

(1t, 0)

FIGURE 1.2.2

=

=

The graphs of the partial sumsfN(O) for N I, 2, 3 of the Fourier series of f(O) 0, -rr < 0 < 1r. FromM. Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications. Reprinted by permission of The McGraw-Hill Companies.

Exercise 1.2.6. Suppose that K E L 1 (1I') is an oddfunction: K(-()) = -K(O). Prove that the convolution operator is skew-adjoint, meaning that for any f E L 1 (1I') and any bounded function g we have j1l'(K *f)g = - J1l'f(K *g).

At this point we can formulate the uniqueness of the Fourier coefficients, following an elementary argument of Lebesgue. The result will also be deduced as a corollary of the summability of one-dimensional Fourier series in Section 1.3.

FOURIER SERIES ON THE CIRCLE

Proposition 1.2.7. Suppose thatf, g for all n E Z. Thenf = g a.e.

17

L 1 (11') have the property that}(n) = g(n)

E

Proof. By the linearity of the mapf--+ j it suffices to prove the result in case g = 0. We first prove the result in case/ is a continuous function. Writing/ = u + iv, we have for any n E Z,

0

= 2rr](n) = =

For any m

i:

=

i:

(u(x)

(u(x) cos nx

+

+ iv(x))e-inx dx

v(x) sin nx) dx

0, 1, 2, ... we apply this to n

i:

f:

=

+i

=

(v(x) cos nx- u(x) sin nx) dx.

m and n

u(x)cosmxdx=O= v(x) cosmxdx

i: i:

0

=

f:

=

-m to conclude that

u(x)sinmxdx, v(x) sinmxdx.

Therefore we are reduced to the case of a real-valued continuous function/ for which (1.2.9)

f:f(x)cosmxdx = 0 = f:f(x)sinmxdx,

m

=

0, 1, 2, ....

Iff is not identically zero, there exists a point Xo where f(Xo) :f:. 0. Replacing f by f(x + x 0 )jf(x0 ), we may assume thatf(x) ~ in a closed interval/ = [ -8, 8], where 0 < 8 < rr. Let t(x) = 1 + cosx- cos8, Tn(X) = t(x)n. Clearly t(x) ~ 1 on I while lt(x)l < 1 on JC, so that Tn(x) ~ 1 on I while Tn(X)--+ 0 on JC. Now Tn is a trigonometric polynomial of degree n, so that from the hypothesis (1.2.9)

4

0

= /_: f(x)Tn(X) dx = 0.

On the other hand, the dominated convergence theorem shows that limn he f (x) Tn (x) dx = 0. Subtracting these, we conclude that limn j;J(x)Tn(x) dx = 0. which contradicts the fact thatf(x)Tn(X) ~ on/. Hencef(x) = 0. Applying this argument separately tof = u and f = v proves the result for any complex-valued/ E C('ll'). Now iff E L 1 (11') satisfies](n) = 0, let F(x) = J~"f(t) dt, a continuous function. Interchanging the orders of integration shows further that for n = ± 1, ±2, ...

4

i:

F(x)e-inx dx = f : e-inx (/_: f(t) dt) dx

=

=

f:

f(t) (frr e-inx dx) dt

11r f(t) (e-inrr-~ e-int) dt 1n

-1r

=0.

=

=

r:,

Hence F(n) 0 for 0 :f:. n E Z. Letting Ao 1j2rr F(x) dx, we can apply the proof in the previous paragraph to the continuous function x· --+ F(x) - A 0 , to conclude that F(x)- Ao 0. But from Lebesgue's theorem on the differentiation of the integral, we have almost everywhere thatf(x) = (djdx)F(x) = 0, which completes the proof. •

=

18

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The above properties of the mapping f ----+ j are algebraic in nature. The following fundamental property is analytic. It is valid for an arbitrary approach to infinity, not restricting n to be integer-value d. Here we note that the Fourier coefficients (1.2.1) are well defined for any n E JR. Riemann-Le besgue lemma: Foranyf E L 1 (1['), limlnl--->oof(n ) = 0 and the convergence is uniform' on compact subsets of L 1 (1r).

Theorem 1.2.8.

We will give two separate proofs of the first statement, then deal with the compactness . First Proof. By making the change of variable

+ JTjn)e-m<+rr(nJ d
+

7r jn,

the integral defining the

= - if(
+ rrjn)e-in d¢.

Adding this to the definition of](n), we obtain 4rr](n) = i

[/(
+

7T jn)

- f( d
Iff is a continuous function, the integrand tends to zero uniformly when In I --+ oo, hence ](n) --+ 0. In case/ is an arbitrary L 1 function, we can find a continuous function g so that II/- gil, < €. Then ](n)

= g(n)

+ if-=8) (n).

The first term tends to zero when In I --+ oo whereas the second term is less than €, by virtue of ( 1.2.4). Thus lim sup, lf(n) I :5 €, which was arbitrary, completing the proof. • Second proof. The result is clearly true for the indicator function of an interval (a, b), since for n I= 0 (1.2.10)

Hence it is also true for the indicator function of a finite union of intervals. Now if E is any measurable set, by the definition of outer measure, there exists a finite union of intervals E so that the symmetric difference Et::J.E has measure less than €. In terms of L 1 norms f ... 11£ - 1t:l < €. By linearity, we have IG(n)l :5 llt:(n)l

+

(l£,.,_E(n)l.

The first term tends to zero while the second term is less than €. Now we can extend to simple functions/ = L a,IE, by linearity and finally to all L 1 functions by appealing to the density of simple functions. For examplefN = Liki:"NZN k2-N Ikz-Ns:"
J...

FOURIER SERIES ON THE CIRCLE

19

This property of uniform convergence is easily applied to show that iff E L 1 (1[') and g is a bounded function, then

L

g(B)f(¢

(1.2.11)

+ (})e-mH d(} ~

0

uniformly for ¢ 1 ::=: ¢ ::=: ¢ 2 when In I ~ cx:l. To see this, note that for any f mapping x ~ fx from 1f to L 1 ('JI') is continuous. In detail lim [ if(x t---j>

t:o

}r;

+ (})- f(xo

E

L 1 ('JI') the

+B) I d(} = 0.

Multiplicatio n by the bounded function g preserves this continuity. The continuous image of a compact set is compact. Hence we can apply the uniform convergence on compacts to deduce that ( 1.2.11) holds uniformly on compact ¢ intervals.

1.2.2

Fourier Series of Finite Measures

The concept of a Fourier series can be easily extended from the class of integrable functions to the class of finite signed measures. Recall that a signed measure on 1I' is defined by a function of bounded variation, which can be represented as the difference of two monotone functions. The sum of two signed measures is the setwise sum: (J.L 1 + f.Lz)(A) = f.L 1 (A) + f.L2 (A). The convolution of two signed measures f.L J. f.L2 is, by definition, the signed measure f.L with the property that for every continuous function h E C('JI')

lx1f

h(x

+ y) dJ.L1 (x) dJ.Lz(y)

= 2n

J

h(z) dJ.L(Z).

The Fourier coefficients of a signed measure are defined by

{l(n) = _I_ { e-inO dJ.L(B). 2n lu Proposition 1.2.1 carries over with no essential change.

Proposition 1.2.9. The space of discrete Fourier transforms is an algebra, as expressed by (1.2.12)

fl1 (n)

+ flz(n)

= (J.LJ+f.Lz) (n),

(1.2.13)

The convolution product is commutative and associative: J.L 1 * J.Lz = f.Lz * J.L , 1 (J.L 1 * J.Lz) * f.L3 = f.L 1 * (J.Lz * f.L3). Furthermore the mapping f.L ~ fl is a contraction, meaning that ( 1.2.14)

~

IJ.L(n)i <

1

2

n Var(J.L).

Exercise 1.2.1 0~ Prove these statements.

20

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The Riemann-Le besgue lemma is not true for signed measures. For example, the Dirac measure 80 , for which 80 (A) = 1 iff 0 E A, has 8o(n) = 1 for all n. One might ask if the Riemann-Le besgue lemma holds for measures that are continuous, but this is not true either. The Fourier coefficients of the Cantor measure do not tend to zero.

Exercise 1.2.11. Prove this statement Hint: FortheCanto rmeasure, /l(n) = n~ 1 cos (2rrn/3k), with/l(3m) = /l(l) form= 1, 2, .... For details, consult Zygmund, 1959, p. 196.

1.2.3

*Rates of Decay of Fourier Coefficien ts

The Riemann-Le besgue lemma provides no further quantitative information about the speed of convergence to zero for an arbitrary L 1 function. We can obtain a convenient upper bound from the representatio n (1.2.15) Hence we have immediately Proposition 1.2.12.

Proposition 1.2.12. Suppose that/

E C(T) has a modulus ofcontinuity w(8) := sup 1x-yJ::;.Sif(x) -f(y)l. Then lf(n)l < !w(:rr/n). More generally, if1 ::;::p < oo and Op(8) := suplhi::;.S 11/h- fliP is the LP modulus of continuity off E LP(T), then lf(n)l < 40p(Jr fn).

For example, iff satisfies a Holder condition with exponent a E (0, 1), we see that}(n) = O(n-a), lnl - 00.

Exercise 1.2.13. Supppose that the LP modulus of continuity satisfies Op(h) < Cha for C > 0 and a > 1. Prove thatf is a constant, a.e. Hint: First show that Qp(h

+ k)

::: Op(h)

+ Qp(k); then iterate this to obtain a contradiction.

If we want to obtain a more precise estimation, we can assume that/ is absolutely

continuous and integrate by parts, as follows:

Proposition 1.2.14. Suppose that f

is absolutely continuous. Then f(n) = (1/in)f'(n); in particular f(n) = o(l/!nl), lnl - oo.A If in addition !'./"::.. ... ,J
Exercise 1.2.15. Prove the above properties by integration by parts. It is difficult to characterize differentiabi lity of a fixed degree in terms of the behavior of the Fourier coefficients. In order to obtain some simple characteriza tions of smoothness, we consider functions that are infinitely differentiabl e. This means that for each m E z+, the derivative j<m) exists and is a continuous function. Then we can

FOURIER SERIES ON THE CIRCLE

21

JT

integrate-by -parts for n "# 0 and write 2rrf(n) = (ljin)m e-inrj<m>(t) dt, to conclude that the Fourier coefficients satisfy a system of estimates of the form ~ Cm 1/(n)l < lnlm

(1.2.16)

0 "# n

E

Z,

m = 0, I, ....

In other wordsf(n) tends to zero faster than any negative power when lnl ---+ oo. Conversely, if the Fourier coefficients off E L 1 ('lf') satisfy ( 1.2.16), then we can repeatedly differentiate the absolutely convergent Fourier series to conclude that/ is a.e. equal to an infinitely differentiabl e function. This is summarized as follows.

L 1 ('ll') is a. e. equal to an infinitely differentiabl e function if and only if its Fourier coefficients are rapidly decreasing, according to ( 1. 2.16).

Proposition 1.2.16. f

1.2.3.1

E

Piecewise smooth functions

The corresponde nce between smoothness of a fixed degree and decay of the Fourier coefficients is not sharp in general. More precisely, the converse of Proposition 1.2.14 is false: there exists a nonabsolute ly continuous / E L 1 ('ll') for which nf(n) ---+ 0 when lnl ---+ oo. In order to obtain sharp results, we consider functions that are piecewise smooth, described as follows. If there exists a subdivision -rr < 80 < 8 1 < · · · < OK < rr so that f is absolutely continuous on each subinterval with a simple jump at the endpoint, denoted 8/(81 ) := /(81 + 0) - f(81 - 0), then we say that/ is piecewise smooth of degree 0. In general we say that f is piecewise smooth of degree k if there exists such a subdivision so that f is absolutely continuous on 1I', together with its first k - 1 derivatives and thatj(01) := j(o1 + 0)- J(Bj- 0).

Proposition 1.2. 17. Suppose that f is piecewise smooth of degree k. Then the Fourier coejJi(:ie~ts satisfy the identity (1.2.17)

(1)/+I(K ) L -;L 8f({)(8j)e-inBJ + 2rr m

f(n) = -

1

k

l=O

J=O

Furthermore the coefficient of 1I n 1+ 1 tends to zero are zero, i.e. jU> is a continuous function. Note that, in case Bo Proof. In case k

= =

-rr, we interpret 8f(80 )

. k+~).

j(k+l)(

(m)

if and only if all of the jumps

= f( -rr) -

f(rr).

0 we do an integration by parts on each interval of continuity:

When we sum the boundary terms and simplify, the sum is written in terms of the jumps and the Fourier coefficients off', which proves the result for k = 0. Iff has k - 1 absolutely continuous derivatives, we can· iterate this to obtain the k terms displayed, together with the Fourier coefficient of j. In order to prove the sharpness, we prove a separate lemma.

22

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Lemma 1.2.18. Suppose that -rr ::::: ()0 < 0 1 < · · · < BK < rr and that Cj are complex numbers so that

Then Cj = Ofor all j. Proof. We use the identity that for 'li'

Let

rn =

L:=O C;e'"

11

J'

so that

rn

e =I= 0

3

~ 0, by hypothesis. Then

lne-tnOm

=

Cm

+L

Clein(fi;-llml

p,Om

Taking N ~ CXl, both the left side and the sum on the right tend to zero, hence required proving the lemma and Proposition 1.2. 17.

Cm

= 0, as

•

Corollary 1.2.19. Let f be piecewise smooth of degree k and let 0 ::::: r ::::: k. Then we have the asymptotic estimatef(n ) = o(lnl-r-l), lnl---+ oo if and only if f E C' (1I'), i.e., f has r continuous derivatives. Proof. The Riemann-Leb esgue lemma implies that the last term in (1.2.17) is o(lnl-k- 1 ). Therefore the asymptotic behavior of] (n) is equivalent to that of the finite sum. Iff E C' ('JI'), then all ofthejump terms in (1.2.17) are zero for I~ r, in particular this sum= o(lnl-r-l). Conversely, iff(n) = o(lnl_'_ 1 ), then the same is true of the finite sum. Applying Lemma 1.2.18 repeatedly shows that 8JU>(()1 ) 0 for I~ r, which proves that/ E C'('JI'). •

=

This corollary takes a particularly simple form in case k = oo, i.e.,f is piecewise C 00 • Within this class of functions we can simply state that f E C' ('f) if and only if f(n) = o(lnl_'_ 1 ), lnl -+ oo. 1.2.3.2 Fourier characteriz ation of analytic functions The Fourier coefficients of an analytic function can be characterize d in terms of the exponential decay of the Fourier coefficients. Recall that a function is said to be analytic if it possesses a power series expansion about each point: fV) = L~o ak(t - t )k, 0 convergent in some interval It- to! < o. From this, it is immediately concluded thatf(t) is infinitely differentiabl e and that the successive derivatives are obtained as t<m> (to) = m!am. Since the series converges at t = t 0 , the terms of the series must tend to zero, and in particular are bounded, from which we conclude that

(1.2.18)

t =to, n = 0, 1, ... , R > 1/o.

F-OURIER SERIES ON THE CIRCLE

23

Equation (1.2.18) is also valid in an open interval it- tol < 813, by noting the followin g system of estimate s: ""-'

f<ml(t) =

L,)k + A=O

(k + m)

f(ml(t) ~ m! = ~

lf I ("m'l ~t)

I)···

m

(k + m)ak(t - to/

ak+mU - to)

k

k=O

:S C

L

rx-

2k+mRk +mit- toik

A=O

C2"'Rm 1 - 2Rit- tol

valid for it - to I < I 13R, which provides the desired uniform estimate . By covering 1r by a finite number of these intervals , we may assume that R, Mare independ ent of to. Convers ely, if an infinitely different iable function f(t), t E 1r satisfies (1.2.18), then the series 'L;:::=of(ml (t0 ) (t- to)"' I m! converge s in the interval It- to I < 1I Rand is therefore an analytic function . Summar izing, we see thatf is analytic if and only if( 1.2.18) l)olds. In order to characte rize analytici ty in terms of the Fourier coefficie nts, first suppose thatf is analytic on 'lr. In particula r f is infinitely different iable and we can integrate by-parts to write for n ¥= 0, 2n:f(n) = [ = ( A

e-mrf(t) dt

i~)

m

i

e-mtf(m) (t) dt

I

lf(n)l < lnlmMm! Rm Since this is valid for all m, we choose the optimal value m = [nl R] and apply Stirling' s formula to conclude that lf(n) I < Me-cini for any c < I I R. Convers ely, suppose that f is an infinitely different iable function on 1r whose Fourier coefficie nts satisfy a system of inequalit ies of the form If (n) I < Ae-cini for some positive constant s A, c. In particula r, by modifyin g f on a null set, we have the converg ent Fourier series f(t) = LnEZ: f(n)eint, which can be different iated m times to obtain f<ml (t) = Lna (in)"'f(n )emt. Applyin g the hypothes is, we have lf(ml(t)i
(1.2.19) for positive constant s c, A.

nEZ

24

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 1.2.21. Carry out the details of the replacement of the sum by the integral.

1.2.4

Sine Integral

In order to treat pointwise convergence of Fourier series and integrals in one dimension, a fundamental role is played by the function (1.2.20)

2

I

Si(x) =

Jr

[x

Jo

sin t -t- dt,

0 <X< CXJ.

Its basic properties are listed as follows: (i) Si(O) = 0, limx--.= Si(x) = 1. (ii) Si(x) S Si (n) = 1.18 ... for all x > 0. (iii) x ._ Si(x) has relative maxima at the points n, 3n, 5n, ... and relative minima at the points 2n, 4n, 6n, ... To prove these properties, we first note that Si ( (n + 1)n) - Si (nn) = 2/n fn<;;+l)rr (sin tjt) dt and that these numbers altemate in sign and decrease to zero in absolute value. Hence the improper integral defining limx Si (x) exists. To compute its value, we note that by the Riemann-Lebesg ue lemma, lim [ [ n

lu

1 2 ] sin(n sin(¢/2) ¢

+

.!..) ¢d¢ = 2

0.

But fu Dn() d¢ = 2n, which shows that

. 1rr

hm n

or equivalently Si(nn

+

~)

._

o

i)

sin(n + ¢ d¢ = n sin(¢/2)

1 when n ._ CXJ.

1.2.4.1 Other proofs that Si(CXJ) = 1 We offer two other proofs of this important improper integral. Proof using complex analysis. Consider the integral of the analytic function ell j z on (counterclockwise ) contour defined by the two semicircular arcs z = €e' 0 , 0 _::: (} _::: z = Re' 0 , 0 .::: (} _::: rr; and the two segments of the real axis defined by € _::: lxl _::: R. Cauchy's theorem the total line integral is zero. When R ~ oo, € ~ 0, the integral on large semicircle tends to zero, while the integral on the small semicircle is € ~

Hence we have lim

t:~..-o.R-oo

f

E
e" - dx X

=

irr,

0.

the rr; By the

25

FOURIER SERIES ON THE CIRCLE

from which we see that the imprope r integral J~ (sinx/x ) dx that Si(oo) = I.

Proof using real analysis . Define S(t)

=

=

1

= e-xt-sin t dt t

0

JT,

from which it follows •

J;(sins )jsds, assumed to converg e to a limit

S(oo). Forx > O,define G(x)

=

=

1""

e-"dS(t ),

X>

0.

0

On the one hand we may differen tiate under the integral for x > 0 to obtain G'(x) = -

"'1 o

e-xt sintdt = -

I I

,,

+ x~

X>

0.

Hence G(x) = C- arctanx for some constan t C. But the estimati on jG(x)l ::=: 1/x shows that 0 = G( oo) = C - rr /2, which shows that C = G(O) = rr /2. On the other hand we can transfor m the integral defining G(x) by a partial integrati on:

rr - arctanx

2

=

G(x)

= {oo xe-xtS(t )dt.

h

The latter integral tends toS(oo) whenx -

1.2.5

0. Hence Si(oo)

=

2/rr S(oo)

=

2/rr G(O)

=

I.

•

Pointw ise Conve rgence Criter ia

We are now in a positio n to prove some criteria for the converg ence of the partial sums of a Fourier series at a given point. All of the results describ ed below will be in the form of sufficie nt conditi ons. It is not possibl e to formula te any effectiv e necessa ry and sufficie nt conditi ons for the converg ence, as we will discuss below. The first step is to recall that the partial sum is express ed as a convolu tion with the Dirichl et kernel: (1.2.21 )

SN/(0) = (f

* DN )(0)

1 2rr

= -

1 1f

sin ( N + .!.2 ) ( ¢) . f(O - ¢) d¢. sm (¢/2)

This formula will be simplif ied in two ways. First, we will show that the factor sin (¢/2) in the denomi nator can be replace d by the simpler functio n ¢ ~ ¢/2. Second ly we will show that the integra l over the circle 1I' can be replace d by the integra l over a small interval about ¢ = 0. The details follow. The Dirichl et kernel is an even functio n and satisfie s the normal ization 1 - -

2rr

so that we can write (1.2.22 )

1

1rr

J1r{ DN(O)d O =

SNf(O) = -2 [/(0 rr o

+

¢)

1,

+ /(0-

¢)]

sin(N + 1 ) (¢) . 2 d¢, sm(¢/2 )

26

INTRODUCTION TO Hll'RIER ANALYSI'> >\ND W\VELiclS

and hence for any constant S (1.2.23)

s

SNf(())-

1 1~

=

2rr

[{(8

+ ¢)

+ j({)- ¢)- 25]

0

sin(N + ~) (¢) . (¢f2) d¢. sm

As a first reduction, we can replace the function l I sin(¢ /2) by the function 2/¢ with an error that tends to zero, uniformly in A. This comes from the fact that ¢~

sin(¢/2)

2 ¢

is bounded and continuous on the interval [ -rr. rr ]. The only possible difficulty is at ¢ = 0, where we can apply !'Hospital's rule to show that the difference tends to zero. Exercise 1.2.22. Show that JT

2 Hint: First show that

lx

~ sin xi S

s:xs:

JT

2

lxl' /6 and then simplify the fractions.

For each (), the function

~

¢

F&(¢) = [ .

l sm (¢/2)

-~]if(() +¢)+/(&-¢)) ¢

is an L 1 function, and the map () ~ F& is continuous from ( -rr, rr) to L 1 (1!'). Hence by the Riemann-Le besgue lemma the integral ofF&(¢) sin (N + 1/2)¢ tends to zero uniformly in() when n ~ oo. Thus we have reduced the problem of pointwise convergence to proving that (1.2.24)

lim N

j

rr

[f(B ()

+ ¢) + j(B- ¢ ) -

sin(N+l) ¢ 2 2S( d¢ = 0.
For the second simplificatio n, we consider the contribution to ( 1.2.24) from the interval 8 < ¢ =:: n. Noting that the function ¢ ~ [j(() + ¢) + j(() - ¢)]/¢ is integrable on the interval [8, n]. therefore by the Riemann-Le besgue lemma this contribution to (1.2.24) tends to zero when N ~ oo. We summarize this as the following necessary and sufficient condition for the convergence of a Fourier series. Theorem 1.2.23. Let f E £I (1!'). A necessary and sufficient condition that the partialsums SNf({j)conv ergetoa!imi tSwhenN ~ ooisthatfor some8 E (0, rr),

(1.2.25)

lim N

1 b

0

[j(()

+

¢) +/(8- ¢ ) - 2Sj

sin

(N + l) ¢ 2 l/J

d¢ = 0.

rOliRIER SERIES ON THE CIRCLE

oE o1 <

27

o

One may note that if ( 1.2.25) holds for some E (0, n ). then it holds for every (0, n ), since the difference of two such integrals is an integral over an interval ¢ ::: <5 2 , which tends to zero by the Riemann-Lebesgue lemma. We can now state and prove Dini's theorem: Theorem 1 .2.24. Suppose that f satisfies a Dini condition at(), meaning that for some 8 > 0 and some real numberS 8

1 ()

Proof. lemma.

Thi~

If(()+¢)+ f(()- ¢)- 2SI --=-------'----=---- ---- d ¢ <

¢

00.

is an immediate application of Theorem 1.2.23 and the Riemann-Lebesgue •

Specific conditions that imply the Dini condition can be obtained from a symmetric form of the Holder condition, as follows: Corollary 1.2.25. Suppose that f satisfies a symmetric HOlder condition at(), in the form

If<() + ¢) + f(() - ¢) - 2f(())l ::: Cl¢1<> for 0 < ¢ < 8, where 0 < a < I. Then limN SNf(O) = f(O). Iff satisfies a Dini condition at() with S =f(()), we say thatf is normalized. The following exercise applies to functions defined on the entire real line, not necessarily periodic. Exercise 1.2.26. Let f E L1~c (JR) satisfy a Dini condition at x where it is normalized. Then if M ¢. z+,

1 lim - N ~ --x, 27r

whereas forM

E

z+

JM DN(x)f(x) -M

dx =

""'

L.._.,

0, ±2n, ... ,

f(2nm)

mEZ.Imi::::OM

we must add tf(2n M)

+

if( -2:rr M) to the right side.

The next result is historically the first theorem on convergence of Fourier series, attributed to Dirichlet (I 829) for monotone functions and to Jordan ( 1881) for the general case. We will use the properties of Si (x). Theorem 1.2.27. Suppose that f is of bounded variation on [() some 8 > 0. Then limN SNf(()) = + 0) + f(()- 0)].

tu<e

o, () +

8] for

28

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Y<e + ¢) +

Proof. This is done by integration by parts: We write m = N + 1/2, F(¢) = ¢), S = F(O + 0); from (1.2.24) and the remarks that follow, we have

Y<e -

= ~ ("

SN/((}) _ S

JT

=

(1.2.26)

lo

1"

(f((}

+ ¢) + f(O- ¢)

_

2

(F(¢)- S) dSi (m¢)

+

s)

sinm¢ d¢

+ o(l)

¢

o(l)

()

1 8

= [F(8- 0)- S]Si (m8) -

Si (m¢) dF(¢)

+

o(l),

where the final integration is with respect to the finite measure defined by this function of bounded variation. Now we apply the dominated convergence theorem and the properties of Si (x) to conclude that

=

lim[SNJ(e) - S] N

[F(8 - 0) - S] - [F(8 - 0) - S] = 0,

•

which was to be proved.

Since every absolutely continuous function is of bounded variation, we obtain the following corollary. Corollary 1.2.28. Iff is absolutely continuous, then the Fourier series converges to f everywhere. It should be noted that the Dini condition and the condition of bounded variation are not comparable with one another. For example, the function¢ -+ 1/ log (1/¢) is of bounded variation but does not satisfy a Dini condition. On the other hand, the function ¢ -+ ¢ sin (1 I¢) satisfies a Dini condition but is not of bounded variation. Finally we remark on the difficulty of finding necessary conditions. The formula ( 1.2.22) shows that the Fourier partial sum at(} depends on the symmetrized function¢- + j((} + ¢) + j((} -¢).If this function is identically zero, then we have limN SNf((}) = 0 regardless of whatever other behavior is present. In particular any odd function (/( -x) = -f(x)) has a convergent Fourier series at(} = 0. The technique used in the proof of Theorem 1.2.27 can also be used to prove the uniform boundedness of the partial sums of the Fourier series of a function of bounded variation. To do this, write

2nSNf(6) =

1rr r +1

[sin

0

(~/ 2 )

-

sinm¢ ¢/2 [j((}

~ Jsin m¢ [j((} + ¢) + j((} -

¢)] d¢

+ ¢) + j((}- ¢)] d¢.

The first term is less than (n 3 /12) max lfl, while the second term can be integrated by parts in terms of the Sine Integral: 2 n

rrr

fo

sin m¢ ¢ f((}

± ¢) d¢

=f((}

± n)Si (mn)-

1rr

Si (m¢) dj((} ± ¢) d¢.

0

The first term is bounded by Si (n) max Si (n) Varf. We summarize as follows:

1/1 and the second term is bounded by

FOURIER SERIES ON THE CIRCLE

29

Proposition 1.2.29. Iff is of bounded variation on 'TI', then the partial sums of the Fourier series are uniformly bounded, in the form

where C 1, C 2 are universal constants and Var f denotes the total variation of the signed measure defined by f. Exercise 1.2.30. Find values for the constants C1. C2. Exercise 1.2.31. Suppose that f is of bounded variation and continuous in a neighborhood of(). Prove that the partial sums converge uniformly in a closed interval containing (). Hint: Without loss of generality assume f monotone increasing. Reconsider the representation ( 1.2.26) of the partial sums and break up the integral into J~ and J;..,. Apply the mean value theorem for integrals to each term and recall that Si (x) :S Si (rr) for all x.

1.2.6

*Integration of Fourier Series

In many applications of Fourier series, f represents the density of mass, charge, or probability. In such situations, the quantity of interest is the integral off over an interval. It is reassuring that, at this level, there are no obstructions to convergence.

Theorem 1.2.32. Suppose! E L 1 ('TI'). Thenforany interval (a, b) we have

(1.2.27)

li~

1b a

(Snf)(B) dB =

1b

j(B) dB.

a

Proof. The partial sum on the left is given by the convolution with the Dirichlet kernel, an even kernel that defines a self-adjoint operator. Thus

Since I
F(B) = /_: (j(¢) - }(0)) d¢. This functi?n is abso}utely cont~nuous with F(-rr) = 0 = F(rr), F'(()) =f(B)- }(0), so that inF(n) = f(n) - 8nof(O). Hence the Fourier coefficients of F are given

30

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

by F(n) = series

f(n)/in for n =j:. 0 and we have the everywhere convergent Fourier

(1.2.28) In particular, for any (a, b)

c [ - l f , n], we have

or equivalently

f bf(8) d(} -

(b- a)f(O) =lim

u

N

fh [SNf((}) -

}(0)] d(}

a

which reduces to ( 1.2.27). Another by-product of the integration identity ( 1.2.28) is the following necessary condition on the Fourier coefficients of an L 1 function. Corollary 1.2.33. For any f

E

L 1 ('II'), the series Lo,.enEZ }(n)jn converges.

This corollary can be used to manufacture trigonometric series that are not Fourier series. Exercise 1.2.34. Prove that function.

Ln::: 2 (sin n(} flog n) is not the Fourier series of an L 1

Hint: Identify f(n) as a suitable odd function.

Later we will prove that the series integrable function.

Ln> 2 (cos n8 flog n) is the Fourier series of an ~

Exercise 1.2.35. Suppose that g is a function of bounded variation on 1!'. Prove that for any f E L 1 (1!'),

lim { g(8)SNf(8) d(} = N~=}y

h{

g(8)f(8) d(}.

1.2.6.1 Convergence of Fourier series of measures The above ideas can also be used to give a quick treatment of the convergence of Fourier series of any finite signed measure on 1I'. The partial sum of its Fourier series is written n

S,J.-L(8) =

L ~n

[l(k)e'kH.

FOURIER SERIES ON THE CIRCLE

31

Proposition 1.2.36. Suppose that 1-L is a .finite signed measure on 'IT'. Then 1iffl

{b

la

SNt-t(B) d(}

=

1

t-L((a, b))+ 2~-t({a})

+

1 2~-t({b}).

Proof. The partial sum on the left is written in terms of the convolution with the Dirichlet kernel. Thus

Since 1 is of bounded variation, SN 1 converges boundedly to 1 (a.b> Applying the dominated convergence theorem completes the proof.

1.2.7

+

~ 1 fa)

+

~ 1 (h). •

Riemann Localization Principle

Fourier series in one dimension have the property that the limiting behavior of the partial sums at a point depends only on the values of the function in a neighborhood of the point, no matter how small. This is expressed as follows. L 1 ('If') is identically zero in an open interval (a, b). Then for any compact subinterval the Fourier partial sums tend uniformly to zero when n -+ oo.

Proposition 1.2.37. Suppose thatf

E

Proof. From formula ( 1.2.24) we have (1.2.29)

SN/(8) = o(l)

+ -I

rr

1" o

[{(8

+ ¢) + f(8

- ¢)]

sin

(N + 4>

!2 )

4>

d¢.

If [a 1, b.J is a subinterval of (a, b), let 28 = min(a 1 - a, b - b.). By hypothesis f(8 + ¢) + /(8 - ¢) = 0 if 8 E [a1, b.J, 4> < 8. Hence the integrand is identically zero when 8 E fa 1, b.], 4> s 8. On the other hand, the integral on [8, rr] tends to zero uniformly when e E [a I, bd, by the Riemann-Lebesgu e lemma. •

The Riemann localization principle allows us to infer that if two functions agree on an interval, then the Fourier series are equiconvergent meaning that limn (Snf1 -S11 f2) = 0 on that interval. This phenomenon is no longer present in higher dimensional Fourier analysis, as we shall see.

1.2.8

Gibbs-Wilbra ham Phenomenon

In the neighborhood of a discontinuity one cannot expect uniform convergence of the Fourier partial sums. The specific form of nonuniform convergence is best illustrated by the example (1.2.30)

f(x) = (rr - x)j2rr. f(x) = -(;r

+ x)j2JT,

0 <X< JT, -JT <X< 0.

32

INTRODUCTION TO FOURIER ANALYSIS AND WAVELI:TS

This is the simplest function of bounded variation that has a single jump of unit size. Now f is an odd function whose Fourier coefficients are

f

~ f(n) = -1

2n i

·~,

f(x)e~Jn'

1rr _n_ _x

sin nxdx

2n

Jr

dx

-i 2nn

so that the partial sum of the Fourier series is SNf(x)

~

=

[sinx

+ · · · + si~Nx].

This may be written as a definite integral by first computing the derivative I 1 (SNf) (x) = -(cosx

7r

+ · · · + cosNx]

1 = -(DN(X)- 1).

2n

Thus (1.2.31)

SNf(x) = -

Defining g(t) =

I

2n

1x

-x 1] dt = -

[DN(t) -

2n

0

1/ sin(t/2)- 2ft,

1-'

+ -1

2n

0

sin(N + 1/2)t . dt. sm(t/2)

we have by a single integration-b y-parts,

fox g(t) sin(N + !) tdt =

(~),

0

N

~

00

uniformly for 0 < x ::::; n, since g is a C 1 function on the interval [0, n]. Thus ( 1.2.32)

-x SNf(x) = -

2n

1x

sin(N + l2 )t dt no t

+ -1

If 0 < x < n, we see clearly that

-x 2n

limSNf(x) = N

To study the behavior when x sup [ SNf(x) O:ox:orr

+ ~ 2n

~

J : : ; _!_2

+-I n

1= - sin t t

0

+0 -x 2n

dt = -

( 1)

-

N

+-1 2

.

=f(x).

0, we note that on the one hand sup Si(x) o:ox<:x,

+0

(_!__) N

= _!_Si(n) 2

+ 0

(_!__) , N

which gives an upper bound to the fluctuations of the partial sums. On the other hand, if XN ~ 0 so that NxN ~ n, then l 1(N+~)x~ sin t

lim SNf(xN) =limN

N

7r

0

These computation s are summarized as follows.

- - dt t

=

1 -Si(n).

2

FOURIER SERIES ON THE CIRCLE

33

Proposition 1.2.38. The set of accumulatio n points of the partial sums SNf(xN) when N -+ OCJ, xN -+ 0 with 0 ::=: XN < n is described as follows: lim supSNf(xN) ::=: !Si (n) = 0.59 ....

(1.2.33)

N

If XN -+ 0 so that NxN -+ n, then (1.2.34)

lim SNf(xN) = lSi (n). N

In particular, any point in the interval [0, lSi(n)] is an accumulatio n point of partial sums SNf(xN) when N-+ OCJ, XN -+ 0. Proof. We have shown the first two statements above. For the final statement, apply the intermediate value theorem to the continuous function SNf· •

1

The number tsi (n) = .09 ... is called the overshoot of the partial sums in the right neighborhoo d of x = 0. We will show below that for any function of bounded variation withf(x + 0) > f(x- 0) we can discuss the overshoot in a right neighborhoo d of x, defined as lim supN~co.xN~x SNf(xN) - f(x + 0), which will be proportional to the jumpf(x + 0)- f(x- 0). The numerical value of Si (n) = 1.18 ... can be computed by expanding sin t/t in a Taylor series and integrating term-by-term . This is carried out in Section 2.3.3. Exercise 1.2.39. Compute limN SNf(kn/N

+

t)fork = 2, 3, ...

Strichartz (2000) discovered a correspondin g behavior with respect to the arc length of the curve {(x, SNf(x)), -n < x < n}. The proof anticipates the behavior of the Lebesgue constants, to be studied in Section 1.5.

Proposition 1.2.40. Letf be defined by ( 1.2.30). Then when N-+ /_: y'1

where

cl

+

[(SNf)'(x)]2 dx = C 1 logN

OCJ

we have

+ 0(1),

is a positive constant.

Proof. From the computations following (1.2.30) we have (SN/)'(x) so that

(1.2.35)

=

(lj2rr)(DN(x ) -1),

::0 I

so that the error term in (1.2.35) is 0(1), N -+ oo. But Proposition 1.5.1 also shows that cl logN + 0(1) for a positive constant cl. •

r:Jr I cosx + ... + cosNxl dx =

Exercise 1.2.41. Suppose thatf(x), -n :::X::: 7r is a piecewise C 1function and that f is continuous on 1T'. Prove that when N -+ OCJ we have

f:

y'1

+

[(SNf)'(x)]2 dx-+

f:

y'1

+

[f'(x)]2 dx.

34

INTRODUCTION TO FOURIER ANALYSIS AND WAVELFT<;

1.2.8. 1 The general case For a general f E L 1 (II'). the Gibbs-Wilbraham phenomenon at a point ¢ E 'II' describes the set of accumulation points of SNf(¢N) when N ~ oo and ¢N ~ ¢. In the above example it is seen that the set of accumulation points consists of the interval ( -0.59 ... , 0.59 ... ), whereas if¢ =1= 0, then SNJ ~ f uniformly in a neighborhood of ¢and the set of accumulation points consists of the single pointf(¢). To study the Gibbs-Wilbraham phenomenon more generally. let f E L 1 ('II') be a function of bounded variation with discontinuities labeled 8 1 , 82, .... Let the jumps be denoted 8f(8;) := f(8 1 + 0) - f(8 1 - 0). Let J(8) = (rr - 8)j2rr for 0 < 8 < rr and extended periodically to the entire line. Then J has a unit jump at 8 = 0. which will be used to study the general case. Form the function "'--'

~

(1.2.36)

J;ump(8) := L(8j)(B1 )Jo, (8) := L(8j)(81 )J(8- 8 1 ) . I=

I

I=

I

The series ( 1.2.36) is uniformly convergent to a function _of bounded variation whose discontinuitites are precisely the points {8 The function! := f - fiump is a continuous function of bounded variation, since all of its jumps are zero. From Exercise 1.2.31, the Fourier series of 1 is uniformly convergent. Therefore the analysis of the GibbsWilbraham phenomenon for f is reduced to that ofJ;ump· Given E > 0, let M be 'so large that L~M 1(8j)(8JIISNJn, II= < E, which is possible since the sum 1 18f(B;)I < oo and the partial sums SNJii, are uniformly bounded by a constant for all i, N. To study the Gibbs-Wilbraham phenomenon near 8 1 , let 8f(B 1 ) > ¢and write 1 }.

L:::

M

SNf(B):::: (8j)(BI)SNJA, (B)+ :L:<8f)(B;)SNJ&,(B) + sN1(B) +E. i=2

From Exercise 1.2.31, the finite sum is uniformly convergent in a closed interval about B,. The Fourier series of1 is uniformly convergent. Therefore when ¢N ~ 0 1 , we have

Letting ¢N

~

0 1 , we have

limsupSNf(¢N):::: ~(8j)(B 1 )Si (rr) + [J(B 1 + 0)- ~(8J)(BJ)] +

E,

N

But E was arbitrary. The same computation applies to any of the discontinuity points B;. We summarize as follows: Proposition 1.2 .42. Let f E L 1 ('II') be a function of bounded variation with a simple discontinuity at¢ E ( -rr, rr ), where (8f) (¢) > 0. The set of accumulation points of the partial sums SNf(xN) when N ~ oo and XN ~ ¢with¢ < XN < rr is described as follows:

( 1.2.37)

FOURIER SERIES ON THE CIRCLE

35

0.5

-6

2

2

6

4

FIGURE 1.2.3 The Gibbs-Wilbraham phenomenon for the functionj(/1) = sgn(&). FromM. Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications. Used by the permission of ThP M(Graw-Hill Companies.

IfxN -+ ¢so that N(xN- ¢)-+ n. then ( 1.2.38)

!imSNf<xN) =f<¢ N

+

0)

Si (n) -

+

1

2

(of)(¢).

In particular, any point in the interval [ ~ (/(¢ + 0) + f(¢ - O)),f(¢ + 0) + (Si (n)- 1)/2)(~{)(¢)] is an accumulation point of partial sums SNf(xN) for some sequence XN when N -+ CXl, XN -+ ¢ with XN > ¢.

1.3 1.3.1

FOURIER SERIES IN L 2 Mean Square Approximation-Pars eval's Theorem

Fourier series are well adapted to deal with the Euclidean geometry of the space L 2 (1I') where the inner product is defined by

<J, g) = _I { f(())g(()) d().

(1.3.1)

2n

Ir::

We can measure the degree of approximation by the mean square distance llf-gll~ = <J-g,f-g).

In particular, if g(()) = .L:~=-N bne 1118 is a trigonometric polynomial then

!If- gil~

=

II! II~ -
=

llfll~-

N

L

+

llgll~

-

(j
+ }(n)bn- lb,1 2 )

n=--N

N

=

L n=-N

Ibn- }(n)l 2

+ llfll~-

N

L n=-N

l}(n)l 2 .

36

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

This has the following consequences. Proposition 1.3.1. Suppose that f E L 2 ('JI'). Then the mznzmum mean square error is attained when bn is the Fourier coefficient bn = }(n). The mean square distance is given by N

(1.3.2)

L

II!- gil~= 11/11~-

l](n)l 2 -

n=-N

In particular we have for each N

E

z+

the inequality

N

L

l}(n)l 2

<

11/11~

n=-N

and in particular Bessel's inequality:

L

l}(n)l 2 < 11/11~-

nEZ

The space L 2 ('JI') is distinguish~d in the theory of Fourier series because of this characterization of the Fourier partial sums. If we consider optimal trigonometric approximation in the space C ('JI') of continuous functions or the space IF ('JI'), p =f:. 2, then the Fourier partial sum no longer provides the best approximation in norm. Fourier series in L 2 (1I') have the remarkable property that Bessel's inequality is in fact true with the sign of equality. This is Parseval's identity, stated as a theorem. Theorem 1.3.2. Parseval's theorem: For any f E L 2 (1I'), the Fourier series converges in L 2 (1I') and we have Parseval's identity (1.3.3) Proof. We first show that the partial sums converge in the norm of L 2 (1r). We have M,N-+ oo

by the Bessel inequality. But the space L 2 (1r) is complete, hence there exists F =limN SNf in the metric of L 2 ('lr). It remains to show that F =fa. e. To do this, we compute the Fourier coefficient by writing

If N > lnl, then the last integral= 2JT](n). Applying Cauchy-Schwarz, we have (N > n).

FOURIER SERIES ON THE CIRCLE

37

Letting N -+ oo shows that F(n) = f(n) for all n E Z. Hence F = f a.e. and we have proved that Snf -+ f in the norm of L 2 ('lf). In particular the L 2 norms converge and we have

IIIII~ = liffl

N

L

/f(n)/

2

,

n=-N

•

which is the statement of Parseval's equality.

Exercise 1.3.3. Suppose that f is absolutely continuou s with f' Parseval's identity to prove that II/'II~ = LnEZ n 2 1f(n)) 2 .

E L

2

(1r). Use

Exercise 1.3.4. Suppose that f is absolutely continuou s with f' E L 2 (1r). Use Parseval's identity to prove that the Fourier series converges absolutely : LnEZ )f(n)) < 00 · Hint: Apply the Cauchy-Sc hwarz inequality.

Theorem 1.3.2 shows that the mapping !--+}. from L 2 (1r) to L 2 (Z) preserves the respective L 2 norms. The next propositio n shows that the mapping! --+ j is onto the set of square-su mmable sequences . Put otherwise , we have an isomorphi sm between L 2 (1r) and L 2 (Z), the set of square-su mmable sequences . Theorem 1.3.5. Riesz-Fis cher theorem: Suppose that {en}. n E Z is a bilateral sequence of complex numbers with LnEZ lcni 2 < 00. Then there is a unique f E L 2 (1r) with](n) = Cn for all n E Z. Proof. Let Tn(B) =

L::=-n Cme'm 11 .

Then Tn is a Cauchy sequence in L 2 (1f) since for = LM
< N,

II TN

- T M Iii

2rrf(n) = /.J(B)e-'" 0 d(}

= L[f((})- TN(8)]e-'" 0 d8 +

.£

TN(8)e-'" 11 d(}.

The absolute value of the first term is less than or equal to 2rr II f - TN 11 • The second term 2 is 2rr x the Fourier coefficient of the trigonomet ric polynomial TN; if N > In 1this is equal to en. Thus we conclude for N >In/. Taking N -+ oo completes the proof of existence. The uniqueness off follows from the uniqueness of Fourier coefficients in the space L 1 (lr). •

Theorem 1.3.5 can be used to identify the smoothne ss of a function from the Fourier coefficien ts, as follows.

Propositi on 1.3.6. Suppose that f E L 2 (1r) has Fourier coefficient s that satisfy Lnezn 2 lf(n)l 2 < oo. Thenf is a.e. equal to an absolutely continuou s/unction, with/' E L 2 (1r) with Fourier seriesj'(() ) .-...- Lne.Z in](n)eino.

38

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. By Theorem 1.3.5, LnEZ inf(n)e'" 11 is the Fourier series of an L 2 function g. Furthermore Jil g((J)d(J = 0. Let F((J) = J~" g(¢)d¢. This is an absolutely continuous function on T with F' = g almost everywhere. Therefore the Fourier coefficients ofF are obtained from g(n) = inF(n). But g(n) = inf(n), from which we conclude that F(n) = f(n) for n i= 0. Hence F- f is a constant a.e. In particular f is a.e. equal to an absolutely continuous with/'= g E L 2 (1i). •

Exercise 1.3.7. Suppose that f E L 2 (1I') has Fourier coefficients that satisfy LnEzn 2 klf(n)l 2 < (X)jorsomeintegerk::: 1. Thenf,f', . .. ,fr~-llareabsolutely continuous with j< 0 E L 2 (1I').

1.3.2

*Application to the lsoperimetric Inequality

We now give an application of Parseval's theorem to geometry. Suppose that we have a closed curve in the xy plane that encloses an area A and has perimeter P. We will prove that P2

:::

4nA,

with equality if and only if the curve is a circle. To do this, suppose that the curve is described by parametric equations x = x(t), y = y(t) where -JT < t :::: JT. The functions x(t), y(t) are supposed to be absolutely continuous with derivatives in the space L 2 (1I') and to satisfy the normalization conditions x(-JT) = x(rr), y(-JT) = y(rr), since the curve is closed. From calculus, the perimeter and area are given by the formulas

P =

~~:

Jx'(t) 2

+ y'(t) 2 dt,

A=

i:

x(t)y'(t) dt,

where x' = dxjdt, y' = dyjdt. By reparametrizing the curve, we may suppose that x' (t) 2 + y' (1) 2 is constant; in fact it must be x' (t)2

+ y' (t)2

p2 = --2. 4JT

Since the functions are real-valued, we work with the original trigonometric form of Fourier series. Since the functions x(t), y(t) are supposed absolutely continuous, they are also of bounded variation, and we have the uniformly convergent Fourier series 00

(1.3.4)

+

L

(an cos nt

+ b" sin nt),

-JT ::::

y(t) =co+ L(cn cosnt

+ d" sinnt),

-JT

x(t) = ao

t ::::

JT.

n=l

ov

(1.3.5)

n=l

< t <

JT.

FOURIER SERIES ON THE CIRCLE

39

The Fourier series of the derivatives are not necessarily pointwise convergent, but they do converge in L 2 and we can apply Parseval's theorem. We have 00

x'(t) ,....._ Ln(-an sinnt

(1.3.6)

+ bn cosnt),

-rr < t < rr,

+ dn cos nt),

-rr < t < rr.

n=l 00

L

y' (t) ~

(1.3.7)

n( -Cn sin nt

n=l

Applying Parseval's theorem, we have

A =

r:

= -1

4

x(t)y' (t) dt

f:rr {[x(t) + y'(t)f- [x(t)- y'(t)] 2 } dt -:rr 00

=

7r

L

n(andn- bnCn)·

n=l

Performing the necessary algebraic steps, we have

p2 2

7r -

00

2A =

7r

L

[n(an- dn) 2

+ n(bn + Cn) 2 + n(n-

l)(a~

+ b~ + c~ + d,;)].

n=l

The right side is a sum of squares with nonnegative coefficients; thus P 2 j2rr - 2A > 0. If the sum is zero, then all of the terms are zero, in particular a~ + b~ + c~ + d,; = 0 for n > 1 and a 1 - d 1 = 0, b 1 + c 1 = 0. This means that (1.3.8)

x(t) = a 0

(1.3.9)

y(t) = c 0

+ a 1 cost- c 1 sin t, + c 1 cost+ a 1 sint,

which is the equation of a circle of radius complete.

-rr < t < rr, -rr

~

t < rr,

.jaf + cf with center at (ao, co). The proof is

Exercise 1.3.8. Show that if x = x(t), y = y(t) describes a plane curve offinite length, then we can re-parametrize with t = t(s) so that (dxjds) 2 + (dyjds) 2 is a constant.

1.3.3

*Rates of Convergence in L

2

Parseval' s theorem allows one to reduce the study of rates of convergence to the estimation of series. The nth Fourier coefficient of SNf - f is zero for In I < N, therefore from(l.3.3),

40

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

the mean square error is (1.3.10) lni>N

This can be used to estimate the mean square error in terms of the smoothness off. If, for example,f E Ci (II'), thenf(n) = O(ini-J) and

L

IISNf- fllz < C

k- 2J = O(N 1 -

21

N ~ oo

).

k>N

which gives an upper bound for the mean square error when N ~ oo. In order to obtain more precise estimates, we introduce the L 2 -translation error, defined by (1.3.11)

llfh- !II~=

2~ lif(x +h)- f(x)i

2

dx,

This can be expressed directly in terms of the Fourier coefficients by using Parseval's identity to write

L

llfh- fll~ =

(1.3.12)

leinh- 11 2 1f(n)l 2 .

nEZ

The next theorem describes equivalent norms to measure the smoothness off. Theorem 1.3.9. Suppose thatf E L 2 (1I'), 0 0. Proof. From (1.3.12) we have for any M

llfi.-

/II~= (

L + L)

lni~M

~

L

2

lemh- ll 1f(n)l 2

lni>M

2

2

n h 1f(n)l 2 +4

lni~M

lf(n)l 2 •

lni>M

If IISnf- /II ~ en-", then the second sum is O(Mby parts, writing En := Likl~n lf(k)l 2 :

L

L

20

).

To estimate the first sum, we sum

M

n2 l](n)l 2

=

M 2 (EM- E 00 ) +

lni:OM

L(2n- 1)(£

00 -

En-d·

n=l

By hypothesis Eoo -En = O(n- 2"

),

so that both of these terms are 0(M 2 -

11/h- /II~ ~ C [h 2

2

M 2 - 2"'

20 ),

+4M- 2"'].

Choosing M = 1/h completes the proof. Conversely, iff satisfies an L 2 HOlder condition, we can write

2 L[1 -cos (nh)] lf(n)l 2 = 11/h -/II~ ~ K 2 h2"'. nEZ

Integrating this inequality over the interval [0, k] and dividing by k, we have " [1L...., nEZ

sin (nk) nk

J l/(n)l ~

2

~

c ea.

k > 0.

therefore

FOURIER SERIES ON THE CIRCLE

41

The terms on the left side are nonnegative. Restricting the sum to indices n for which lnlk ~ 2, we have

c ea ~

L: nEZ.Inlk~2

[t - sinn~k) Jll
L:

ll
neZ:Inlk~2

The proof is complete.

•

We can paraphrase Theorem 1.3.9 in terms of equivalent norms. On the one hand we have the L 2 Holder norm Na(J) :=sup llfh- fllz/lhl"'. h#O

On the other hand we have the normalized mean square error, defined by Ra(J) := supncxiiSJ- fllz. n~O

Theorem 1.3.9 asserts that for 0 < ex < 1, there exists a constant C = Ca so that C- 1Na(f) < Ra(f) < CNa(f).

In case ex next example.

1 one cannot expect the above equivalence to hold, as shown in the

Example 1.3.1 0. Let f be defined by the absolutely convergent trigonometric series eikx

oo

L

f(x) =

k3/2.

k=l

Then llf -

1

oo

L

SJII~ =

I

k3 < 2n2.

k=n+l

On the other hand, from (1.3.12) lifh- fll~ = 4

L

00

k=l

> 4

L

1

3 sin 2 (kh/2) k 1 k 3 sin 2 (kh/2)

k<7r /h

> 4

I:

k13 (kh/7r)2

k<7r/h

4h 2

> -

Jl" 2

log (Jr jh),

which shows that f cannot satisfy an L 2 HOlder condition with ex = 1. Exercise 1.3.11. Suppose that f satsi.fies an L 2 Holder condition with ex _ Prove that /If- SJIIz = O(ljn).

1.

42

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Hint: Go back through the steps of the second part of the proof of Theorem ( 1.3.9) with a = 1.

Exercise 1.3.12. Suppose that f satisfies an L 2 Holder condition with a = 1. Prove that LnEZ lnl 2 1f(n)l 2 < oo. Hint: Apply Fatou's lemma to formula ( 1.3.12).

The relation between the L 2 Holder condition and the pointwise Holder condition is not symmetrical. Indeed, if lf(x +h) - f(x)l S Cha for some 0
-;r

< x <

;r,

where 0
We will show that the L 2 Holder condition is a strict improvement of the pointwise Holder condition. To see this, we first compute the Fourier coefficients: 2rrf(n) = /_: lxla cos (nx) dx =

21Jr xa cos (lnlx) dx 2

= - a {Jr lnl Jo =-

2a I lnl +a

sin (lnlx) dx

xa-I

11n1Jr ya-I sinydy. o

But the final integral converges to a constant when n -- oo as shown by another partial integration. Therefore we have the asymptotic formulaf(n) = C/lnl 1 +a(l + o(l)), lnl -- oo. Applying the formula (1.3.12), we have 11/h -/11~ =

L leinh -11 1f(n)l L (nh) 1f(n)l + 4 L 2

2

nEZ

(1.3.13)

S

2

2

lnl::oM

lf(n)l 2 •

lni>M

We consider three cases. Case 1: a > ~- The first sum in (1.3.13) is convergent when M-- oo, while the second sum is O(M- 1 - 201 ). Taking M = 1/h, we see that llfh- /11 2 < Ch 2 , hence the L 2 modulus satisfies II fh - f ll2 < Ch, irrespective of a. Hence we have an improvement of the Holder exponent in the amount 1 - a. Case II: a = ~. In this case the first sum in ( 1.3 .13) .-...- h 2 log M, whereas the second sum= 0(1/M 2 ). Again taking M = 1/h, we have 11/h- /112 < Ch.jlog (ljh), thus an improvement of the Holder exponent by nearly

!.

Case Ill: 0 < a < ~. In this case the first sum diverges, asymptotic to a multiple of M Again taking M = ljh, the two terms are now balanced and both are asymptotic 1 2 - a.

FOURIER SERIES ON THE CIRCLE

43

to a constant multiple of h 2 cx+l. Thus in this case we have llfh -!liz ::;: Chcx+4, an improvement of the Holder exponent by ~. These examples give a concrete indication of the disparity between the L 2 Holder classes and the pointwise Holder classes. A more systematic approach is contained in the proof of Corollary 1.3.19 below. The next two exercises give a relation between the mean square error and the fractional Sobolev classes on the circle. Exercise 1 .3.14. Suppose that f E L 2 (If) and that for .some LnEZ lnl 2 cxl](n)l 2 < oo. Prove that IISNJ- fllz = o(N~cx), N ~ oo.

a > 0,

Hint: Begin with ( 1.3.10).

Exercise 1.3.15. Suppose thatf E L 2 (1I') and that for some a > 0, IISNJ-!liz = O(N~a), N ~ oo. Prove that LnEZ lni 2 .Bi](n)l 2 < oojor any f3
Llnl:oM

lnl 2 fl l](n) 12 .

At the beginning of this section it was noted that if a function has additional smoothness, then we may expect that the mean square error decays more rapidly when N ~ oo. The following exercise gives an extension of Theorem 1.3.9 to higher derivatives. Exercise 1.3.16. Let k E z+ and 0 < a < 1. In order that the mean square error satisfy the estimate IISNJ - fllz = O(N~(k+cx>), N ~ oo, it is necessary and sufficient that f, f', ... , J be absolutely continuous and that J liz < Kh 01 •

1.3.3. 1

Application to absolutely convergent Fourier series

We can also use the L 2 Holder condition to give a sufficient condition for f to be represented as an absolutely convergent Fourier series. Theorem 1.3.17. Bernstein: Suppose that f satisfies an L 2 Holder condition with exponent a > Then the Fourier series is absolutely convergent:

i·

LnEZ

if(n)i <

00.

Proof. We estimate dyadic blocks by the Cauchy-Schwarz inequality. Thus

By Parseval's identity, we have for any m

nEZ

44

INTROD UCTION TO FOURIER ANALYS IS AND WAVELE TS

Writin g lernh- 11 2 = 4sin 2 (nh/2), we see that if h 2m :::, lnl < 2m+ I, so that

L

11/h- Ill~~

l](n)l

~rr2-m, then lemh- 11 ~ 1 for

2 •

2m~lnl<2m+l

Now we apply Cauchy -Schwa rz to each dyadic block and sum to obtain

1 3

00

< "'""'2
-

2

m=O

'

00

= c <~r L:2~(1-2a)

< oo.

m=O

•

since a> ~-

Exerci se 1.3.18 . Suppo se that f satisfie s an L Holde r condit ion with expon ent I that LneZ lnl fJ 1/(n)l < oc;for any f3 < ot- z· 2

ot > ~-Prove

A

Iff is of bound ed variati on, then the absolu te conver gence of the Fourie r series holds under any pointw ise HOlde r condit ion, accord ing to the next corolla ry.

Corol lary 1.3.19 .

Zygm und: Suppo se that f

BV(1I') and that f satisfie s a < Clyla for 0 < ot < 1. Then

E

lf(x + y) - f(x)l

pointw ise Holde r condit ion: LneZ lf(n) I < OCJ.

Proof. Letting V denote the total variatio n of f, we can estimat 1 e the L 2 modulu s of continu ity by writing

11/,..;3N -fib2 =

1 2Nrr

< V1 c

b jTr If ( X+ N

krr)

3N

2

-

f

(

X+

(k- 1)rr) 1

3N

d.x

(..!!_)" 3N

-

N

=

Nl+ar·

c

Theref ore/ satisfie s the L 2 Holder conditi on with expone nt (l + cx)/2 > at least along the values h = rr j3N. But this is sufficie nt to apply the proof of Bernst ein's theorem and thus conclu de that Lnez l](n)l < oo. •

!.

Iff satisfie s only a Holde r condit ion with ot < 1 /2, the Fourie r series is not absolutely conver gent in genera l. There exist many examp les in the literatu re. An alterna tive treatm ent is to look at random Fourie r series and to prove that almos t every realiza tion

FOURIER SERIES ON THE CIRCLE

45

is not absolutely convergent, but satisfies a Holder condition with a < 1/2 (see Kahane (1968)).

1.4

NORM CONVERGENCE AND SUMMABILITY

The tools introduced thus far do not permit us to deal with the norm convergence within a Banach space of functions. Indeed, the oscillatory properties of the Dirichlet kernel will allow us to show that there exists a continuous function whose Fourier series diverges at a point. Hence one cannot prove uniform convergence, for example, within the class of continuous functions. Furthermore, we would like to deal with convergence in the norm of If, where p > 1. This turns out to be impossible in the space L 1 , but can be dealt with nicely if p > 1. In order to launch a systematic theory, we consider the Cesaro averages of the Fourier partial sums. These are called the Fejer means and defined as the arithmetic means 1 (1.4.1) aNf(8) = (Sof(8) + · · · + SN/(8)), N = 0, 1, 2, ... N+ 1 We also consider the Abel means 00

(1.4.2)

Arf(8) = (1 - r) L

rnSJ(8),

O
n=O

The Abel means can be written as a Fourier series by proving the identity 00

(1.4.3)

(1 -

r)

L

rnSJ(8) = Lf(n)rlnlein8'

n=O

O
neZ

Exercise 1.4.1. Prove (1.4.3 ). Exercise 1.4.2. Prove that ifthe Fourier partial sums converge pointtvise, then the

Fejer means converge pointwise. Prove that if the Fourier partial sums converge uniformly, then the Fejer means converge uniformly. Prove that if the Fourier partial sums converge in LP, p > 1, then the Fejer means converge in If. Hint: Use the triangle inequality and the fact that if a numerical sequences, satisfies lim, s, = s, then lim,(s0 + · · · + s,)jn + I = s.

Exercise 1.4.3. Prove that if the Fourier partial sums converge pointwise, then

the Abel means converge pointwise. Prove that if the Fourier partial sums converge uniformly, then the Abel means converge uniformly. Prove that ifthe Fourier partial sums converge in LP, p > 1, then the Abel means converge in LP. Hint:

First prove that if r) L:,o r" s, = s.

lim,_~ 1 (I -

1 .4.1

a

numerical

sequence

s,

satisfies

lim, s, = s,

then

Approximate Identities

In order to deal systematically with these procedures, we define the general notion of

approximate identity.

46

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Definition 1.4.4. An approximate identity on the circle defined for(}

E 1I'

1I' is a function k(r, B)

and r in some directed index set/, so that

1 lim - - { k(r, (}) d(} = 1 r 2rr Jv:

(1.4.4) (1.4.5)

llk(r, 0)1 d(} ::S C,

lim {

(1.4.6)

r

Yr

lk(r, (}) d(} = 0,

E

I

\18 > 0.

}1&1>8

Here C is a constant independent of r. In case k(r, (}) > 0, then ( 1.4.5) is superfluous and we can take C = 2rr.

By definition, a directed set is a set I together with a collection of subsets {A;} with the property that for each (i,j) there exists k with Ak C A; nAJ. In case I= [0, 1), the subsets can be taken in the form Ak = (1 1). A complex-valued function! on a directed set has a limit L, by definition, if for each E > 0, there exists a subset A so 1 that lf(x) - Ll < E for all x E A;. With this definition it is immediate that limits obey the usual laws for sums, products, and composition of functions.

t,

Remark. We choose the formulation with a general directed set in order to have maximum flexibility in the applications. For example, for the Fejer means we have the index set {1, 2, ... } with n --+ oo, whereas for the Poisson kernel associated with the Abel means we have the index set [0, 1) with r--+ 1. In the first case we may take Ak = (k, k + 1, ... ) whereas in the second case we take Ak as the open interval ( 1 1).

f,

Example 1.4.5. The Poisson kernel Pr((}) is an approximate identity.

JT

Indeed, we showed in (1.1.39) that Pr((}) dB = 2rr. Since Pr((}) > 0, the second property is automatically satisfied. To prove the third property, note that for IB I > 8 the denominator 1 + r 2 - 2r cos(} = (1- r) 2 + 2r(l- cos(}) > 2r(1- cos 8). Therefore in this interval we have Pr((}) < (1 - r 2 )/2r(l -cos 8), which tends to zero when r--+ 1. The fundamental use of approximate identities is described as follows:

Proposition 1.4.6. Suppose that k(r, (}) is an approximate identity. • If¢> E L 00 (1I') with limo--+O ¢> ( (}) = L, then

(1.4.7)

1 lim - - { k(r, B)¢>(B) dB = L. r 2rr }T

• If, in addition, for each 8 > 0 sup 101 ::: 8 1k(r, B)l--+ 0, then (1.4.7) holdsforall ¢> E L 1 (1I') with limo--+o ¢>((}) = L. Proof. For any 8 > 0, we have

-:}-- f 7r

J.r

1 k(r, 0)<1>(0) de-L= ( 2 7r l101>o

f

+ f

Jllils.o

)

k(r, 0)(<1>(0)- L) dO+ o(l).

FOURIER SERIES ON THE CIRC~.E

47

The first integral tends to zero, for any o > 0. Given E > 0, the second integral can be made less than E by taking o sufficiently small, which proves the first statement. To prove the second statement, note that the first integral is bounded by sup 1111 >J lk(r, 8)1 x (L + 11<1>11 d. which tends to zero by hypothesis. The second integral is bounded byE x j, lk(r, 0)1 de, which completes the proof. •

Exercise 1.4.7. Suppose that the approximate identity k(r, 8) has the additional property that k is even: k(r, 8) = k(r, -8)forall8 E 'Jr. Suppose that E L 00 (1r) with lim8 ..... 0 [<1>(8) + <1>(-8)] = 2L, for some complex number L. Prove that formula (1.4.7) holds. Exercise 1.4.8. Suppose that the approximate identity k(r, 8) is even and has the property that for each 8 > 0, sup 181 :o:.li lk(r, 8)1 ~ 0. Suppose that E L 1 (1r) with lim8 ..... o[(8) + ( -8)] = 2L, for some complex number L. Prove that formula (1.4. 7) holds.

In order to apply approximate identites to norm convergence, we recall the notation !¢>for the translate off E L 1 (11), defined by /¢>(8) =f(8 -¢>).The following definition is essential.

Definition 1.4.9. A subspace B c L 1 ('lr) with norm 1!·11 8 is called a homogeneous Banach subspace ifwe have II/III < 11/lla. the mapf ~ fo is B-normpreservi ng. and the map 8 ~ fo is continuous in the B norm. In detail we require that 11!8 1! 8 = llfllaforallf

E

Band al/8

E 'lr,

and that limo ..... o

lifo- fila~ Oforallf

E

B.

Example 1.4.1 0. The space C('lr) with the supremum norm is a homogeneous Banach subspace. The space LP (11) for 1 < p < (X) is also a homogeneous Banach subspace. Exercise 1.4.11. Prove these properties. Then prove that L 00 ('lr) with the supremum norm is not a homogeneous Banach space.

This notion is very effective for dealing with norm convergence, when we represent the convolution of two functions as a vector-valued integral. If K E L 1 (11'), we can write (K *f)(8)

= 2~

L

K(,P)f(8- ¢>) drp

= 2~

L

K(,P)f<J>(()) d,P.

The final integral is a vector-valued integral, defined as a limit in norm of Riemann sums. In particular, iff E B, then K *f is an element of Band we can estimate the B-norm of the vector-valued integral by the inequality

which follows from the triangle inequality for finite sums. Similarly

48

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

which can be analyzed by the more elementary techniques of Proposition 1.4.6. We formalize this as follows.

Theorem 1.4.12. If B is a homogeneou s Banach subspace and k(r, fJ) is an approximate identity, then

li~ II 2 ~

i

k(r,

~)fq, d~- f t

= 0.

Proof. The required norm is less than or equal to

2~ i

ik(r, cf>)lll/ot>- fiiBdcf>,

•

which tends to zero by Proposition 1.4.6.

The first application of approximate identities is to the sequence of Abel means of a Fourier series. To make the connection between Abel means and the Poisson kernel, we recall the basic identity of Fourier reciprocity, Proposition 1.2.3, which states in this case that for any f E L 1 ('lr) _1

2rr

rf(f}- ~)P,(~) d~ = L

J1r

Theorem 1.4.13. Iff

rlntj(n)einB.

nEZ

C('lr), then the Abel means converge uniformly to f. If f E U'('lr), 1 < p < oo then the Abel means converge to fin the norm of U'. If f E L 1 ('][') has right and left limits at f} E 'lr, then the Abel means converge to Hf
Proof. We have shown that the Abel means are defined by the Poisson kernel, which satisfies the conditions of an approximate identity. Hence the first two statements follow immediately from Proposition 1.4.12. The third statement is a direct application of Proposition 1.4.6.

•

We will prove in the next section that the Fejer means are also represented by an approximate identity. This will allow us to prove the norm convergence properties for Fejer means also. Theorem 1.4.12 admits a sort of converse, expressed as follows:

Proposition 1.4.14. Suppose that B is a homogeneou s Banach subspace ofL 1 ('][') and k(r, f}) is an approximate identity with the property that for some f E L 1('lr), k * f E B for all r E I and k * f converges in the B norm. Then f E B. Proof. Letting g = lim, k * f (in the B norm). we must also have g = lim, k * f in the L 1 norm. But from Theorem 1.4.12,/ =lim, k *fin the L 1 norm. Hence/= g a.e. But the space B is a closed subspace of L 1 ('Ir), therefore/ E Bas required. •

Applying this to the Poisson kernel, we have the following useful converse statements:

Corollary 1.4.15.

Suppose that f E L 1('lr) is such that P, f converges in the norm of LP ('][')for some 1 < p < oo. Then f E U' ('lr). (i)

FOURIER SERIES ON THE CIRCLE

49

(ii) Suppose that/ E L 1 ('JI') is such that Prf converges uniformly. Thenf E

C(1I'). Proof. It suffices to remark that for any f particular a member of the space LP (1!').

E

L 1 (11'), Prf is a continuous function, in •

We will see later that condition (i) can be weakened to boundedness in the LP norm. However condition (ii) cannot be weakened to uniform boundedness; consider the example of the Poisson integral of a bounded but discontinuous function. 1.4. 1.1 Almost everywhere convergence of the Abel means We have shown that the Abel means of an L 1 function converge at every point where the right and left limits exist. This condition can be weakened to the existence of the limits of the averages:

Le = lim - 1 10+€ j(y) dy.

(1.4.8)


2€

0-€

From the fundamental theorem of calculus (Lebesgue's differentiation theorem), it is known that these limits exists with L 0 = j(()) except for a B-set of measure zero. The almost everywhere convergence of the Abel means will be deduced as a corollary of the following general theorem on a class of approximate identities on [0, 1r]. The definition of the latter simply amounts to replacing 1I' by [0, n] in the original definition of approximate identity.

Theorem 1.4.16. Suppose that k(r, ()) is an approximate identity on [0, n], with the property that k(r, ()) ::: 0 and () ~ k(r, ()) is absolutely continuous with k'(r, ()) := (dkjd()) < Ofor 0 < () :::; 1r and all r E /.Suppose that] E L 1[0, n] satisfies limo--o o-I I:](¢) d¢ = L. Then limr(l/JT) J; k(r, ())j(()) d() = L. Proof. We will show that 0 -+ -Ok'(r, 0) is an approximate identity on [0, rr]. To see this we first note that, since k' ~ 0, we have for 0 < 8 1 < 82 ~ 1r,

which shows that lim, k(r, 0) I+--1 7r

1" 0

=

0 for 0 ~ 0 ~ rr. From the normalization of k, we have

k(r,O)dO =k(r,rr)-1 7r

1"

Ok'(r,O)dO

0

which shows that lim, f 0" -Ok'(r, 0) dO= 1. Furthermore, for any 8 [" !Ok'l dO=-[" Ok'(r, 0) dO= 8k(r, 8)- rrk(r, rr)

which shows that()

-+

-(}k'(r, ())is an approximate identity.

+

1"

E

(0, rr),

k(r, 0) dO-+ 0

50

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Now define F(O) = fc~[j(cp) - L] dcp, a bounded function on [0, rr] with F(0)/0 = 0. We integrate by parts, replacing k by c,k with lim, c, = 1. thus

lim 11 ~ 0

1"

k(r, 0)[](0)- L] dO= k(r, rr)F(rr)-

=

o(l)

1"

F(O)k'(r, 0) dO

+ {" F(O)[-Ok'(r,O)]dO.

Jo

0

•

The final integral tends to zero by Proposition 1 .4.6.

Corollary 1.4.17. Fatou: TheAbelmeansP,f(O)convergetof(O)whenr- + 1 whenever the limit ( 1.4.8) exists. Proof. satisfy 0 E 1r with L

On the interval 0 ::: 0 _::: rr, the functions 2P,((}) form an approximate identity and the conditions of Theorem 1.4.16. Now define](¢)= klf(O + cp) + j(O- ¢)].If satisfies (1.4.8), then lim~oF(cp)jcp = 0, so that we may apply Theorem 1.4.16 =f(O) to conclude that lim, PJ(O) =f(O), as required. •

Alternative (explicit) proof. One can avoid Theorem 1.4.16 and work directly as follows. Define F(cp) = J~ (j(u + 0) - j(O)) du. Then the Poisson integral off can be integrated by parts as follows:

I

{"

p ,f ( 0 ) - f ( 0 ) = 2rr ]

0

I -- 2rr

1" 0

1 - r2 -I-+----,r2,__-2_r_c-os_cp_ (j ( 0 + cp) + f ( 0 - cp) - 2! (cp) dcp I - r

I+ r 2

-

2

2rcoscp

I I - r 1 = - + F(rr) + 2 JT 1 2 JT r

1" o

dF(cp) F(cp)

K,(cp)~dcp SID 'I'

where

• is an approximate identity on [0, rr], since K,(cp) l -

JT

1rr o

2

r 2 ) sin 0 dO- 1 2 (l + r - 2rcos0) 2 ' (l -

~

0 and we have from (1.1.38)

. 1xm sup

r~ I IJ';:J (1

(1 - r 2 ) sin 2 (}

+ r2

-

2r COS ())2

= 0.

Exercise 1.4.18. Show that Theorem 1.4.16 can be generalized as follows. Instead of assuming absolute continuity, simply assume that() -+ k(r, ()) is monotone, decreasing for each r. By suitably applying integration-by-parts and suitably modifying the definition ofapproximate identity to include a sequence ofmeasures, show that the conclusion holds exactly as stated.

FOURIER SERIES ON THE CIRCLE

1 .4.2

51

Summability Matrices

Closely related to the notion of approximate identity on 1f' is the notion of a summability matrix, which is the discrete analogue for sequences of real or complex numbers. In this section we give the basic notions of summability, which includes the Abel and Fejer means of the Fourier series as special instances of this general notion. Definition 1.4.19. A summability matrix is a doubly infinite array of real numbers amn definedfor m, n:::: 0 with the following properties:

(i) limm_,.ooamn = Oforeach n = 0, 1, 2, .. . (ii) L~oamn = 1foreach m = 0, 1, 2, .. . (iii) L~o iamn I < C for some constant C and all m = 0, 1, 2, ....

A summability matrix defines a linear transformation on the space of bounded sequences CXl

s---+ A(s),

Am(s) =

L

amnSn.

m=O

The basic property of consistency is expressed as follows: Proposition 1.4.20. .lflimn Sn = s, then limm Am(s) = s. Proof. We use (ii) to write 00

Am(s)- s =

L

amn(Sn- s).

n=O

Given E > 0, choose N = n(E) so that lsn - sl < E for n > n(E). Then /Am(s)- s/ ::::;

~ lamnllsn- s/ ::::; (~ +

The last sum is less than C E. Now we can let m ---+

<Xl

f;)

lamnllsn- s/.

and use (i) to conclude that

lim sup lAm (s) - sl ::::; CE. m

But

E

•

was arbitrary, which completes the proof.

Exercise 1.4.21. Prove that the conclusion of Proposition 1.4. 20 still holds condition (ii) is weakened to the relation that limm->oo L~o amn = 1.

if

Basic examples of summability matrices are provided by the Cesaro means and Abel means:

m = 0, 1, ... CXl

A,(s) = (1 - r)

L

n=O

rnsn,

O:S:r
52

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

In the first case we have amn = 1/ (m + 1) for n < m and zero otherwis e. In the second case we pick a sequenc e of r m ---+ 1 , for example r m = 1 - 1/ m thus defining amn = r::,(l - rm). In order to work with Abel means, it is useful to note the followin g transform ation formula, proved by summati on-by-pa rts for 0 < r < 1: 00

(1.4.9)

(1 -

r)

L

00

rnsn =

n=O

L

rnan

n=O

where sn = a 0 + · · ·+an for n = 0, 1, .... This identity allows one to go back and forth between a sequenc e an and its partial sums. The sequenc e {sn} is Abel-su mmable to L if and only if we have limr---+ 1 L::::o anrn = L.

Exercise 1.4.22. Prove (1.4.9). The followin g concrete example s are useful for referenc e.

Exampl e 1.4.23. The negative binomia l series is 1/(1 + r)k = L~o (-;,k)rn for any 0 < r < 1 and k is any real number. Hence the numeric al series L~o (:k) is

Abel-sum mable to the value 2-k for any real number k. For example , with k = 0, 1, 2, we have

-1- = 1

+

r

1 (1

+ r) 2 1

(1

+ r)3

1- r + r

2

3 n - r + · · · :=::::::} Abel ~ L,_, ( -1) n=O

= -1 2

~ + n = 1 - 2r + 3r 2 - 4r3 +···:=:::: ::}Abel L,_,(n 1)(-1) =

41

n=O = 1- 3r

+ 6r 2

~ (n+ l)(n+2 )

-lOr 3 +···:=:::: ::}AbelL ,_,

n=O

2

n 1 (-1) = -.

8

On the other hand, the series L:~o (-;,k) is Cesaro-s ummabl e if k < 1 but not fork > 1. To see that k > 2 does not yield a Cesaro-s ummabl e sequenc e, consider the followin g exercise .

Exercise 1.4.24. Suppose that the sequenc e sn is Cesaro- summab le to s. Prove that Sn = O(n), n ---+ 00 Hint: Letting Un = Cn(s), check that Un --+ s implies that Sn = (n is violated for the example in case k > 2.

+

l)un -nun- I = O(n). This

It is natural to compare different methods of summab ility. We say that method A is stronger than method C if the matrix A can be factored in the form A = BC where B is another summab ility matrix. This ensures that any sequenc e that is C -summa ble is also A-summ able.

FOURIER SERIES ON THE CIRCLE

53

The previous examples suggest that Abel-summabili ty is stronger than Cesarosummability. Even more is true: Proposition 1.4.25. For any k there exists a summability matrix Bk so that the Abel matrix can be factored as A = BkCk. Otherwise put, Abel is stronger than any power of Cesaro. Proof.. We factor the Abel matrix as follows: 00

(1- r) Lr"sn = ( I - r) [ao

+ r(2al- ao) + · · · +

r"((n

+

Oan- nan-J)

+ · · ·]

n=O

= ( l - r)

[0- r)ao + · · · +

(n

+

l)(r"- r"+ 1 )an

+ · · ·]

00

= (1- r) 2 L ( n

+

l)anr",

n=O

so that the matrix B 1 is defined by the coefficients (1 - r) 2 (n + l)r", which satisfy the conditions of a summability matrix. Continuing inductively, we write the second Cesaro means a,: 2 > = (a0 + · · · + an)/(n + 1) and its "inverse" an= (n + I)a~ 2 >- na~~~ to obtain 00

( I - r)

L

00

r"sn

= (1- r) 3 L

n=O

ap>(n

+

I)(n

+ 2)r"j2.

n=O

In general we show by induction that 00 00 (1-r)~r"sn =O-r)k~

(-k) n

(-l)"a,:k-l>r",

•

which exhibits the matrix Bk explicitly.

Remark. The matrix Ck defines the kth order Holder means. This summabillity method is distinct from the kth order Cesaro means, usually denoted (C, k). For details see Hardy ( 1949), p. 94 ff. Figure 1.4.1 shows the relation between Abel and Cesaro summability.

Cm(s) = s 0

+ ... + sm m + 1

c~;(s) =

C 0 (s)

+ ... +

m + 1

C"'(s)

A,(S) =(I - r)L,r"sn 0

FIGURE 1.4.1 Relations between Abel- and Cesaro-summability . Stronger methods are to the right of weaker methods.

54

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Any theorem that affirms a method of summability is called an Abelian theorem. Theorems in the converse direction are called Tauberian theorems. The following is a simple Tauberian theorem. Proposition 1.4.26. Suppose that L~o an is Cesaro-summable to s and that the terms satisfy limn nan = 0. Then the series converges to the same sum s. Proof. In terms of the original sequence we can write O'n

=

So

+ · · · + Sn = n+ 1

~

L._, k=O

(

k 1 - - - ) ak. n+ 1

Hence we have 1

Lkak. n+ 1 n

Sn- a n = - -

k=O

But this is the average of a sequence which tends to zero, hence also tends to zero, which completes the proof. •

Exercise 1.4.27. Suppose that the terms satisfy klakl :;: M for some constant M and k = 0, 1, 2, .... Show that ian- snl < M for all n = 1, 2, .... Exercise 1.4.28. The Riemann means of a series L~o an are defined by Rh(a) =

z~~

(

si:;h) 2 am,

where 0 < h --* 0. Show that this is defined by a summability matrix that is not positive. Hint: Write Rh(a) = L::;:'= 0 [K(mh) - K((m + 1)h)]sm where K(x) K(O) = 1, K(oo) = 0 and L:,o IK(nh)- K((n + l)h)l < oo.

1.4.3

Fejer Means of a Fourier Series

The Fejer means are defined by

Writing this out in detail, we see (N

+

N

l)aNf =

n

L L

f(k)eikO

n=O k=-n

N

- L k=-N

(N

+

1 - lki)f(k)eikO,

(sinxjx) 2 ; check that

FOURIER SERIES ON THE CIRCLE

55

which gives the useful representation

a-N(f) =

(1.4.10)

t

1..=-N

The Fejer kernel is

t

KN(()) :=

(1 -

lkl N

(1 -

1..=-N

+

)J(k)e'w

1

lkl ) N

+

e'kfJ.

1

This is a finite Fourier series of a discrete convolution. To see this, define IM (j) = 1 if ljl < M and zero otherwise. Then by counting points along the 45 degree lines in the square of side M, we see that (IM * IM )(j) = (2M+ 1 - ijl)/zM+l (j) if Ul < 2M and zero otherwise. This will allow us to factor KN in case N is even. To deal with the case of N odd, define (j) = 1 if j = ± t .... , ±(M - t) and zero otherwise. Again we count the points in the square to see that I%tdd * I%tdd = (2M- iji)hM· In either case KN has been written as the Fourier series of a self-convolved sequence, hence it must be the square of the trigonometric sum obtained from the original sequence. In detail, we have

r,;r

(2M+ l)

K2M(8)

2M K2M _ , (8)

~ ,)~;M (2M+ [ - lki)I,M+' (k)e''" ~ (.f~ IM(k)e'" ~ ,}:M (2M -

lk I)I,M+, (k )e'"

r'

~ Cf~ 1'/J" (k)e"8 )'

The trigonometric sum L~M IM (k)eike was evaluated as the Dirichlet kernel sin[(M + tWJ/sin (f)/2), which gives a closed form for KzM· But KzM-l can also be expressed in this form by doing a finite geometric sum: sinMt sin(t /2)

(1.4.11) We conclude that for all N we have the formula (1.4.12)

K

=

() N (

)

(sin[(N + 1)()/2]) N + 1 sin(() /2) 1

2

Exercise 1 .4.29. Prove the trigonometric identity ( 1.4.11 ). Figure 1.4.2 shows the graph of the Fejer kernel with N = 8. The above computations permit us to conclude

Proposition 1.4.30. KN(()) is an approximate identity. Proof. From the definition we see that 2 ~ Jy KN = I. From the formula ( 1.4.12) we see that KN ~ 0, so that the L 1 norms are bounded by I. Finally, we see from (1.4.12) that for

56

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

y

6

-3

~

-1

-2

2

3

FIGURE 1.4.2 The Fejer kernel with N = 8

any 8 > 0, KN(()) < Ij(N when N ---+ CXJ.

+

1) sin 2 (8/2) whenever 101 > 8. Hence

J;

81

>

8

KN(()) d() ---+ 0 •

This leads us to a general statement of Fejer's theorem. Theorem 1.4.31. Iff E LP(1f'), 1 _::: p < CXJ, then the Fejer means converge in U'(1f'). Iff E C(1f'), then the Fejer means converge uniformly. Iff E L 1 (1f') has right and left limits f(() ± 0) at a point 8o, then (anf)(8o) -+ Hf(8o + 0) + f(8o- 0)] when N-+ CXJ. If, in addition, f E L=(1f'), then lan(f)l < llfii<XJProof. The first two statements follow immediately from the fact that KN is an approximate identity and that L''(1r) and C(1r) are homogeneous Banach spaces. For the third statement, note that the kernel KN is even, so that we can write En(()o) :=aN(f)(Bo)- t£J(Bo =

-1

2rr

+

0)

+

f(Bo- 0)]

17T (j(Bo + ¢) + f(Bo- ¢)- f(Bo + 0)- f(Bo- 0)) KN(¢) d¢. 0

The convergence now follows by Proposition 1.4.6. Finally, iff

which completes the proof.

E

L 00 (1r), then

•

Remark. The proof of pointwise convergence of the Fejer means only requires that the symmetrized function¢ -+ f(() + ¢) + f(() -¢)have a limit when¢ -+ 0. This

principle applies more generally to operations that are defined by an even kernel, thus

FOURIER SERIES ON THE CIRCLE

57

expressed in terms of the symmetrized function. Such extensions of the basic results are helpful and can be deduced when necessary.

Corollary 1.4.32. The space oftrigonometric polynomials is dense in the spaces C('ll') and l.J'('ll') for 1 < p < ex::>. Proof. The nth Fejer mean un(f) is a trigonometric polynomial of degree nand converges tof in the norm of C('I[') and in the norm of U'('lr). •

This leads to a new proof of the uniqueness of the Fourier coefficients, as follows:

Corollary 1.4.33. Iff

E

L 1 ('ll') has Fourier coefficients identically zero, thenf =

Oa.e. Proof. From formula (1.4.10) we see that un(f) corollary.

=

0, hencef = 0 a.e. from the previous •

Exercise 1.4.34. Suppose thatf is bounded above and below: m < f(O) all 0 E 'll'. Prove that m < Un (f)(O) < M for all n = 1, 2, ... , 0 E 'll'.

~

M for

Exercise 1.4.!35. Suppose thatf is a continuous[unction with nonnegative Fourier coefficients: f(n) ::: 0, n E Z. Prove that Lnezf(n) < ex::>. Hint: Apply Fatou's lemma to the Fejer means.

Exercise 1.4.36. Suppose that f is continuous on a closed subinterval I. Prove that the Fejer means converge uniformly on I.

1.4.3.1

Wiener~

closure theorem on the circle

Fejer's theorem can be used to discuss the L 1 closure of the set of transl,tes of a given f E L 1 ('ll'). This is identical to the closure of Mt := {f

* g: g

1 E L (1I')}.

Proposition 1.4.37. Mt is dense on L 1 ('ll') if and only iff(n) :f:. 0, Yn E Z. Proof. If }(no) = 0 then for any g e L 1 (1r) we have if* g)(n0 ) = 0. Therefore M 1 !_ies in the proper closed subspace consisting of {F e L 1 ('lr) : F(n0 ) = 0}. Conversely, if j(n) ¥- 0, Vn e Z, let F E L 1 ('lr) be given. By Fejer's theorem, uN-l (F) ---+ Fin L 1 ('lr) when N ---+ oo. Let gN((}) = L,'!_N F(k)/}(k)(l - (lki/N))eik 9 e L 1 ('lr). Computing the Fourier coefficients, we see that/* gN = uN-I (F). Taking N ---+ oo completes the proof.

•

1.4.4

*Equidistribution Modulo One

The density of the trigonometric polynomials can be used to give a direct treatment of a simple model in ergodic theory. The integer part and fractional part of a real number x are denoted [x] and (x) respectively; thus x = [x] + (x) with [x] E Z and 0 < (x) < 1.

58

lNTRODUCTrON TO FOURIER ANALYSIS AND WAVELETS

If a

E

(0, 2n) is a real number, we consider N,(a, b)= card{O < k < n - 1 : (kaj2n)

where (a, b)

c

E

(a, b)j2n}

[0, 2n]. We propose to show the following.

Proposition 1.4.38. If a j2n is an irrational number and 0 < a < b < 2n, then N,(a, b)/n-+ (b- a)j2rr when n-+ CXI. Proof. To do this, we first write the counting function as n- I

Nn(a, b)= Lf(ka) 1.=0

where f is the 2rr -periodic extension of the indicator function 1
(1.4.13)

l

n-1

lim - Lf(ka) = n~'Xl n k=O 2rr

1

/({))dB.

T

We will first prove ( 1.4.13) for trigonometric polynomials. If/(B) = 1 n-1 l n-1 - Lf(ka) = - L n k=O n I.=O

1e'ml.a

e'me,

with m =1= 0, then

e'mna

=

n(1-e'm")

-. 0,

since the numerator is less than 2 and the denominator is a nonzero multiple of n. Now if f(B) = 2:_"_N ake'kl! is a trigonometric polynomial, we see immediately that 1 n-1 1 - Lf(ka)-. ao = n k=O 2rr

1

f(B) dB =J
n-. oo.

"

Now letf be any continuous function on 'li', extended 2rr-periodical ly to~- By the corollary to Fejer's theorem there exists, for any E > 0, a trigonometric polynomial g so that If- gl < Eon the real line. Therefore l n-1 - Lg(ka)n k=O

l E

l

n-1

n-1

:::0- L f(ka) :::0- Lg(ka) n k=O n <=O

+E.

Taking n -. oo, the extreme members tend to g(O). Since this holds for any E, we conclude that lim,~oo ~ 'f:.Z"~~f(ka) =f(O) whenever/ is continuous and periodic. Finally, if f = l
Taking j -. oo we have b- a . . f Nn(a, b) . N,(a, b) b-a - - < 11m m :::0 11m sup - - - - <

2rr

-

which completes the proof.

"

n

n

n

z;:-·

•

FOURIER SERIES ON THE CIRCLE

Corollary 1.4.39. /faj2rr is irrational, then the set {(ka/2rr): k in [0, 1].

E

59

Z} is dense

Remark. It's clear that this result is true only if aj2rr is irrational. If aj2rr is rational, the set of {(ka/2rr) : k E Z} is a finite subset of (0, 1). In this caseNn(a, b)= 0 if (a, b) is in the complement of this finite set.

1.4.5

*Hardy's Tauberian Theorem

Having proved the convergence of the Fejer means, we can obtain results on the convergence of SJ by means of Tauberian techniques. We now prove the Tauberian theorem of Hardy, which applies to any sequence of complex numbers an with partial sums and Cesaro means denoted 1

n

an:=-n + 1

Ls

1

J=O

1 = -n + 1

n

L(n + 1- j)a1 .

J=O

Theorem 1.4.40. Suppose that {adk;::O is a sequence of complex numbers such that kjakl < M for some constant M. Then the convergence of the Cesaro means implies the convergence of the original sums: limn (sn - an) = 0. If ak depends on a parameter x so that the convergence of ak is uniform in x and lkak I < M for all x, then Sn converges uniformly. Proof. For n < m we study the expression (m

+

l)um- (n

+

m

I)un-

L

(m

+

1- j)a1

;=n+l n

=

L (m + I -

n

j)a1 -

;=0

L (n + I -

j)a1

;=0

n

= (m -n) Lai j=O

= (m- n)sn. Subtracting (m- n)un from both sides and dividing by m- n, we have the useful identity (1.4.I4)

Sn -Un

= -m+I -(Um m-n

m+ILm -Un)- m-n

n+l

( 1-} -) m+l

a·. 1

The first term tends to zero whenever m, n ~ oo with (m + 1)/(m - n) bounded, for example if m/n ~ K > I. To examine the second term, write (1.4.15)

(m

+

1)

i: (1- m+ __j__l) n+l

laJI :::0 M [<m

+

1)

log (m/n)-

(m- n)].

Now we let m, n ~ oo so that mjn ---+- (1 +a) and use the Taylor expansion of the logarithm; we see that the final term in (1.4.15) is less than (m + 1)8 2 , so that when we

60

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

divide by (m- n) the sum is eventually less than 28. Thus we have limsuplsn -ani ::S 28.

•

But 8 was arbitrary, so that we conclude sn - O"n ~ 0, as desired.

How did Hardy think of the representa tion (1.4.14) for Sn in terms of O"n? Although we cannot say with certainty, we can surely provide a natural motivation in terms of the continuou s-paramet er analogue. Suppose that three functions a(t), s(t), a(t) are related by the formulas

1

11

1

s(t) =

a (t) = -

a(x) d.x,

t

1

s(x) d.x.

0

To study s(t) in terms of a(t), a(t), we can use Taylor's formula with remainder : t2a(t2)- tw(t,)

=

U2- t!)s(t,)

Solving for s(t 1 ), we obtain s(t 1 )

-

a (t,)

=

+

12

1,,

a(t2)- a(t,)

t2 - - - - t2- t,

(t2- x)a(x) d.x,

fr: (t2 2

x)a(x) d.x

t2- t,

which is the exact analogue of ( 1.4.14) in the continuou s-paramet er context.

Exercise 1.4.41. Suppose thatf(t), t > 0 is absolutely continuou s with an absolutely continuou s first derivative and that f"(t) = 0(1/t), f(t)/t --* s when t --* oo. Prove that f' (t) --* s when t --* oo. Hint: Takej(t)

=

ta(t)

above.

We can now reap some consequen ces of Hardy's theorem.

Corollary 1.4.42. Suppose that f is continuou s on 1I' and that its Fourier coefficients satisfy f(j) = 0(1/ljl), Ul --* oo. Then the Fourier series off converges uniformly on 1I' to f. Proof. We let ao = f(O) and ak = f(k)eik." + f( -k)e-ik1J fork = 1, 2, .... The Fejer means converge uniformly, so that O"n satisfies the hypotheses of Hardy's theorem, together with a, = O(lji)l) when)~ oo, uniformly in (J E 'lr. •

In particular, Hardy's theorem gives a new proof of the uniform convergen ce of the Fourier series of a continuou s function of bounded variation.

Corollary 1.4.43. Suppose that f is continuou s and of bounded variation on 'Jr. Then the Fourier series off converges uniformly on 1I' to f. Proof. We need to check the Fourier coefficients . By partial integration , we have f(n) = _21 { j((J)e-•nlld( J = __ 1_ { e-mBdj((J) = 0

(_!_).

J.r 2mrr J.r n Here we used the fact that the Fourier coefficients of a finite measure are bounded. 1T

•

FOURIER SERIES ON THE CIRCLE

1.5

61

IMPROVED TRIGONOMETRIC APPROXIMATION

1.5.1

Rates of Convergence in C('TI')

We now consider the Fourier approximation in the space C('JI'). In contrast with the space L 2 (1I'), the Fourier partial sum is not the closest trigonometric polynomial in this norm. In order to study the rate of convergence of the Fourier partial sum, we recall the representation of the Fourier partial sum in terms of the Dirichlet kernel: (1.5.1)

Suppose that gN is another trigonometric polynomial of degree N. Then clearly 8N = SNgN so that we have SNf- f = (SNf- SNgN) + (gN -f), from which we obtain (1.5.2)

Therefore the discrepancy ISNf - fl is measured in terms of the best trigonometric approximation and the L 1 norm of the Dirichlet kernel, which we now estimate. The Lebesgue constants are defined by the integrals (1.5.3)

Ln

= -1

2n

1

I sin (n + ~)tl frr . dt. sm (t/2) 2n -rr

= -1

IDn(¢)1 d¢

1r

The asymptotic behavior of the Lebesgue constants is provided as follows.

Proposition 1.5.1. Whenn--+ oo, Ln = (4lognjn 2 ) + 0(1). FurthermoreLn < 4 +log nfor all n ::: 1. Proof. Recall that on the interval (0, rr) the function t

~

-,--sin (t/2)

2 t

is bounded. Therefore we can write

1

1:rr I sin. (n + 4)tl dt

Ln = rr o

21:rr

=JT

0

sm (t /2)

I sin

(n

t

+ ~ )ti -

21
=-

dt+0(1)

+ 0(1).

*

We are reduced to examining the integral of I sin vII v on the interval 0 ::::; v ::::; (n + )rr. This is the sum of the integrals on intervals over (krr. (k + 1 )rr) for 0 ::::; k ::::; n pi"Us a term over the last half interval, which tends to zero. The separate terms, apart from sign,

62

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

are of the form

-1<<+tm sin v --dv krr

=

V

1
-dcosv

kn

V

(-l)l.+l

(-l)k+l

rrk

rr(k+1)

---------,-- + 2(-1)k+l

+0

rrk

(

+

1(k+l)rr krr

COSV

--2 v

dv

1 )

k2

'

so that 2 L, = -

L<=t" ( -rrk2 + O(k __ -)

4 log n ' + 0(1). 7r rrTo obtain the upper bound, we underestimate the denominator of the integrand in (1.5.3), replacing sin (t/2) by tjrr and considering separately the integrals on 0 :::; v :::; 1 and 1 < v :::; mrr, m: = n + ~. • 0

)

=

Referring to formula (1.5.2), we obtain the following useful fact. Proposition 1.5.2. Letf E C(1l). Then the maximum discrepancy between! and its Nth Fourier partial sum is bounded by log N times the best trigonometric approximation off by any trigonometric polynomial of degree N: lf(x)- SNJ(x)l < [5

1.5.2

+

logN] x inf."EPNmax1rlf- gl

Approximation with Fejer Means

In parallel with the L 2 theory developed in Section 1.3.3, we wish to develop corresponding results on the speed of trigonometric approximation in the supremum norm on the space C (1l). The Fejer means are an efficient device for obtaining the first results of this type. In this section we will develop results for Lipschitz and HOlder continuity. Higher order smoothness will be treated in the subsequent sections. To begin, we write the Fejer approximation of order m - I (to simplify the formulas): 2 /rr (si~m% ) (f(x+t) -f(x)]dt 2 rrm -rr t sin. mu ) [f(x + 2u) - j(x)] du - 1 f rr 1 rrm -rr/2 u - I 1rr/ (sinmu) . [f(x + 2u) + j ( x - 2u)- 2f(x)] du.

(am_J!)(x) -f(x) = -

2

1

Sin

2

2

=

(

Sin

2

2

(1.5.4)

=

rrm

o

s1n u

Using this form of the approximation, we can now state and prove some properties of the Fejer approximation. Theorem 1.5.3. Suppose thatf is Lipschitz continuous with constant K. Then

ilan
logn

< C,K-n

FOURIER SERIES ON THE CIRCLE

63

where C is an absolute constant. More generally, iff satisfies a Holder condition with exponent a < 1 and Holder constant Ka. then

flloo

llan(f) -

~

CaKan-a

where Ca depends only on a. Proof. Beginning with formula (1.5.4), we integrate by parts, writing (y) u)- :lf(x)l du. From the hypothesis of Lipschitz continuity, we have ( y) ::; K 1 y 2 • Now the Fejer kernel is bounded everywhere by m and by rr 2 j4mu2 on the interval u > 1/m. Therefore we can write

J6 !f(x + u) + f(x-

rr!am-l
~

m

(

t/m + r/2) (si~mu)2 d(u).

Jo

Jl/m

SIO U

The first integral is estimated as

~ 111/m (s!7n':uY dj::: m (~)

(1.5.5)

while the second integral is estimated by

~I 2 mrr

rr/2

}l/m

csinmu)2 d(u)/ < sin u -

rr/2 }1/m

_1 d(u) mu 2

= (4/mrr 2 )<1>(rr/2)- m(l/m)

(1.5.6)

+

J:

rr/2 2<1>(u)

l/m

mu 3

du.

The first two of these terms is 0(1/m), m -+ oo, and the last integral is O(logm/m), completing the first part. In case f is only Holder continuous, we have (u) ::; Kaul+a. Appealing again to (1.5.5) and (1.5.6), we see that the desired conclusion holds. •

Remark. One may inquire on the choice of the cutoff level 1/m in the above proof. It can be checked that this is optimal in terms of balancing the size of the two error terms. Exercise 1 .5.4. Suppose that f satisfies the symmetric Holder condition lf(x + u) +f(x- u)- 2f(x)l ~ Kalula for some K, whereO 2, for a suitable constant Cz.

t-

The above transformati ons can be used to prove a theorem of Lebesgue. Theorem 1.5 .6. Suppose that f 1

E

L 1 (1r) satisfies the condition that

rh if(x + u) + f(x- u)- 2SI du-+ 0,

h Jo for some x

E

T, S

E

h

---+ 0,


64

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. Using the same notations as above, we let ct>(h) so that ct>(h)

=

-1

1!/m (sinmu)

m

~ I {'r/2 1 11 m

lf(x + 2u) + j(x- 2u) - 251 du,

o(h), h---+ 0. Referring to (1.5.5) and (1.5.6) we have

(1.5.7)

mn 2

1h

=

.

o

2

dct>(u) .:::: mct>(IIm) .::=: E,

SID U

I

(sin mu )2 dct>(u) < {rr/2 _1 dct>(u) sin u - 11/m mu 2 -4-0 ct>(rrl2)- mct>(llm)

(1.5.8)

mn-

+

!"

12

1~

2ct>(u) du. --,mu

1

The first term is O(m- ) and the second term is o(l). The final integral clearly tends to zero if we change the notation and write ct> ( u) I u = E (1 I v) ---+ 0, so that

f

1/m

I ~m

ct>(u)

rr/2

du = m

- -3

mu

•

E(llv) dv---+ 0.

2/rr

Corollary 1.5.7. The Fejer means of an L 1 function converge almost everywhere. Proof. The strong form of Lebesgue's differentiation theorem states that for almost all x E 1r, limh ..... o01h) foh lf(x + u) - j(x)l du ---+ 0. Therefore on this set we can take S =f(x) in the previous theorem. •

Corollary 1.5.8. The Abel means of an L 1 function converge almost everywhere. Proof. If a sequence {sn} is Cesaro-summ able, then it is also Abel-summab le to the same sum, from Proposition 1.4.25. Since CTn(f) converges almost everywhere, the same is true of the Abel means P, f. •

We close this section with a negative result, showing that the Fejer means have an inherent limitation in their ability to approximate functions to a higher order of approximati on. Proposition 1.5. 9. Suppose that f E C ('II') satisfies n ~ ex::>. Thenf is a constant, almost everywhere.

II an (f) - f II=

Proof. Recall that

Hence lklf(k) = _I_ n + I 2rr ~

lkll/(k)l .::::

n

+

r

J.K' 1

~

[f(e)- CT,;(j)]e-lkli de,

r lf(e)- an 1f

1

(f) I de,

lkl lkl

.::=: n,

.::=: n.

o(l/n),

65

FOURIER SERIES ON THE CIRCLE

For any fixed k, the right side tends to zero when In I -- oo, hence](__k) = 0 for all k By the uniqueness of Fourier coefficients, we conclude that a. e. f f (0).

=

Exercise 1.5.10. Suppose that f E L 1 (1I') satisfies n ---+ oo. Prove that f is a constant, almost everywhere.

¥:-

0. •

llan(j)- /Ill = o(I/n),

Finally, we note that the Fejer approximation holds with the rate O(n- 1 ) for wellbehaved functions. Exercise 1.5.11. Suppose that the Fourier coefficients of f E L 1 ('ll') satisfy Lnez. lnll](n)l < oo. Prove that llan(f)- /II= < C/nfor some constant C. Exercise 1.5.12. Suppose that the Fourier coefficients off E L 1 ('ll') satisfy Lnez lnll](n)l < oo. Prove that the uniform limit of n(an(f) - f ) exists and compute its Fourier series.

In the next two sections we introduce other approximate identities to obtain higherorder trigonometric polynomial approximations.

1.5.3

*Jackson's Theorem

Iff has additional smoothness properties, we can obtain quantitive estimates forf- gN by working with the Jackson means. The Jackson means of order four are defined by

JNf(x) = Zh1

1:rr/Z (sinNu)

N

. stnu

o

4

+ 2u) + f(x-

(f(x

2u)) du

where hN :=

l

2

:rr/ (sinNu)

0

4

du.

.

SinU

By examining the transformations in (1.5.4), it is clear that JNf is the convolution of f with the square of the Fejer kernel, a trigonometric polynomial of degree 2N- 2. Therefore J Nf is a trigonometric polynomial of degree 2N - 2. Recall that/ satisfies a Lipschitz condition if there exists a constant K so that

1/(x)-f(y)l

< K!x-

Theorem 1.5.13. Iff has Lipschitz constant K

IJNf(x)- f(x)l ::::

yl. 1,

cl

then Kl

N '

where C1 is a universal constant. If, in addition, the derivative f' is Lipschitz continuous with Lipschitz constant K 2 , then Kz IJNf(x)- f(x)j :::=: Cz N 2

where

c2 is a

universal constant.

,

66

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. We write 2hN

iJNf(x)- f(x)i :::0 - 1

4

2

JNf(x) - f(x) = - 1

2hN

1trl (sinNu) -.(f(x + 2u) smu

o

irr/ 0

2

+ f(x-

2u) - 2f(x)] du

4

(sinNu) -.4Ktlul du. smu

The denominator is estimated by hN

4

{tr/ 2 (sin Nu ) ~ Jo --u-

3

{Nrr/

2

du = N Jo

4

(sin v ) -v-

dv . . __ constN

3

•

To estimate the numerator, note that, on the interval [0, 7r /2), sin u is bounded below by 2u/7r. Making the change of variable v = Nu, the integral is no more than 4

7r

Kt 16

1rr 0

12

(sinNu)

4

4

1 2 7r -KN udu = 16

u

1N:n:f 0

2

4

(sin v) v dv, --

v

which proves the first statement and identifies the constant as 7r

4

Ct = 32

f 000 (sin vjv) 4 v dv J0 (sinvjv) 4 dv

=-'-';;=---'----00

If, in addition, f' is Lipschitz, then the mean-value theorem provides the estimate if(x + 2u) + f(x- 2u)- 2f(x)l :::: 8K2iul 2, which gives the improved estimate iJNf(x) - f(x)i :::0 - 1

hN

1rr; 0

2

4

(sinNu) --.8K2Iul 2 du. Sln U

Again we make the change of variable v = Nu and find the required estimate, with 7r 4

c2 = -

2

f 000 (sin vjv) 4 v 2

dv fo (sinvjv) 4 dv

::....::...,.,oo,--------

•

Exercise 1.5.14. Suppose thatf is Holder continuous with exponent et: lf(x) f(y)l ~ Klx- yla for some 0 < Ot < 1. Prove that IJNf(x)- f(x)l ~ C 01 KN- 01 for a universal constant C 01 • Exercise 1.5.15. Suppose that f is absolutely continuous and that f' is Holder continuous with exponent Ot, 0 < Ot < 1. Prove that IJNf (x) - f (x) I < c~ KN-I-Ot for a universal constant c~.

1.5.4

*Higher-Order Approximation

If r E z+, the space Cr (Ir') consists of functions whose rth derivativef(r) is a continuous function. If r E z+ and 0 < Ot ~ 1, the space cr,Ot ('Ir) consists of functions I E cr ('][') such thatj satisfies a Holder condition of order Ot. Since any differentiable function is Lipschitz continuous, we have the inclusion cr+ I ('Ir) c cr,l ('Jr). If 0 < Ot < 1, we often write, by abuse of notation, cr,Ot ('II') = cr+at ('Jr). It is natural to expect that iff E C('II') has derivatives of higher order, then we will obtain an improved rate of approximation by suitable trigonometric polynomials.

FOURIER SERIES ON THE CIRCLE

67

To make this concrete, consider for any even integer 2k, the difference operator 2k (2k) . ll.zkf(x;u )=L . (-1)'f(x+ u(k-j)) }=0

(1.5.9)

=f(x

J

+ ku)-

2kf(x

+

- 2kf(x- (k- l)u)

(k- l)u)

+ f(x-

+ ··· +

(-1) k(2k)r k (x)

ku).

The coefficients are those that occur in the binomial expansion of (ei 0 vanishes to order 2k at()= 0. Hence iff E C 2k(1r), the derivatives are

(::UY ll.zkf(x; u)lu=O (::U) 2k ll.zkf(x; u)lu=O

+ ···

= 0,

-

1 ) 2 k, which

0 < j < 2k- l,

= (2k)!f(2kl(x) ,

so that we have the bound

More generally, we can apply Taylor's theorem with remainder to prove that when u

~

0

u~o

==> ll.zkf(x; u) = O(u2k-l+a), f E C2k-1(1r) ==> Dozkf(x; u) = O(u2k-1), f E C2k-2,a ('][') ==> Dozkf(x; u) = O(u2k-2+a), f E C2k-l,a(1r)

u ~ 0,

0
u ~ 0 u ~ 0,

0 < a ::5 l.

In order to construct improved trigonometri c approximati ons, we consider Ezk.mf(x) :=

i

rr/2

0

ll.zkf(x; u)

(sinmu)2k+ 2 . du. SlnU

Noting that (sin (mu)j sin u) 2 k+Z is a trigonometri c polynomial of degree (m- l)(k + 1), we see that the same is true for each term that figures in the definition of E 2 k,mf, save for the middle term with}= k. Iff E C 2k(1I'), we have (1.5.10)

!Ezk.mf(x)! < C

rr /Z

1 0

(1.5.11) (1.5.12)

i

< C

1

(1.5.13)

o

o

=mC

(sin mu) Zk+Z . du SinU

rr/2

< C -

u 2k u 2k

(sinmu)2k+ 2 du 2ujrr

=u 2k (sinmu)2k+ 2 du

i

o

2ujrr

oo

v 2k

(

sin u ) 2k+2 dv 2ujrr '

--

68

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

while Ckm:=

1

rr/2

0

(sinmu)2k+2 d . U Sln U

: : : 1rr

/2 (sinumu) 2k+2 du

= m2k+l

mrrf2 (sin V )2k+2 -dv. 0 v

1

These two estimates are combined to prove the following. Proposition 1.5.16. Suppose that f E C 2k ('li'). Then there exists a sequence of trigonometric polynomials fm of degree (m - 1) (k + 1), so that

Proof. It suffices to set fm(X) =f(x) _ E2k.mf(x) •

(1.5.14)

Dkm (1.5.15)

2

Dkm = (-1)k( : )

1"

12

2

(s:7n':uYk+ du =

<-1)k(~)ckm.

Notethatfm is a trigonometric polynomial of degree (m-l)(k+ 1). ThenDkm(f(x)-fm (x) = (E2k,mf)(x). Applying the above estimates gives the result. •

In the general case of functions with a HOlder continuous derivative, we have the following general result. Theorem 1.5.17. Suppose that f E cr,o: ('JI'). Then there exists a sequence of trigonometric polynomials fm of degree < m(r + 2) so that

where the norm is defined as llfllr,o: = suplf(x)l

+ ·· · +

lf(r)(x)l

+

XE1!'

sup lf(rl(x) -f(r)(y)l IX - ylot

x#yE1l'

and the constant C depends only upon rand at. Exercise 1.5.18. Prove the estimates (1.5.10). Exercise 1 .5 .19. Complete the steps of the proof of Theorem 1. 5.17.

It is often convenient to have a universal sequence of trigonometric polynomials, which can be used at every level of differentiablity. These are provided by the de la Vallee

FOURIER SERIES ON THE CIRCLE

69

y 12

FIGURE 1.5.1 The de Ia Vallee Poussin kernel with N

=

8

Poussin means (Figure 1.5 .1 ), which are defined in terms of the Fejer means by Tn(X)

= 2azn-l (x) -

O"n-1 (x).

Let En (f) = inf TEP., II f- T II 00 be the sup-norm distance between f and the trigonometri c polynomials of degree n. Theorem 1.5.20. The de la Vallee Poussin means satisfy the estimate

llrn - flloo

:S 4En(f).

Proof. The above infimum is attained by some (possibly nonunique) T; E Pn. Indeed, any minimizing sequence Pk must be uniformly bounded, in particular have uniformly bounded Fourier coefficients. But these reside in the finite dimensional space C 2 n+l, where one can apply the Balzano-Weie rstrass theorem to obtain a convergent subsequence, for which the corresponding trigonometric polynomials are uniformly convergent to some T; E Pn. Now we write j(x)

=

T;(x)

+ R(x)

The Fourier partial sum operators

Sk

where

and Fejer means

ak-l

k

SJ'(x)

=

k

LJ(J)e'lx, J=-k

I

ak_J(x)

= k

are defined by

k-1

LS..J(x), }=0

SkR(x) = L j=-k

R(j)eijx

70

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

If k -> n we have

S, T*n =

T*, so that n l

S,J =

T,:

+ SkR and hence

2n-l

l

2n-1

n

k=n

- L:s,J = r: +- L:skR. n

k=n

which can be written in terms of the delayed Fejer means:

But the Fejer kernel is a contraction in L 00 (1l'), thus

lr,(x)- T,:(x)l

=

Ia~.

(R)I .:::: IIRIIoo .:::: E,(j). In particular

12a2n-J(x)- a,_J(x)- T,:(x)l .:::: 2E,(j)

+

E,(j)

=

3E,(j)

so that

llr, -ill"".:::: II<,- T;ll:x.. +liT; -ill= .:::: 3E,(j)

=

+

E,(j)

4E,(j),

•

and the proof is complete.

The de la Vallee Poussin approximation can be written in terms of the Fourier coefficients as follows:
2 L (1- ~~)f(j)eiJX- L (1- I~I)J(j)e'Jx L f(j)eijx + 2 L (1 - W.)J(j)e'JX. IJI:"2n-1

· IJI::"n-1

IJI:"n

ns 1·1 1 < 2n

2n

In terms of the Fejer kernel, we have 2K

2 ( sin nt ) 2 n-l (t) - Kn-l (t) = 2n 2 sin t/2

2

2 ( sin nt /2 ) - ;; 2 sin t/2

2

2 sin 2 nt - 2 sin 2 (nt /2) 4n sin 2 (t /2)

cos nt - cos 2nt 4n sin 2 (t /2)

Unlike the Fejer kernel, the kernel of de la Vallee Poussin is not positive. However it does satisfy the general properties of approximate identities.

Exercise 1.5.21. Prove directly that the de la Vallee Poussin kernel satisfies the three properties for an approximate identity.

1.5.5

*Converse Theorems of Bernstein

Bernstein showed that the rates of convergence of trigonometric approximation can be used to characterize the degree of smoothness of a function. The key to proving these converse theorems is the following inequality for trigonometric polynomials.

FOURIER SERIES ON THE CIRCLE

71

Lemma 1.5.22.

sup 1/'(0)1 <2M x sup 1/(0)1.

(1.5.16)

8ET

8ET

Proof. Recall the Fejer kernel

= M .2._

KM_,((})

(sinM0/2 ) sin0/2

2

=~ ~ ( 1 _ ~) e'*e. M

This will allow us to represent/' ((}) as an integral transform off. We begin with the Fourier coefficients

-jJr KM_ 1(f/J)e-•*<1> df/J = -Jr

1 A(k) := 2rr

+

Hence A(k- M)- A(k (1.5.17)

1 -k = M 2rr

M) = k/M for lkl

1Jr

{

~- lki/M

lkl :S M lkl >M

:=:: M. Thus

KM_ 1(f/J)e-•kz isinMf/Jdf/J,

lkl :SM.

-1r

Multiply (1.5.17) by iMakeikB and sum for-M::; k :=:: M: M

f' ((})

=

L

ikakeikB

k=-M

=iM 2rr

1Jr

-Jr

Ml"-Jr

= -

2rr

KM-1(4J)2 isinMfjJ

(L M

ake•*> )

d¢

k=-M

KM- 1(f/J)( -2 sin MfjJ)f(O - f/J) df/J

which is the desired representat ion. Hence

1/'(0)1 :=:: M

2rr

(1Jr_,..

2KM-l (l/J) df/J) x sup 1/(¢)1, ¢eT

=2M sup 1/(f/J)I ¢eT

which completes the proof.

•

To prove converse theorems in the supremum norm, it is convenien t to work with trigonome tric sums of order 2k, known as dyadic sums. This subsequen ce is useful because of the following simple estimates that pertain to tail sums and finite sums:

To study the approxima tion in detail, letf E C('II') and Tn(x) be the trigonome tric polynomia l of degree n, which achieves the best approxima tion: En(f) =II/- Tnlloo·

72

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Theorem 1.5.23. If En(j) < en-a for C > 0, 0 < a < 1, thenf satisfies a Holder condition: lf(x +h) - f(x)l s Kha. Proof. We examine the limit in terms of dyadic sums. For any no E z+, f(x)

=

00

lim T2n(x) n---+00

L

= T2no(x) +

,(x)

r=no+l

where ,(x) = T 2 , (x)- T2 ,_, (x). From the triangle inequality we have l;(x)l ::5 3C2-"'. Now define m = m(h) E z+ by the inequalities 2m- I < ljh ::5 2m. Then

~~~>

,(x)l :::;

•~> 3Cr'"

For the remaining sum we use the mean value theorem and Bernstein's lemma (1.5.16) to write m(h)-1

L

m(h)-1

L

l;(x +h)- ;(x)l ::5 h

r=no+l

sup l;(x)l

r=no+l xET

m(h)-1

L

::5 h

C2' 2-cri

r=no+l

::5 hCa2m(h)(l-a) ::5 Cah". Hence lf(x +h)- f(x)l :5 IT2 no (x +h) - T 2 no (x)l

+ Cah" as required.

•

Exercise 1.5.24. Use Theorem 1.5.23 and the proof of Theorem 1.5.3 to show that iff satisfies a symmetric Holder condition: lf(x +h)+ f(x- h)- 2f(x)l < c ha for some 0 < a < 1, then f satisfies the usual one-sided Holder condition: lf(x +h)- f(x)l s Cha. (For a = 1 this is false; see Zygmund (1959).) If a = 1, the above estimates break down. Indeed, it is not generally true that En (f) < C In implies that f E Lip(1I'). The difficulty is in the estimate of m(h)-1

L

m(h)-1

l;(x +h)- ;(x)l < h

L

i=no+ I

i=no+ I

sup l;(x)l xe1'

m(h)-1

< h

L

cziz-i

i=no+1

shCm < Ch log (1/h).

Thus we have the general implication

c

En (f) < -

n

~

lf(x +h)- f(x)l < Kh log (1/h).

FOURIER SERIES ON THE CIRCLE

73

Bernstei n's inequali ty can also be used to characte rize the different iability off in terms of rates of converg ence of En(f). Indeed, suppose that En(f) < Cjnf3 for some f3 > 1. Then the triangle inequali ty gives l;(x)l < 3C2-f3i and Bernstei n's inequali ty (1.5.16) shows that for r E z+, r < {3, l~r\x)l < 2<•+l)r3 c2-f3i = Cr2i(r-f3 ). Hence the series"" = ;(x) can be different iated term-by- term, since the numeric al ~•=no+ 1 series L~I 2i(r-f3) converge s. We formaliz e this discussi on as a theorem . Theorem 1.5.25. Suppose that En (f) < C /nr+a where r E z+ and 0 < a < 1. Thenf E Cr('JI') andf(r) satisfies a Holder conditio n of order a. Proof. From the above discussio n, we have the uniformly converge nt series

j(x) =

d ( dx

)r

Tno(X)

+

L

oo

~r>(x).

i=no+l

The first term is infinitely differenti able, hence Holder continuou s. The second term is handled exactly as in the proof of Theorem 1.5.23, replacing ; by <1>?>, to which the same

estimates apply.

Exercise 1.5.26. Suppose that En(f) < Cn-r where r E z+. f E c
•

Show

that

Exercise 1 .5 .2 7. Suppose that En (f) .:::: C 1/log n for all n > 2 for some constant C 1. Prove thatf satisfies the continui ty conditio n lf(x+h) -f(x)l .:::: C2flog (1/h) for 0 < h < Compar e with the result of Exercise 1.5.5 .

!.

1.6

DIVER GENCE OF FOURI ER SERIES

In this section we turn to some negative results, which have been instrume ntal in the developm ent of harmoni c analysis. In 1873 du Bois-Re ymond showed that there exists a continuo us function whose Fourier series diverges at a point. This was further develope d to show that any preassig ned set of Lebesgu e measure zero can be the set of divergen ce of the Fourier series of a continuo us function (Kahane and Katznels on, 1966). Meanwh ile, in 1915 Lusin had posed the problem of proving the almost-e verywhe re converg ence of the Fourier series of an arbitrary f E L 2 (1I'). This was proved by Carleson (1966) and extended by Hunt (1968) to the class U' ('JI') for p > 1. Another proof of Carleson 's theorem by C. Fefferm an (1973) has been useful in more recent developm ents of harmonic analysis. Many years earlier Kolmog orov (1926) had proved the existenc e of an L 1 function whose Fourier series diverges at every point of 'JI'. These results and counterexam ples are beyond the scope of this book. We will prove, by two different methods , the existenc e of continuo us function s with Fourier partial sums unbound ed at a point. We will also construc t L 1 function s whose Fourier series do not converge in the L 1 norm. In Chapter 3 we will prove the theorem of M. Riesz (1927) that for any function in ll'('JI'), 1 < p < oo the Fourier series converge s in the U' norm. The upshot of these results and countere xamples is that LP, p > 1 is a good space for one-dim ensional Fourier series, both in the a.e. sense and in the sense of norm

74

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

convergence. The space L 1 is bad in both senses. The space of continuous functions is bad for norm convergence and for convergence at a preassigned point, but good for almost-everywh ere convergence.

The Example of du Bois-Reymond

1.6.1

Proof. By suitable grouping of terms, we will construct a continuous function whose Fourier series diverges at a preassigned point. Without loss of generality, we will do this at () = 0. The desired function will be sought in the form (1.6.1)

where

aJ e'19

and where the integers mk. Nk will be chosen. Bk(()) is the partial sum of a Fourier series of a function of bounded variation with a jump discontinuity at the point () = 0 and otherwise smooth. For example, one may take f(O) = rr - () for 0 < () < rr, extended as an odd function. In this case ak = -i/k. Since/ is of bounded variation, the partial sums are uniformly bounded: IBk(O)i ::= M for a constant M, simultaneously for all() E T, k = 1, 2, .... Hence the series ( 1.6.1) is uniformly convergent to a continuous function, by the Weierstrass M test. In order that the different blocks Bk involve different frequencies, we will choose the integers mk> Nk so that

Although ( 1.6.1) is not written as a Fourier series, we claim that the nth Fourier coefficient of is given by the coefficient of em9 in the series. Indeed, since the series ( I.6.1) is uniformly convergent, when we multiply by e-mfJ we still obtain a uniformly convergent series that we can integrate term-by-term:

f

Since the terms of Bk contain different frequencies, all of the integrals will be zero, save for the value of k satisfying in - Nk I :5: mk> if there is one with an-N* :j:: 0. In that case the integral is 2rran-Nk and zero otherwise. Hence the nth Fourier coefficient is given by an-NJe or zero, which completes the required identification. Now we examine the partial sum at level Nk:

The first sum converges by the Weierstrass M test. The second sum can be evaluated exactly in the case ak = -ijk:

L-;-1I = mk

logmk

+ 0(1).

j=l

We now choose mk so that (log mk) j k 2 -+ oo, for example mk = 2k 3 will do. Having chosen mh we choose the sequence N"- so that Nk+ 1 - Nk > mk + m"-+l which is possible, for example by taking N1 = I and Nk+ 1 = r=::~ (m1 + m 1 + 1 + I) for k = I, 2, .... The proof is complete. •

FOURIER SERIES ON THE CIRCLE

75

The example of du Bois-Reymond depends ~ritically on the fact that the one-sided sums defined by L:J=o ajeiJ8 are divergent when k ---+ CX), whereas the corresponding two sided sums defined by Bk(()) are convergent when k ---+ CX).

Exercise 1.6.1 . Suppose that f is a function of bounded variation that has a jump discontinuity at () = 0 and is otherwise of class C 2 on the circle. Prove the asymptotic formula j(n) = C/n + O(l/n 2 ) and identify C in terms of the jump. 1.6.2

Analysis via Lebesgue Constants

In this section we re-examine the questions of convergence and divergence in a more general setting. A Banach space is a complete normed linear space. For example C('li'), LP (1I') for p ::: 1 are familiar Banach spaces. A mapping T : B 1 ---+ B2 is a bounded linear operator if it satisfies the conditions that (1.6.2)

T(f +g) = Tf

+

Tg,

T(cj) = cT(f),

IITflls2 < Kllflls,

Here cis any complex number and K is a positive real number. For example, if B 1 B2 = LP (1I') and g E L 1('II"), the convolution Tf = f * g defines a bounded linear operator. This is immediate from the B-valued norm computation

11Tflls2 = lllf(x- y)g(y) dyt = 2 <

i

lg(y)lllhlls, dy

=

Ill

g(y)fv dyt,

IIIIIs, lig(y)l dy

=

IIIIIs, llgllvcn·

The operator norm of a bounded linear operator is the smallest number K so that ( 1.6.2) holds. Equivalently,

IITII

=

sup 11Tflls2 -

llfll :51

If Tn is a sequence of bounded linear operators, the pointwise convergence to a limiting operator T is defined by the requirment that lim IIT,J- Tflls 2 = 0 n

for allf

E

B1. The following condition is clearly sufficient.

Proposition 1.6.2. Suppose that Tn is a sequence of bounded linear operators and T is a bounded linear operator so that (i) For some dense set V, lim T ,J = Tf for allf E V. ( ii) There exists a constant K so that II Tn II < K for all n and II T II < K. Then the sequence converges pointwise on all of B.

76

INTRODUCfiON TO FOURIER ANALYSIS AND WAVELETS

Proof. Iff

E

B, there exists g E V so that

+

<E. Now

+ (Tg- Tf), TJIIB 2 ~liT,./- TngiiB2 + IITng- TgiiB2 + IITg- TJIIB2 ~ Kllf- giiB1 + IITng- Tg1IB2 + KIIJ- giiB1·

T,.f- Tf IIT,J-

II/- gliB 1

=

(T,.f- Tng)

(Tng- Tg)

Hence limsupn UT,J- T/11 82 S 2KIIf- gll 8 , proof.

~ 2Kt:.

But

E

was arbitrary, completing the •

This may be applied to give a new proof that the Abel and Cesaro means converge in LP (1I'): In each case we can take K = 1 and note that we have convergence on the space of trigonometric polynomials, since in each case we have T ,J = TnJ for n :::;: no, the degree off. It is surprising that the converse is true: If a sequence of linear operators converges pointwise, then the operator norms remain bounded. Th~s is embodied in the uniform boundedness principle, stated as follows. The proof is given in an appendix to this chapter.

a

Theorem 1.6.3. Suppose that Tn is sequence of bounded linear operators so that for eachf E Bt. supn IITJIIB 2 < oo. Then the sequence of operator norms remains bounded: supn II Tn II < oo.

This is applied to study the Fourier partial sum operators on C(1I'): N

L

f ~ SNf =

}(n)einO

n=-N

The supremum norm is estimated by

In fact this bound is sharp, since we can take the bounded functionfo(¢) = sgn IDn(¢)1 and achieve equality. This choice of fo is not continuous, but can be approximated boundedly be a sequence of continuous functions, for example by using the Fejer means to definefn = K,Jo. Then 2rrSNfn(O) = f1r fnDN--+ f1r foDN = f1r IDNI when n--+ oo. We now recall the estimation of Proposition 1.5.1, which states that IISN II "" (4 log N) I 1r 2 • This allows us to infer, without any computations, that there exists a continuous function whose partial sums do not remain uniformly bounded, especially not uniformly convergent.

Proposition 1.6.4. There exists a continuous function whose Fourier partial sums do not remain uniformly bounded. For any() E 1I', there exists a continuous function whose Fourier partial sums are unbounded at().

FOURIER SERIES ON THE CIRCLE

77

Proof. The first statement is evident from the previous discussion where we take B 1 = B 2 = C(1l'). For the second statement simply take B1 = C(1I') and B2 =
Therefore, by the uniform boundedness principle, there exists a continuous function whose Fourier series is unbounded at (). • The following exercise shows that the Lebesgue constants can be used to estimate the rate of divergence of the Fourier partial sums for a function that has a local average.

Exercise 1.6.5. Supppose thatf E L 1 ('Ir) andfor some() E 'IT', s := lim¢_,. 0 (f(B + ¢) + j(B- ¢)]/2 exists. Prove that SNJ(B) = o(logN) when N---+ oo. Hint: Given E > 0, choose 8 > 0 so that If(()+¢)+ j(B- ¢)- 2sl < 2E for 0 <¢<E. From the representation of partial sums we have

~

ISNJ(())- sl =

~

11" DN(¢)(]e(¢)-

s) d¢1

8

E

~ E

IDN(¢)1 d¢

1

IogN

+

11,. DN(¢)(]11(¢)- s) d¢1

+ o(l).

For a general f E L 1 ('IT'), one can establish the above rate of divergence at almost every () E 'IT'. The proof is very similar to the proof of almost-everywhere Cesaro-summability in the space L 1 ('IT').

Proposition 1.6.6. Let f o(log N), N ---+ oo.

E

L 1 ('IT'). Then for almost every ()

E

'IT', SNf(B)

J:

-

Proof. Let Fe(¢)= lf(B + u) + f(B- u)- 2f(8)1 du. From Lebesgue's differentiation theorem, we have for almost every e E 1!', F 0 (u)ju --+ 0 when u ---+ 0. Now we write s = 2j(()) and, as above,

SNJ(B)- j(()) = -I (1"/N 1f

0

+

1" ) rrjN

DN(¢)(]e(¢)- s) d¢ = /1 +h.

On the first integral we use the fact that IDN (¢) 1 ~ 2N l/d

~

2N

+

rr

1

+

F11(rrjN) = o(l),

On the interval rrjN ~ ¢ ~ rr we can write DN(¢) ~ rr 24

21"

lhl~-+-

rr

rr/N

I, 0 < ¢ < rr; thus

N---+oo. 1r 2 /24

+ 2/¢. so that

F'(¢) --d¢.

¢

A partial integration shows further that

1" rrjN

I" + 1"

F'(¢) F(¢) - - d¢ = - fjJ fjJ rr/N

rrjN

F(¢) d¢ 2

¢

.

78

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The first term is bounded when N -+ <Xl. On the other hand we write E(
rr 1rr(N

F'(
= 1rr rr(N

= o(logN),

€(
N-+ <Xl.

•

The proof is complete.

1 .6.3

Divergence in the Space L

1

The analysis of divergence in the space L 1(IT') is entirely parallel. First we outline a qualitative argument to prove the existence of an L 1 divergent Fourier series, then we proceed to construct a class of examples. In order to exploit the principle of uniform boundedness, we need to compute the norm of the operator S N, which maps L 1(1I') to L 1(1I'). On the one hand, for any f E L 1(1I'), we have

so that the operator norm IISN 11 1. 1 is bounded by the Lebesgue constant LN. On the other hand, if we take the Fejer kemelf = Kn with n > N, then by the properties of the Fejer kernel n---+ oo,

since for any fixed N, an(DN) converges boundedly to DN when n ---+ oo. Hence we conclude that the operator norm is given by l1SNII1.1 = IIDNIII = LN = -

4

7r 2

logN

+ 0(1).

By the uniform boundedness theorem, we conclude that there exists f E L 1 (1I') such that IISNflil is unbounded when N ---+ oo. This completes the qualitative argument and proves the following proposition. Proposition 1.6.7. There exists f E L 1 (1I') whose Fourier series diverges in the L 1 norm: supN IISN/III = +oo.

We now use Lebesgue constants to construct explicit examples of functions in the space L 1(1I') whos~ Fourier series do not converge in the L 1 norm. We begin with a sequence {an} of positive real numbers tending to zero so that the second differences are nonnegative; in detail (~ 2 a)n := an+l +an-I - 2an ::: 0. Such a sequence is termed convex. Lemma 1.6.8. For any convex sequence, we have limn n(a11 -an+ I) = 0 and the series L~o n(~ 2 a) 11 = ao < oo. Proof. Let b, =a., -an+ I· From the convexity condition we see that b, ~ b,+ 1 for all n. Also bn -+ 0, since an -+ 0. To prove that b, ~ 0 for all n, we assume that b,0 < 0 for some n 0 . By convexity we have b, ::5 b., 0 for n ~ no. Then an - an 0 ~ (no - n)bno• which implies a, -+ <Xl, a contradiction.

FOURIER SERIES ON THE Cil
79

Given E > 0, > ~N -an = bN+l

there exists N(E) so that an < E, bn < E for n > N. Thus + · · · + bn 2: (n - N)bn, which proves that lim supn nbn .::5 E, which was to be proved. To prove the convergence of the series of second differences, use summation by parts to write E

N

L

n(6 2 a)n = ao- aN- NbN.

n=l

•

When N--+ oo, the right side tends to a 0 , as required.

We can use these techniques to construct trigonometri c series in L (1['). Let {an} 1

be a convex sequence and consider the sequence of functions N

SN((}) = ao

+

2

L

an

cos (nB).

n=l

Recall the Dirichlet kernel and Fejer kernels: Dn(B)

= l

+ 2cosB + · · · + 2cos (nB),

K (B) = Do(B) n

+ · · · + Dn(B). n+l

Inversely, 2cos(nB) = (f::t.D)n(B), Using summation by parts we can write N-1

(1.6.3)

SN(B) = L

n(!::t. 2 a)nKn-l (B) - NKN-1 (B)(f::t.a)N

+ aNDN(B).

n=l

For any B =I= 0, the last two terms tend to zero, while the first sum remains bounded in 1 • We define f e L 1 (II') by the L 1 convergent sum

L

00

(1.6.4)

f(B) = Ln(!::t. 2 a)nKn-I(B). n=l

From the above lemma and the normalizatio n of Kn we see that this series of nonnegative terms converges in L 1 , hence the sum is finite almost everywhere and defines an L 1 function. It remains to compute the Fourier coefficients. To do this we multiply j(B) by e-imB and integrate term-by-term , by dominated convergence . Thus we have 00

f(m) = L

n(!::t. 2 a)nKn-l (m)

n=O

the last step being the result of summing by parts twice. Having established that alml =f(m), we can investigate the L 1 norms.

Proposition 1.6.9. Suppose that {an} is a convex sequence and letf be defined by ( 1. 6.4 ). Then the partial sums s N remain bounded in L 1 if and only if {an log n} is a

80

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

bounded sequence. The partial sums are convergent in L 1 if and only if an log n ~ II 0 when n ~ <Xl. Proof. Returning to (1.6.3) we see that the first two terms are bounded in L 1 • Therefore if sN also remains bounded in L 1 the same must be true of the final term anDN. But its L 1 norm is asymptotic to ( 4 j rr 2 )aN log N, which proves the first statement. To prove the second statement, write SN(O)- /(0) =

L=

n(~ 2 a)nKn-l (0)- NKN-1 (O)(~a)N

+

aNDN(O).

n=N

=·

The first sum tends to zero in L 1 when N --+ as does the second term. Therefore the convergence to zero of llsN- /11 1 is equivalent to the convergence to zero of llaNDNII~> which is equivalent to aN log N --+ 0, which completes the proof. •

1.7

*APPENDIX: COMPLEMENTS ON LAPLACE'S METHOD

1.7.0.1

First variation on the theme-Gaussian approximation

The proof of Laplace's method can be modified at no expense to handle integrals of the form C(t) =

1b A(J-L)erB(iJ-)ei~-'-c

dJ-L.

Assume that A(J-L) and B"(J-L) are Lipschitz continuous. Without loss of generality, we may assume that the maximum of B is attained at f.-L = 0 E (a, b) and that B(O) = 0. Thus we apply Steps I, 2 and 3 of Section 1.1.5 to reduce to 2 C(t) = A(O) /_: e-tk11- ei11-c dJ-L

Applying Step 4 we replace the limits by so that C(t) = A(O)

i:

-<Xl

+

o( ~) .

< f.-L < <XJ and incur an exponential error

e-rk~-'- 2 eiw df.-L +

o( ~) .

But this is the Fourier transform of the Gaussian density, hence we have the asymptotic formula C(t) = A(O)Ifte-c2/4kr

+

o( ~).

All of the error estimates are independent of c, so that this can be used in cases when

c depends on t. Of course, to provide useful information, it is only interesting when c is restrained, for example c = 0(.../i); if cis too large the exponential term will be smaller than the error term when t

1.7.0.2

~ <XJ.

Second variation on the theme-improved error estimate

If we have additional information on the function B(J-L), we can refine the error estimate

to O(ljn 3 12 ), which comes up in many problems. Specifically, assume that A(J-L)

=

1,

FOURIER SERIES ON THE CIRCLE

and that B(JL) is four times differentiable with B' (0) for some b 3 real. We begin with C(t)

lb

=

=

1"

e 18 (J.L) df.L

+

-k < 0, B

111

(0)

= ib3

etB(J.L) df.L.

Assuming that the maximum is attained at f.L reduce to, C(t)

= 0, B" (0) =

81

=

0 and that B(O)

O(e-c

1

),

C >

=

0 we immediately

0.

-.5

Now we use the inequality (1.1.41) with

ZI

= B(JL), zz = -kf,L 2

+

if,L 3b3/3

-8 < f.L <

o.

The integral of the remainder term is

1

.5 tC4f.L4 e-tkJ.L2 /3 df,L -.5

<

1oo

tC4J-!'4 e-tkJ.L2 /3 df.L

= t;:z'

-CX)

as required. Now we use the inequality leia- 1 - ial < a 2 /2 for a real to write le-t(kJ.L 2/2-iJ.L 3 b 3 )

_

e-tkJ.L 2/2 _ if,L3b e-tkJ.L 2!2I < t2f,L6b~e-tkJ.L 2 /2. 3

The integral of the cubic term is zero, since this is an odd function over a symmetric interval. The integral of the remainder term is 8

12b23

1

11

,.-

2 6e-tkJ.L /2 d" < t2b2 ,.-

3

1CX)

-{j

11

,.-

6e-tkJ.L 2 !2 d"

,.-

-CXl

canst = _ _ t3!2

.

Finally we replace the integral over -8 < f.L < o by an integral over -oo < f.L < oo with an exponential error. Thus we have shown that (1.7.1)

1.7.1

t--+ 00.

*Application to Bessel Functions

The modified Bessel functions lm(t) are obtained from the absolutely convergent trigonometric series CXl

etcos9

=

L

eim9 Ilml (t),

m=-CXl

or explicitly as the Fourier coefficients m=0,1,2, ....

Laplace's method can be applied here with the choice A(B) = e-ime, which is Lipschitz continuous and B(B) = cos B, which is twice differentiable with B" Lipschitz continuous.

82

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

B has a unique maximum at () = 0 with B(O) Applying ( 1. 7.1) gives the asymptoti c result fm(t) =

1 .7 .2

=

1, B' (0) = 0, B" (0)

~ [1 +0(7)].

*The Local Limit Theorem

t--+

=

-1, 8

111

(0)

=

0.

00.

of DeMoivr e-Laplac e

Laplace's method is naturally adopted to problems in the theory of probability . In the simplest probabilit y model, one considers independe nt trials of an event whose probability p of success is assumed known with 0 < p < I. For example, if we have a fair coin it is natural to take p = 1/2. If we are rolling dice then it is natural to take p = 1/6, if success correspon ds to a given face showing. More general systems of probabilit y distributio ns will be studied in Chapter 5, where we prove a more general form of the central limit theorem, generalizi ng the theorem of DeMoivre -Laplace. Assuming n trials, this random experimen t has 2n possible outcomes, consisting of strings of zeros and ones, where I correspon ds to success and 0 correspon ds to failure. The probabilit y of a given string with k successes is defined to be pk(I - p)n-k and the number of such strings is the binomial coefficien t = n!jk!(n - k)!. This is summariz ed as the statement

G)

The probabilit y of k successes inn trials is Pk.n := (;)pk(I - p)n-k. When we try to compute Pk.n we find that the maximum value occurs at the integer closest to np, and that this maximum tends to zero when n--+ oo. The local limit theorem of DeMoivre -Laplace is the following quantitativ e asymptoti c statement:

Theorem 1 .7 .1 . Let x = (k - np) I ..Jnp( 1 - p ). Then for n --+ oo, uniformly in O
This theorem shows that the individual probabilit ies tend to zero, while approxima ting a bell-shape d curve. In particular if p = I /2 and x = 0, we see that the probabilit y of an equal number of successes and failures is asymptoti c to ,.)2jnn when n --+ oo. Proof. First we represent the binomial probability function as the Fourier coefficient of an elementary function. Writing q = 1 - p, we letf(O) = p e' 0 + q. Then from the binomial theorem (1.7.2) (1.7.3)

j((})n

=

(pe'o

=

t

k=O

+ qf

(n)p*qn-kerkfJ, k

FOURIER SERIES ON THE CIRCLE

from which we conclude (1.7.4)

Pk,n

=

G)pkqn-k

= 2~

i:

e-lkf!j((:))n

83

d~.

We cannot apply Laplace's method directly, since the integrand is not presented as an exponential. To find an equivalent exponential form, we note first that for(:) =I 0, j(f)) lies on the segment joining two distinct points of the unit circle, hence lf(f))l < 1 for 0 < If) 1 :::: rr. Therefore, at the expense of an error which is exponentially small, we can restrict attention to the integral over ( -8, 8). The Taylor expansion ofj(f)) is j((:)) = I+ ip(:)- p(:) 2 /2- ip(:) 3 /6 + 0((:) 4 ),

(:)---+

0.

Choosing 8 so that 11 - j((:))l < 1/2 for If) I < 8, we can define the logarithm logj((:))

= =

ip(:) - p(:) 2 /2- ip(:) 3 /6- (l/2)[ipf)- p(:) 2/2] 2 ip(:) - pq(:) 2 /2- i((:) 3 j6)pq(q- p)

+

(lj3)[ip(:)- p(:) 2 /2] 3

+ 0((:) 4 )

+ 0((:) 4 ).

The Fourier representation (1.7.4) gives, up to an exponentially small error Pk,n

= - 1 1~ exp( n[ip(:)- pq(:) 2/2- i((:) 3 j6)pq(q- p) + 2 JT

-~

1 = - - {~ exp [ -npq(:) 2 j22rr 1-~

0((:)

4

)]-

i(:)(np

+ x~))

d(:)

ix(:)~- i(n(:) 3 j6)pq(q- p) + O(n(:) 4 )] d(:).

We can apply the first variation of Laplace's method to obtain the Gaussian approximation with c = x...Jnfiii and obtain the required result with an error of 0(1/n). To obtain the sharper error, we argue directly as follows: If we ignore the error term in the exponent, we incur an error of at most

i:

4

2

n(:) exp [ -npq(:) /2] d(:)

=

0 (n:/2) ,

To dispose of the cubic term we again employ the inequality leia - I - ial :::: a 2 /2 with a = (:) 3 j3. The integral involving (:) 3 is zero and the new error term is at most

j_:

n

2

(:)

6

exp [ -npq(:) 2/2] d(:)

=

0 ( n:12 ).

Having done all this, we have 2

Pk.n = /_: exp [-npq(:) j 2 -

ix(:)~]d(:) + 0

(n:/ 2

).

Finally, we can replace the integral over ( -8, 8) by an integral over ( -oo, oo) at the expense of an error {

1101~8

exp [ -npq(:) 2 /2 - ix(:) ~] d(:) = 0 (e-npqfJ 2 ).

From Chapter 2, Example 2.2.7, we borrow the Fourier transform of the Gaussian density:

roo exp [ -npq(:) /2 2

1-x

which completes the proof.

ix(:)

~] d(:) = ~e-x f

V;p;j

2 2

•

84

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 1.7.2. Show by direct calcuation that

lf(l9)1

< 1for 0 <

1191 =::

7r.

Exercise 1.7.3. Suppose that x = (k- np)/ ,.ji1iiij---+ oo, so that x 2 /2 -log n---+ -oo. Show that

Pk,n'"""' r,;r=== .j2n:npq in the sense that the ratio tends to 1 when n ---+ oo.

1.8

APPENDIX: PROOF OF THE UNIFORM BOUNDEDNESS THEOREM

The norm of a linear operator L is defined as (1.8.1)

ILfl } IlLII := sup{ILfl: lfl = I } = sup { T.fT :f #- 0 .

An operator is bounded if IlLII < oo. The uniform boundedness theorem is the following statement. Theorem 1.8.1. Suppose that £ is a collection of bounded linear operators from a Banach space B to a normed linear space Y with the property that for eachf E B (1.8.2)

Then sup{IILII : L

sup{ILfl : L E £} < oo. E £} <

oo.

To simplify the proof, we first prepare a lemma that will allow us to make a proof by contraposition. Specifically, we assume that (1.8.3)

sup{IILII : L

E

£} = +oo.

Lemma 1.8.2. Suppose that formulas ( 1.8.2) and ( 1.8.3) both hold. Thenforeach n > 1 there exist Ln E £ andfn E B so that (1.8.4)

Ifni= 4-n

(1.8.5)

ILnfnl > ~IILnlllfnl

(1.8.6)

ILnfnl > 2(Mn-I

where Mo = 1 and fork> 1, Mk = sup{IL(fi

+ n)

+ · · · + fk)i

:L

E

£}.

Proof. From (1.8.3), there exists L 1 E £with IIL 1 11 > 24. From the definition (1.8.1), there existsft E B with lf11 = I and ILtftl > ~liLt II· Setting/1 =fd4 shows that (1.8.4), (1.8.5) and ( 1.8.6) are all satisfied with n = 1. Assuming that/1, ... ./n-t, L 1 , ••• , Ln-l, have been defined, choose Ln E £so that IILnll > 3 · 4n(Mn-t + n), which is possible by hypothesis (1.8.3). With this choice of Ln. there exists]n E B so that Ifni = 1 and iL,J,i > iiiLnll·

FOURIER SERIES ON THE CIRCLE

Settingfn

=Jn/4n, we clearly have Ifni= 4-n, proving (1.8.4). Now IL,J;,I > ~4-niiLnll > ~4-n · 3 · 4n(Mn-1 + n) = 2(Mn-I +

85

n),

which proves (1.8.5) and (1.8.6) for the value n. This completes the proof of the lemma by mathematic al induction. To complete the proof of the theorem, we let/ = E:;':tfn, which is a well-define d element in the Banach space B, by virtue of (1.8.4). We will show that supn ILnfl = +oo. To do this, first note that

so that by the triangle inequality we have

ILnfl = ILn

(~Jk +fn + k~ 1 h

)I

~ ILJni-ILn (~~k)I-ILn (~ 1 h

)I

~

ILJnl- Mn-1 - ~ IILnll Ifni ~ ILJnl- Mn-1- ~IL,J;,I = fiLJ,.I- Mn-1 ~

which proves that

1.9

n,

•

supn ILJI = +oo, the desired contradicti on.

*APPEN DIX: HIGHER -ORDER BESSEL FUNCTI ONS

Higher-or der Bessel functions of integral order are easily constructe d beginning with the series representa tion of I 0 (2r). Differenti ating term-by-te rm, we find (1.9.1)

L

d

-/o(2r) =

dr

oo

2nr2n-1

n=l

n! 2

1 = -

17r cose e 2rcos(J de.

7r

-1r

The right side can be integrated by parts to obtain the identity

~ rzn ~ n!(n + 1)! =

I 7r

17r sm. -tr

2

e

2rcoso de

e

I, (2r)

:=

r

.

Proceedin g inductivel y, we find for each m > 1 the identity (1.9.2)

f. n=O

I

r2n

n.(n

I

+ m).

= Cm

2.n-

!7r sin2m ee2rcos0 de

for a suitable choice of the constant Cm.

-1r

== lm(2r) rm

86

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 1.9.1. Prove that Co= 1, Cm+t/Cm = 2/(2m + 1) and conclude that Cm = m!2 2 m/(2m)!form = 0, 1, ....

Equation (1.9.2) provides a definition of the Bessel function Im form = 0, 1, 2, ... as a power series convergent in the entire complex r-plane. If we interpret the factorial in terms of the gamma function, the definition can be extended to all complex values ofm.

Exercise 1.9.2. Prove that for r > 0 we have the inequality llm(2r)l < Cmrme 2r.

The higher-order Bessel functions can be recognized from the trigonometric series for the even functionf(B) = e 2rcosB. We begin with the power series expansion ~ (2r cos B)k e2rcos8 = ~ k=O k!

From the binomial theorem we have

leading to the absolutely convergent double series e2rcos8 =

f

r~

t

c.)ei(2j-k)B =

k=O k. j=O

f

t .,

r~

k=O j=O J .(k

. I ei(2j-k)B. j).

If m = 0, 1, 2, ... , the coefficient of e-imB is obtained by summing over those indices (j, k) for which 2j- k = -m. This is a line of slope +2 in the (j, k) plane, written as k = 2} + m. Thus oo r2j+m k Loa r2J = ~ = r = lm (2r) •t(k- .), ~ .,(j + )I .,(j )I (J.k):2J-k=ml· 1 · J=O 1· m · J=O J. + m ·

~ ~

rk

m = 0, 1, 2, ..

Noting thatf is even, if m = -1, -2, ... we obtain the same result with lml in place of m. Hence we obtain the absolutely convergent trigonometric series e2rcos8 = z=~lml<2r)eim8

r

E

C.

mEZ

From Parseval's identity we obtain the unexpected dividend that for r real

L mEZ

/lml (2r)2

=

_1_ { e4rcosB de

2n

JT

= lo(4r).

Bessel functions will be useful in Chapter 2 when we consider Fourier transforms in JRn.

1.10

APPENDIX: CANTOR'S UNIQUENESS THEOREM

Trigonometric series can be considered apart from the framework of Fourier series, as we have indicated in the opening section of this chapter. In most of our work we operate in

FOURIER SERIES ON THE CIRCLE

87

the framework of Fourier series. Nevertheless it is still useful and instructive to explore a larger context. The next result is a basic theorem in this direction. Theorem 1.1 0.1. Suppose that we have two trigonometric series LnEZ Ane"'-' and LnEZ Bneinx that converge to the same sum for every x E 1!'. Then An = Bn for all n E z.

By considering the difference, we can immediately reduce to the case Bn Thus we are given

0.

N

(1.10.1)

L

Ane'nx ---+ 0,

N---+

CXJ, X E

1!',

n=-N

and we must prove that An

= 0.

Lemma 1.1 0.2. Cantor-Lebesgu e theorem: If LnEzAnemx converges for all x in a set E of positive measure, then lim1n1~ XJ An = 0. Proof. Since the series converges, we have limn~"'- [Ane'nx + A_ne-""] = 0. By taking the real and imaginary parts we obtain two terms that may be written in the forman cos (nx+l/>n), for which we must prove that an --+ 0. Assume not; then there exists E > 0 and infinitely many indices n 1 < n 2 < · · · so that Ian, I 2:: E. Dividing by Ian, I. we conclude that cos (nkx+ l/>n•) --+ 0. Squaring this and using the double-angle formula, we have 1 + cos (2n* x)) ----+ 0 onE, with lEI > 0. This is a sequence of uniformly bounded functions, which we can integrate and take the limit. But if we apply the Riemann-Lebesgu e lemma to h, we conclude that the second integral tends to zero, thus lEI = 0, a contradiction. •

!(

Riemann introduces the function (1.10.2)

F(x) = Aox2 -

2

L

A~ em<.

n;tO n-

Since An ---+ 0, this is an absolutely and uniformly convergent trigonometric series, especially a continuous function whose Fourier coefficients may be retrieved by integration. In order to implement the basic hypothesis of convergence, Riemann considers the second difference quotient (1.10.3)

F(x +h)+ F(x- h)- 2F(x) - - - - - - - - : : : - - - - - - = Ao h2

+

L n,.OO

Ane

,u

(sinnh/2) nh/2

2

From exercise (1.4.28), the right side defines a regular method of summability; given (1.10.1) we see that (1.10.3) tends to zero when h ---+ 0. To complete the proof we develop some ideas of convexity to prove the following lemma. Lemma 1.1 0.3. Suppose that F is a continuous function defined on some interval so that limh-+o(F(x +h)+ F(x- h)- 2F(x))j2h 2 = 0. Then F is a linear function: F(x) = ax+ b for suitable constants a, b.

88

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. Introduce F, (x) = F(x) + Ex 2 , a continuous function that satisfies lim[F,(x +h)+ F,(x- h)- 2F,(x)]jh 2 = 2~:: > 0.

(1.10.4)

~-+0

We will prove that F, (x) is a convex function. Suppose not; then there is an interval (a, h) and a linear function g(x) = rx + s so that F, (a) = g(a), F, (b) = g(b) and F. (x0 ) > g(x0 ) for some x 0 E (a, b). The difference F, (x) - g(x) has a positive maximum at some point X max E (a, b). At this point we must have the inequality F. (x +h) +Fe (x - h) - 2F. (x) :::; 0 for small h. But this contradicts ( 1.1 0.4 ), so we have shown that F. is a convex function. But F is the limit of the sequence of convex fum:tions F., hence also convex. Applying the same reasoning to -F, we conclude that -F ~~also convex, hence F must be a linear function. To complete the proof of the uniquene~s theorem, from (1.10.3) for any h =1= 0, we may retrieve the Fourier coefficients as (sinnh/2)

A n

2

nh/2

_ _1_

2n

-

whereas

{ F(x +h)+ F(x- h)- 2F(x) h2

Ir

_ _I_ { F(x +h)+ F(x- h)- 2F(x)

A

o-

2n

J.r

h2

-in:c

e

_

dx -

o

(n =I= 0)

_ dx - 0.

•

The proof is complete.

Exercise 1.1 0.4. Suppose that LnEZ Aneinx converges to zero for all x {Xi, ... , xk}. Modify the above proof to show that An = Ofor all n E Z. Hint: F(x), defined by (1.10.2), will be piecewise linear.

E

"'I'\

CHAPTER

2 FOURIER TRANSFORM S ON THE LINE AND SPACE

2.1

MOTIVATION AND HEURISTICS

In parallel with Chapter 1, we can motivate the theory of the Fourier transform on the real line by considering an absolutely convergent trigonometric integral (2.1.1)

where

L1ccn1 d~

< oo.

In order to retrieve the coefficient function C (~) we multiply (2.1.1) by e-i 11·' and integrate over x E [ - T, T];

One cannot immediately take the limit T-+ oo without further hypotheses. If, for example, C(~) satisfies a Dini condition, then one can show that the right side converges to 2.nC(~) so that (2.1.2)

C(~) =

lim - 1 2rr

T-.oo

1T

.

f(x)e-'4 d~,

-T

which can be used to motivate the definition of the Fourier transform. The following exercise shows that C(~) can be retrieved under the hypothesis of continuity.

89

90

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 2.1.1. Show that

~loT (!~

1

Conclude that T-+ oo.

L

e-iqxf(x) dx) dt = 2

1

~;s!.~~ 2- ry) C(~) d~.

-

if C is continuous, then the right side converges to

Example 2.1.2. Let

C(~) = e-YI~I

j(x) =

for y > 0, x

E

2:rcC(ry) when

JR. Then

L

e-yl~le-lX~d~

= 2 Re

~a= e-Y~ e-ixl;

1 =2Re-y+ ix 2y

= x2

+

y2 ·

Apart from a constant, this is the Poisson kernel in the setting of the real line. The normalization is obtained from the elementary calculus integral for JTR dx I ( 1 + x 2) = :rc. Thus we have the normalized Poisson kernel · 1 Py(X) = 1T

v

y

2.

2

y > 0,

X E

+X

JR,

which has the properties of a positive approximate identity: (2.1.3)

Py(x) =::: 0,

L

Py(x)dx = 1,

{ Py(x) dx-+ 0 Jlxl>li

(y ..j... 0).

Exercise 2.1.3. Use the previous example andformula (2.1.2) to compute { e-ix~

lrrt

1 dx 1 +x 2

~ E

JR.

A more classical motivation, which goes back to Fourier, is to begin with a Fourier series on the interval [ -L, L]: (2.1.4)

f(x) ,.....,

L

Cnein:rcxfL'

nEZ

(2.1.5)

Cn =

2~ /_:_J(x)e-in:rcx/L.

These formulas will now be rewritten in terms of the functions

FOURIER TRANSFORMS ON THE LINE AND SPACE

91

Letting f.Ln = mrjL, we have (D..JL)n = rr/L so that we can write (2.1.4) as a formal Riemann sum (2.1.6) Formally taking L f(x)

---+ ~

oo, we find -1 2rr

1 lR

.

F(J-L)e' 1H dJL,

In order to have a more symmetrical theory, we let JL = 2rr I; in the first integral, to obtain (2.1.7)

f(x) ...__

i

F(2rr!;)e2rrtl;x dl;,

This symmetrical form will be in force in the systematic approach beginning in Section 2.2. In case f is real-valued, (2.1. 7) can also be written in terms of real-valued functions by writing F(JL) = HA(JL) - iB(J-L)] to obtain (2.1.8)

f(x)

'""1=

[A(2rrl;) cos(2rrl;x)

+ B(2rrl;) sin(2rr/;x)] dl;.

The above transformations are purely heuristic, with no pretense of rigor. We will show in the following sections that they can be systematically developed to obtain a powerful theory of Fourier analysis on the real line and in Euclidean space.

2.2

BASIC PROPERTIES OF THE FOURIER TRANSFORM

'

In order to formulate an unambiguous theory, we begin with a complex-valued , Lebesgue integrable function on Euclidean space, denoted by f (x), x E Rn. The Fourier transform is the complex-valued function](/;), I; E Rn defined by the integral (2.2.1)

f(l;) = (Ff)(!;) := {

f(x)e-2rril;·xdx.

jTP/'

The Euclidean dot product is I; · x = L.j= 1 t;1x1 . The basic elementary properties of the Fourier transform are summarized in the next statement. Proposition 2.2 .1 . Let f E L 1 (R n). Then

• Continuity: I; 1--4 f(l;) is a uniformly continuous function. • Contraction: The mapping f ---+ J is norm-decreasing from L 1 to L 00 , in the sense that

(2.2.2)

IFf(!;)! <

l.

lf(x)l dx := !Ifill.

• Linearity: If F1 is the Fourier transform off1 and F 2 is the Fourier transform of h. then a1F1 + a2F2 is the Fourier transform of a 1 f 1 + a 2 f 2 , for any choice of the complex constants a 1 , a 2 .

92

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

• Translation and Phase Factor: If F is the Fourier transform off, then e-Zrria·l; F(~) is the Fourier transform off(x- a), and F(~ +b) is the Fourier transform of e-Zrrib·xf(x). • Multiplicatio n and Convolution : If F 1 is the Fourier transform offJ and Fz is the Fourier transform ofh, then F 1 F 2 is the Fourier transform of(fJ * fz), where the convolution of two functions is defined by the integral ( /1

* f2)(x)

f f 1 (y)f2(x- y) dy. JJR"

=

• Differentiati on and Multiplicatio n: If the partial derivative (8f j8x ) exists and 1 is in L 1 (lRn), then 2ni~;}(~) is the Fourier transform of(8f jax ). 1 If x1 f E L 1 (JR."), then (aj ;a~1 )(~) exists and is the Fourier transform of -2nixJf(x). • Fourier Transform of J!adial Functions: If f(x) = cp(lxl) for some cp E L 1 (JR+; rn-Jdr), then f(~) = 1/F(Igl) for some 1/F E C(JR+). Restated, the Fourier transform of a radial function is again a radial function. Proof. To prove the uniform continuity, we write

f(l.; +h) - f(t;) =

1

e-2rrt<·-"]f(x) dx

[e-2lTt(l:+h)·.• -

IR"

If(~+ h)- f(~)l ~

11e(1 + 1 ) 2

"'h·x -

111/(x)l dx

IR!'

=

txl:sM

le- 2"ih·x

-

111/(x)l dx.

txi>M

Given E > 0, the second integral can be made less than The first integral is majorized by 27Tihl {

ftxt:sM

E

by taking M sufficiently large.

lxllf(x)l dx.

Therefore with the above choice of M, we have lim sup sup If(~ +h) -f<~)l ~E. h-+0

~ER''

But E was arbitrary, which proves the uniform continuity. The linearity and phase factor properties are direct computations . The existence of the convolution of two L 1 functions is guaranteed by the Fubini theorem, when we consider the functionf1 (z)/2(y) in JR 2n and make the substitution z = x - y:

oo

>

{

}Jil211

l/2(y)/J(Z)Id zdy= {

}fi(ll

=

l/2(y)! ( {

}IRn

1 ( { lfi an

lr.-t'

1

1/J(x-y)!dx )dy

(x- y).f2(y)l

dy) dx.

FOURIER TRANSFORMS ON THE LINE AND SPACE

93

The joint measurability of the product/1 (z)f2(y) is established by writing.fi (z) = limN Lkezkz--N1k2-NS"< z-N and similarly for fz(y). The product is then written as a pointwise limit of simple functions, especially jointly measurable. Therefore the convolution is finite almost everywhere and defines an L 1 function with II/I *h II ::::; 11/IIIIII/z III· Now we can compute Fz (5 )FI (5)

=

l

.. 2

=in =

ln

e-zrr.~-h ( y)/I (z) dz dy

e-2rrtl; Y!z(y)

(in

e-2rrtl; (x-y)II (x- y)

e-ZrrtH (LJI (x- y)fz(y) dy)

dx)

dy

dx.

The differentiation and multiplication properties will be proved below in a more amplified context. Finally, iff is a radial function, then we consider a rotation 'R in the ~-space, making ihe change-of-variabl e y = 'R' x with IYI = lxl, dy =

dx:

]CR5) = { ~(lxl)e-2rrtRI; x dx

Jan

= which shows that

{

}'Jtn

= { ~(lxl)e-2rrii;-R'x dx

Jan

~(lyl)e-zrr,~-y dy =f(~).

J is invariant by rotations, hence a function of 1~ 1.

•

We will also need a form of the Fourier reciprocity formula. Lemma 2.2.2. Suppose thatf E L 1 (IRn) and 1/r E L 1 (JRn) are integrable functions with Fourier transforms j and ~. Then we have the identity (2.2.3) Proof. We use the Fubini theorem to write { 1/f(~)f(5) d~ = { 1/f(~) ( { f(x)e-Zrril; xdx) d~ JlR, Jan }R.n

= =

which was to be proved.

1n (1.n 1/f(~)e-27ril;xd~) f(x)

{

}Rn

f(x).(f,(x)

dx

dx,

•

This applies in particular to a convolution operator with respect to a kernel which is written as an L 1 Fourier integral. In detail, if K(x) = JR.n k(ne2rri~·x d~ with k E L 1 (JR"), then (2.2.4)

94

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 2.2.3. If 1-L is a finite Borel measure on JR.", its Fourier transform is defined as jl(l;) := fl!?.n e- 2 7Ti~ x !-L(dx). Prove that if /-L. v are.finite Borel measures on JR.", then we have the Fourier reciprocity formula fii?.n v(x) !-L(dx) =fiR" fl(l;)v(dl;).

2.2.1

Riemann-Lebe sgue Lemma

The most basic analytic fact about the Fourier transform is the Riemann-Lebesg ue lemma, expressed as follows.

Theorem 2.2.4. For any f

E

L 1 (JR.")

lim f(l;) = 0,

I~ I~=

and the convergence is uniform on compact subsets of L 1 (JR."). Proof. From inequality (2.2.2), we need only prove this for a dense set of functions in the L 1 norm. From the dominated convergence theorem,

1

IJ(x)l dx-+ 0,

1-M

1

lf(x)l dx-+ 0

{x·IJ (x)l >M}

when M -+ =, so that we can approximate f in the L 1 norm by a bounded measurable function 1, which is supported on the cube [- M, M]". By rescaling we may suppose that M = 1. Now can be uniformly approximated by the simple function

1

2N

j=

L

k2

-N l{(k-1)2-N 4-sk2-N)

k=-2N

111-111

so that 1 :::: 2 x 2-N. Now we will prove that the theorem holds for any simple function,!= 2:~= 1 ck IE, where Ek are measurable subsets of [ -1, 1]" and Ck are arbitrary complex numbers. In case of an indicator function j(x) = n;=J l
/;; =1- 0, which clearly tends to zero if at least one of the coordinates 1;1 -+ =·By Proposition 2.2.1, the same is true i f f is the indicator function of a finite union of intervals. Now if E is any measurable subset of [ -1, 1]", there exists a finite collection of open cubes {C1 } with union U = U1 Cj so that the Lebesgue measure of the symmetric difference E ~U is less thanE. In particular Ill u - 1£ II 1 < E so that IFOu)(l;)- F(l£)(01 ::0 lllu- lEIIt <E.

Therefore, Fl£(1;) -+ 0 when 11;1 -+ =·This extends immediately to finite sums/ = L~= 1 c, 1£, and the proof is complete. To check the uniform convergence on compact subsets. follow the same reasoning as in the proof in Chapter 1. •

For a general L 1 function, there is no universal rate of decrease to zero. However, if wt asssume some differentiability, we have the following fact.

FOURIER TRANSFORMS ON THE LINE AND SPACE

95

Proposition 2.2.5. If the partial derivative (aj jaxi) exists and is in L 1 (lRn), then 2rri~1 j(~) is the Fourier transfonn oj(aj ;axJ)(x). In particular

lim ~1 ](~) = 0.

I~

1--+oo

More generally ifot = (ot 1 , ••• , otn) is a multiindex, and the mixed partial d"rivative Irf E L 1 (lRn), then (2rri~yxJcn = :F(Dcxf)(~). Proof. Without loss of generality suppose thatj = 1. First, note that by Lebesgue theory, = .{yx fx, (z, x2, ... , Xn) dz, which shows that/ must have a limit when x 1 ~ oo, which can only be zero, since/ E L 1 (.!Rn). Now we apply partial integration for ~ 1 =I= 0:

f(x, x2, ... , Xn)- f(y, x2, ... , Xn)

j(~) = {

lntn

M 1 -M

f(x)e-2'nt; x dx

f(x)dx 1

(

=

lim

{

f(x)e-2rrit; x dx

M---CX'J 1/xi/.:;::M

e-2rrtl; x ) e-2rril;·x . =f(x) .

-21rl~I

-21rl~I

00

~

1

-oo

fx 1 (X)

IM 1M /x, -

-M

(x)

-M

e-2rri/;·x . dx1

-21rl~I

e-2:n:t/;·x . dxt. 2rrl~t

Integrating this over x 2 , ..• , Xn proves the identity 2rri~J(~) = :Fif.,, )(~). Repeating this procedure and using mathematical induction shows that for any multiindex ex, (2:rri~)'xf
•

A dual property is afforded by the next statement, where loti := ot 1 +

Proposition 2.2.6. Suppose that

fiR"

· · · + otn.

lxlklf(x)l dx < oo. Then] is differentiable

to order k and

loti< k. Proof. To prove the formula fork = I we apply the dominated convergence theorem, using the inequality Ie;(! - II ~ 10 I for 0 real. The general result follows by mathematical induction .

•

Example 2.2.7. A basic example with n = 1 is provided by the Gaussian density function f(x) =

2

e-rrx '

To see this, we differentiate the Fourier transform:

96

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

But the Gaussian density satisfies the differentia l equation/ ' = -2]'( xf(x), so that we may apply Propositio n 2.2.5 and integrate by parts to obtain }'(I;) =

l( l

= i

-2]'( ix)e-2rrit;x f(x) dx e-2rrit;xj' (x) dx

= -2]'(1;

l

e-2rril;xf(x ) dx

= -2]'( ~; Jc~; >.

The unique solution of this equation with the condition }(O) = 1 is}(/;) = the example is complete.

2

e-rr/;

,

and

Exercise 2.2.8. Show that}(O) = 1. Hint: Do the double integral off (x) f (y) in polar coordinates .

This example may be immediate ly extended to lRn, to obtain the Fourier pair (2.2.5) The Gaussian density provides a concrete example of Fourier inversion. (2.2.6)

Exercise 2.2.9. Check the previous statement from the formulas above. The Gaussian density is a simple example of a rapidly decreasin g function. To define this notion in general, introduce the seminorm s

llfllk.m := sup(l

+

lxl)ki.D"f( x)l

xERn

where at= (otl, ... , otn) is a multiindex , Da = (ojoxl)a' · · · (ojoxn)'"", and m =loti:= + · · · + otn. The Schwartz class S of rapidly decreasing functions is defined as the set of complex-v alued functions that are infinitely differentia ble and for which 11/llk.m < oo for all m, k. From Propositio n 2.2.5 and Propositio n 2.2.6, the following proposj.tio n is immediate . ot1

Propositi on 2.2.1 0. The Fourier transform maps the space S into itself In the next subsection we will show that this mapping is 1: 1 onto the space S. The Gaussian density example in n = I can be transform ed into additional examples by successive differentia tion of the Fourier transform .

FOURIER TRANSFORMS ON THE LINE AND SPACE

Thus

l i

i

97

e-rrx2 e-2rril;x dx = e-:r/;2.

e-:rrx~ ( -2TCix)e-Zrril;x dx

2

= ( -2TCI;)e-JT/; ,

e--rrx 2(-2TCix) 2 e-zrr,l;xdx = [(-2TCl;) 2

-

2TC]e-rr1;

2 ,

and so forth. This is closely related to the Hennite functions, which will be discussed in Section 2.4.5.

2.2.2

Approximate Identities and Gaussian Summability

On the circle we had the Abel and Cesaro means to regularize the convergence of Fourier series. Suitable analogues of these exist in the context of one-dimensional Fourier transforms, and will be treated. However there is a more natural and symmetrical approximate identity, the Gauss- Weierstrass kernel, which applies in higher dimensions as well as in one dimension. From the example of equation (2.2.5), we replace x by x / .../4ifi to obtain the identity

(2.2.7)

This is also called the heat kernel, since it is a solution of the equation aH ;at =

LJ=I a 2 HjaxJ. It has the following three basic properties: (2.2.8)

H 1 (x) > 0,

1

H 1 (x) dx-+ 0

(t-+ 0).

[x[>8

Exercise 2.2.11. Prove that H,(x), defined by the integral (2.2.7), satisfies the n-dimensional heat equation. Exercise 2.2.12. Check the three properties (2.2.8).

Notation. We will use the notation HJ for the convolution H 1 *f. It will always be clear from context whether we are dealing with the kernel or with the convolution operator. Applying the Fourier reciprocity formula (2.2.4) to (2.2.7) yields a useful identity forHJ:

98

INTRODUCTION TO FOURIER ANALYSIS AND WAVELFTS

Definition 2.2.13. An approximate identity in IR.n is a family of functions k,(y) defined fort in some directed index set, with the following three properties:

r lk,(y)jdy < lim r k,(y) dy = 1 }TR_n lim r lk,(y)l dy = 0. lu.vl sup

(2.2.10)

f

(2.2.11)

CX)

}Rn

1

(2.2.12)

r

>8J

These properties are clearly satisfied by H,, for example. Exercise 2.2.14. Show that the Fejer kernel, which is defined on JR. by kT(x) (1 -cos Tx)jnTx 2 , is an approximate identity, where T E (0, cx:>) the limits are taken as T-+ cx:>. Assume known that jiR(l - cosx)jx 2 dx =Jr. Exercise 2.2.15. Show that the Poisson kernel, which is defined on IR by ky(x) = yj[n(y 2 + x 2 )], is an approximate identity, where y E (0, cx:>) and the limits are taken as y - 0. Assume known that fiR 1/(1 + x 2 ) dx =Jr. Exercise 2.2.16. Suppose that k 1 (x) is an approximate identity. Prove that iff is a boundedfun ction with limx---.of(x) = L, then lim, JR" k 1 (x)f(x) dx = L. The examples of Gauss, Poisson and Fejer can be abstracted as follows. Exercise 2.2.17. Suppose that K E L 1 (lR.n) with K(x) ~ 0, JR" K(x) dx = 1 and set k 1 (x) = t-n K(x/t). Prove that k, is an approximate identity, where the limits are taken as t -+ 0. A homogeneou s Banach space B is a Banach space of complex-val ued functions on !Rn whose norm satisfies the properties

11/yliB

= 11/lls,

'Vy

E

IR.n

and

lim llh

y-+0

-fils= 0

where_{y is the translation off, defined as_h.(x) =f(x- y). The following examples of homogeneou s Banach spaces occur frequently. Example 2.2.18. B = LP(.lJ.~.n), 1 :::: p < cx:> is a homogeneou s Banach space. This follows from the translation invariance of the Lebesgue integral and a density argument, beginning with continuous functions with compact support. Example 2.2.19. B = Buc(IR.n), the space of bounded and uniformly continuous functions on IR.n with the supremum norm, is a homogeneou s Banach space. Clearly the norm is translation invariant. The continuity of y -+ h is equivalent to the definition of uniform continuity of f.

99

FOURIER TRANSFORM S ON THE LINE AND SPACE

Example 2.2.20. B = C 0 (IR"), the space of continuou s functions vanishing at infinity, is a homogene ous Banach space. As a closed subspace of Buc(IR"), B is a homogene ous Banach space. In the following theorem we will assume, without loss of generality , that the limit in the directed index set is taken as t ~ 0.

Theorem 2.2.21. Suppose that B is a homogene ous Banach space and k, is an approxima te identity. Then f'J<'.n k,(y)f... dyE B whenever fEB and Jlii" k,(y)fy dy ~fin the norm of B when t ~ 0. Proof. fan k,(y)f.. dy can be computed as the limit of Riemann sums. But any finite sum is a linear combinatio n of elements of B. hence in B. The B-norm of any finite Riemann sum is bounded by 11/lls times the Riemann sum for fan lk,(y)l dy which is uniformly bounded. From this it follows that the Riemann sums converge in the B-norm. To study the convergenc e when t ~ 0, we write

{ k,(y)if... II JIR,

-f)dyll

.:::0 B

{

]!R.n

k,(y)llf.

= ( { J{ly)Sii)

-!IIndy

+ {

) lk,(y)lll/, - !IIndy.

J{)\)>li)

The integral over IYI > 8 tends to zero for any 8 > 0. Given E > 0, we choose 8 > 0 so that II f.. - f II 8 < E for /y I < 8. Then the first integral is less than C E. With this choice of 8, we have lim sup t---+0

for any

E

II f

'JR."

k,(y)(J; - j)

dyll

.::=: CE B

•

> 0. The proof is complete.

Corollary 2.2.22. 'i): Iff E Buc(IR"), then Hrf converges uniformly tof: IIHrf- flloo ~ 0 when t ~ 0. ii): Iff E U'(IR"), I < p < oo, then Hrf converges in LP: IIHrf- fliP~ 0 when t ~ 0. Proof. These follow immediatel y from Theorem 2.2.21 when applied to the heat kernel on the spaces B = B"' (lR") and B = LP(JR"). •

Corollary 2.2.23. Iff

E

L 1 (IR") and]= 0, thenf = 0 a.e.

Proof. By Fourier reciprocity, the heat kernel convolution operator can be written

(2.2.13) If]= 0, then

HJ = 0 and by Corollary 2.2.22 f = 0 a.e., which was to be proved.

•

Exercise 2.2.24. Let B be the space of complex-v alued functions for which frr<.(lf(t)l )/(1 + t 2 ) dt < oo. Consider the Poisson kernel operator P,: f ~ r · n:- 1 JRYf(t)d tj(y 2 + ( x - t) 2 ). Show that P, maps B to Band that Pvf ~ f ~~y~Q

.

100

lNTRODUCIION TO FOURlER ANALYSlS AND WAVELETS

2.2.2. 1

Improved approximate identities for pointwise convergence

In the case of one dimension, we can obtain pointwise convergence at additional points under additional conditions on the functions k,, assumed to be of the form r 1K(xjt). From Exercise 2.2.17 any such kernel is an approximate identity, hence we have convergence at points of continuity of a test function/. The following theorem formulates supplementary conditions for convergence at additional points. Theorem 2.2.25. Suppose that K is absolutely continuous with K(x) JJRK(x)dx = 1, K(-x) = K(x) andxK'(x) < 0. Then

>

0,

• (i): x-. ( -2xjt 2 )K'(xjt) is an approximate identity on [0, ex:>) . • (ii): lim,_.o t- 1 JR K(xjt)j(x) dx = L iflimx----o0/2x) f~x f(t) dt = L. Proof. By partial integration, for any M > 0 we have

(2.2.14)

MK(M) =1M K(x) dx +1M xK'(x) d-..:.

When M ~ oo, the first term on the right tends to 1/2 and the second term has a limit E [ -oo, 0). But. K(x) 2: 0, hence there exists C = limM _. 00 MK(M) E [0, oo). Since J00000 K(x)dx = l/2 < oo, we must have C = 0. Taking M ~ oo in (2.2.14) shows that j 0 xK' (x) dx = -1/2. Now for any 8 > 0,

1

-

X 1 -K 2

lxl;>:8 f

which tends to zero when t

~

(X) - dx = f

1

lvl;>:~/1

yK 1 (y) dy

0. In addition,

-i ~K' (~)

dx = -

i

yK'Cy)dy = 1

J;

which proves (i). Toprove(ii), wedefineF(x) = f~x f
1

1 -K

IRI

(X) - f(x) d-..:= t

1

= -

X) dF(x)

00

} -K (-

o

t

t

{oc ~ K' (~) F(x) dx

lo

r-

x

t

where we have used the bounds on F(x) and K(x) to discard the term at the limits. But F(x)/x -~ 2L and ( -2xjt 2 )K1 (x/t) is an approximate identity on [0, oo), so that (ii) follows from the basic properties of approximate identities (Exercise 2.2.16).

Replacing /0 by f(x theorem.

+ ·),

•

we obtain a more generally applicable form of the

Corollary 2.2.26. Under the above conditions on K, we have for anyf . 1 hmt--->0 t

1

K

R

(Y) - f(x + y)dy = l i ml- lx+h f(u)du t

h-~o

2h

x-h

wherever the latter limit exists, in particular for almost all x

E

JR.

E

L 1 (.IR),

FOURIER TRANSFOR MS ON THE LINE AND SPACE

101

Note that the set of admissib le x is precisely the set of points where the fundamental theorem of calculus applies, in the sense of the symmetr ical limit. In general this is strictly larf?er than the Lebesgu e set, defined as the set of points x where limh---*o (lf2h) if(u)- f(x)i du = 0.

J:::

Exercise 2.2.27. Show that the Poisson kernel, with K(x) = 1/.rr(l +x 2 ) satisfies

the conditio ns of Theorem 2.2.25. In particul ar

t f(y) dy = lim - 1 1x+h f(y) dy 2 r~o .rr J'R + (x - y) h--o 2h x--h whereve r the latter limit exists, in particul ar almost everywh ere.

lim

2_ {

t2

Exercise 2.2.28. Prove that the one-dim ensional Gauss kernel satisfies the conditions of Theorem 2.2.25, in particul ar lim r--o

1

{ e-<x-y) 2 / 41f(y)dy =lim__!_ _ 1x+hf( y)dy

....l41rf }!R

2h

h---*0

x-h

whereve r the latter limit exists, in particul ar almost everywh ere.

Exercise 2.2.29. Prove that the Fejer kernel does not satisfy the conditio ns of Theorem 2.2.25. Theorem 2.2.25 does not apply directly to the Fejer kernel, where K(x) = (1 - cosx)j.rr x 2 • In order to deal with this and other oscillato ry kernels, we introduc e the notion of monoton e majoran t. This is a function k E L 1 (IR), which satisfies (2.2.15)

IK(x)i < K(x),

K(-x) = K(x),

xk'(x) < 0.

J'R

Note that we do not require that K(x) dx = l, so that we cannot expect, for example, that r 1K(xjt) be an approxim ate identity. Howeve r we have the followin g useful replacem ent for Theorem 2.2.25. Lemma 2.2.30. Suppose that k x > 0 with limx-+O E(x) = 0, then lim t---+0

1

=

0

E L 1 (IR)

satisfies (2.2.15).

X - (X) - €(X) dx =

K' 2 t

t

If IE(x)l

0.

Proof. Followin g the steps of the proof of Theorem 2.2.25 with K replaced by that limx-~oo xk(x) = o.- .f/:0 xK'(x) dx ~ c < ex>. For any o ~ o

(2.2.16)

-

8

i

x-,(x) 2K - dx = -K 8t t t

00

t

t8

t

Taking = 0 shows that - f 0 (x/t )K'(xjt) right side of (2.2.16) tend to zero when t ----*

IJr= ~ k' (~) 0

First take

t -

f

t

ex> and then 8 -

2

E(x)

dx!

k, we see

8-(8) +-1[ K-(X) - dx. dx ~ C. For any o 0, both terms on the 0. Therefore

00

00

< M for

~

c sup IE(x)l + M O:;ox~8

0 to complete the proof.

>

lioo ~k' (~) 8

t2

dxl,,.

f

•

102

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

This allows us to reformulate and prove Theorem 2.2.25 as follows.

Theorem 2.2.31. Suppose that K defines an approximate identity with a monotone majorant satisfying (2.2.15). Then

(2.2.17)

(~)f<x + y)dy t

lim.!._ { K t---+0

t

}[I;

at every xfor which limh-o0/2h) J~h lf(x Proof. Without loss of generality, we takex

write

~

L

K (

~) f (y) dy =

L

+ y)

= L

- Ll dy = 0.

= 0 in the proof. Since JR K(x) dx =

+

~

L

K (

~) (j (y)

I, we can

- L) dy.

It remains to show that the last term tends to zero. For this purpose, define F(x) = J~. if(u) -LI du, so that F(O) = 0, F(x)/ixl -. 0 when x-. 0 and IF(x)l :::: 2Lixl + llfll•· Then

1 = -t2

1=- (y) K'

t

0

F(y)dy

where we have used the bounds on K(x), F(x) to discard the terms at the upper limit in the integration-by-par ts. Now we use Lemma 2.2.30 with €(y) = F(y)jy: _;. ("" K' t-

Jo

(~)

F(y) dy =

t

_!_2 {oo yK' (~) t

Jo

t

F(y) dy-+ 0, Y

completing the proof.

•

Exercise 2.2.32. Show that for the Fejer kernel, with K(x) = (1- cosx)jrrx 2 we may take

k

(x) = I/ ( 1 + x 2 ) as a monotone majorant.

Hint: Check that 1 - cosx:::: rrx 2 /(1

+ x 2)

for all x. Consider separately lxl :::: 1 and lxl > L

Corollary 2.2.33. If the approximate identity defined by K(x) has a monotone majorant, then for any f E L 1 (1R), and x E Leb(f) lim_!. { K t---+0

t

}[I;

(~)f(x + y) dy =f(x). t

We now return to the development of the Fourier transform.

2.2.2.2

Application to the Fourier tra"sform

Theorem 2.2.34. The Fourier transform is a 1:1 map from the spaceS onto itself In particular every f E Scan be recovered from its Fourier transform as f(x) =

{

lnt"

e 2 "'~·xj(~) d~.

FOURIER TRANSFORMS ON THE LINE AND SPACE

103

Proof. Since the Fourier transform is a linear map, it suffices to show that] = 0 implies

f = 0 a.e., which was just proved in Corollary 2.2.23. To prove the onto property, note that since] E S, in particular in L 1 , so that we can take the limit under the integral in (2.2.13) to obtainf(x) = JR.J(s)e 271 ' ds, which represents/ as the Fourier transform of the • reflected function x--+ ]< -x). This proves the onto property. The almost everywhere convergence of the heat kernel transform cannot be proved from the abstract properties of approximate identities. The one-dimensional case is covered by Exercise 2.2.28. For then-dimensional case the statement and proof follow:

L 1 (JR.n), then lim~-+o HJ(x) =f(x) where limr-..o r-n ~y-xl'5./f(y)- j(x)) dy = 0, in particular almost everywhere.

Proposition 2.2.35. Iff Proof. We let (r) :=

1

E

(f(y)- j(x)) dy =

Jy -xl:sr

r{

Jo

(f(x

+

pw)- f(x))pn-l dpdw,

Jsn-1

a continuous function of bounded variation. Clearly l(r)l ~ J'fif!' lfl hypothesis (r)/r" --+ 0 when r--+ 0. Now we can write H,f(x) - f(x) =

1

(f(x

+ y)

- f(x)) e

1 = 1

(47rt)n/2

0

d r

( )

re-r 2 f4t

oo

(r)

0

dy

e-r2f4t

00

-

and by

(47rt)

IR"

-

-lvl 2 /4t n/ 2

+ Cnrnlf(x)l

2t(47r t)n/2

dr.

To prove that this tends to zero, let E(r) = (r)jr", a bounded function that tends to zero when r--+ 0. Then we make the change of variable r = st 112 to obtain

1

re-r2f4t

00

IH,f(x)- f(x)l ::S

0

foe

= Jo

rnjE(r)l

~

2t( 47r t)nf-

dr

e-•214

IE(sv'f)lsn+l 2(4.7r)n/2 ds.

The integrand tends to zero and is bounded by an integrable function, completing the proof.

•

The Lebesgue set of a function/ (2.2.18)

Leb(j)

=

{x

E

E

L 1 (JR.n) is defined by

lf(y)- f(x)l dy IR.n : lim r-n { Jly-xl'5_r r-+O

= 0}.

Clearly X E

Leb(j)

==>

!~ r-n I-xl'5.rj(y) dy =

C,J(x)

where the dimension constant Cn is the volume of the unit ball in JR.n. But the converse is not true. in general. [Consider f(x) = lr-t.tJ sgn (x)/../lXT at x = 0 in JR.] In this

104

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

terminology . Proposition 2.2.35 implies that Ht.f(x) --* f(x) for all x E Leb(f). This example is an instance of the fact that many of the almost-every where results in harmonic analysis are proved on the Lebesgue set.

Exercise 2.2.36. Extend the result of Proposition 2.2.35 to f p <

E

LP (IRn) where 1 <

00.

Hint: Use the HOlder inequality to check the bound on (r).

We can use Gaussian summability to extend Fourier inversion beyond the spaceS.

Proposition 2.2.37.

(i) Suppose thatf E L 1 (IRn) has an integrable Fourier trans-

form: j E L 1 (1Rn). Thenf is almost everywhere equal to a continuous function, and we have f(x) =

(2.2.19)

{

}JRn

(ii) Conversely, supposethat S(x)

f(x) a.e.

j(~)e2rri~·x d~

a.e.

= limM ~~l:sMj(~)e 2 rri~·x d~ exists. ThenS(x) =

Proof. From Fourier reciprocity, we have

Appealing to Proposition 2.2.35 shows that the left side tends to f(x) almost everywhere, whereas the right side converges by the dominated convergence theorem. For the converse, letS, = J;~ ~,f(~)e 2 :n:i~·x dl;, the so-called spherical partial sum. 1 Using S 0 = 0 and Soo = S(x), we can write

1oo = 1oo

HJ(x) =

1

00

HJ(x) - S(x) =

e-4:n:2rr2 dS, 8Jr2tre-4:n:2rr2 S, dr 21 2 81r 2tre- 4:n: ' [S, - S(x)] dr.

It is easily checked that r ---+ k,(r) := 81r 2tre- 4:n: 2'' 2 has total integral 1 and that for any M > O,lim,_.o foM k,(r) dr = 0, from which the result follows immediately. •

Proposition 2.2.38. Suppose that f E L 1 (lRn) has a nonnegative Fourier transform: j(~) > 0 and 0 E Leb(f). Thenj E L 1 (lRn), in particular f is a. e. equal to a continuous function.

Proof. Take x = 0 in (2.2.13) and apply Fatou's lemma.

•

Warning. The alert reader may note the difference between the notion of

"almost everywhere equal to a continuous function" and the weaker notion of "continuous almost everywhere. " For example, the indicator function of an interval is continuous

FOURIER TRANSFORMS ON THE LINE AND SPACE

105

almost everywhere, but it is not a.e. equal to a continuous function. Continuity a.e. is a local notion, whereas the stronger concept is a global notion.

Exercise 2.2.39. Suppose that f E L 1 (JRn) has integrable partial derivatives of order n + 1: Daf E L 1 (JRn) for any multiindex with Ia I < n + 1. Prove that J E L 1 (lRn) and hence Proposition 2.2.37 applies to give Fourier inversion. We will see in the next section that pointwise Fourier inversion holds with n/2 derivatives instead of n + 1. The heat kernel convolution operator, which was initally defined on the space L 1 (lRn), can be naturally extended to functions satisfying either of the growth conditions (2.2.20) or (2.2.21)

Proposition 2.2.40. Suppose that f satisfies either (2.2.20) or (2.2.21). Then u = H ,f is defined for 0 < t < B /4 and is a solution of the heat equation u, = Uxx for which lim~-+o H,f(x) =f(x) at every point of continuity of f. Exercise 2.2.41. Prove these statements, noting that one must work directly with the heat kernel transform, since the Fourier transform is not applicable to this class offunctions. The heat kernel can be used to prove that S is dense in each of the LP spaces, 1 < p < oo. First note that the set of LP functions with compact support is dense in LP. Now iff has compact support, the Fourier transform] is infinitely differentiable with bounded derivatives. For such anf and any multiindices a, f3 we can write

H,f(x) =

D~H,f(x)

=

(2rrix) 13 D~H,f(x) =

{

e2lfix·q(~)e-4Jr 2 ti~i 2 d~

{

(2rri~)ae2lfix·q(~)e-4Jr2ti~i2 d~

}Rn }Rn

l"

D:

(e2lfix·~) [ (2rri~)af(~)e- 47f 21 1~1 2 ] d~

= (-1)1/311."

e2lfix·~v: (< 2 rri~)aj(~)e-4Jr 2 ti~i 2 )d~,

which is a bounded function. This proves that H,J E S. But we proved that for any LP function IIH,J- fliP-+ 0 when t-+ 0. We summarize as follows.

Proposition 2.2.42. The spaceS is dense in each LP space, I < p < oo. Note that the case p = oo is excluded, because we cannot assert that HJ -+ f for every f E L 00 •

106

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

Exercise 2.2.43. Letf = lr-I.IJ with n = 1. Show that UHJ- /lloo does not tend to zero when t -+ 0 One should not get the impressi on from the above proof that the heat kernel operator maps LP into S. We expect that for small t the decay of Hr.f at infinity should mimick that off.

Exercise 2.2.44. Letf be the Poisson kernel inn= l:j(x) = lj:rr(l +x2 ). Show that HJ(x) ,. . __ Cjx 2 , !xi-+ oo,fora positive constant C. Hint: Write

Integrate by parts twice and identify the constant.

When we pass out of the space S the problem of pointwis e Fourier inversion becomes nontrivia l. In the next section we will treat the one-dim ensional case, in parallel with the treatmen t of Fourier series in Chapter 1. We will return to the higher-d imension al case in a separate section.

2.2.2.3

The n-dimen sional Poisson kernel

Then-di mension al Poisson kernel is defined as the absolute ly converg ent Fourier integral (2.2.22)

y > 0,

X

E

:R".

By different iation under the integral sign, it is immedia te that u = P(x, y) is a solution of the Laplace equation Uyy + L7= 1 Ux,x, = 0 in the half space {(x, y) : x E R", y > 0}.

Exercise 2.2.45. Suppose that FE L 00 (R"). Prove that

is a solution of Laplace 's equation

Uyy

+ L7= 1 Ux,x,

= 0 in the half space.

We have already shown, in case n = 1, that P(x, y) is an approxim ate identity, by an explicit computa tion. We now obtain an explicit formula for the n-dimen sional case. The general idea is called Bochner 's method of subordin ation, which allows us to obtain new kernels as suitable transform s of the heat kernel in the t-variabl e. In terms of operatio nal calculus , the heat kernel operator H, is the exponen tial of t times the Laplace operator , whereas the Poisson kernel operator Py is the negative exponen tial of y times the square root of the negative of the Laplace operator . Since the exponen tial is a simple and basic function , it is natural to expect that other kernels can be obtained by suitably transform ing the heat kernel. The method will also be applied in later sections to compute the Newtoni an kernel associat ed with Laplace 's equation and the more general Riesz kernels.

FOURIER TRANSFORM S ON THE LINE AND SPACE

107

To compute then-dime nsional Poisson kernel, we begin with the Laplace transform of the one-dimen sional heat kernel:

=

1

e-1~1 2 f4r

e-1~1...;;:

--==-- e -AI dt = - - = A > 0, l; E JR. o v'4iri 2--fi. This is proved by taking the one-dimen sional Fourier transform of both sides, which is justified by the Fubini theorem. Indeed, for the left side we have (2.2.23)

[

JTR

e2n:t~x

roo (e-1~12/4! e-At dt) dl; =

lo

v'4iri

{CX' e-Ale-4rr2rx2 dt =

lo

1 A+ 4rr 2x 2 '

whereas the Fourier transform of the right side is {

JTR

e2n:i~xe-llil...;;: 2--fi.

dl; =_I_ ( 1 ) +_I_ ( 2--fi. -.fi. - 2rr ix 2--fi. 1 A+ 4rr 2x 2 '

1

-.fi. + 2rr ix

which proves (2.2.23). Now we apply (2.2.23) with A 112 = 2rry and l; (2.2.24)

e-ll;l2n:y - --- = 4rr y

1oo

e-1~1 /4r

o

v'4iri

E

)

JR.n to obtain

2

e -4JT2},2! d t,

Finally, we compute the n-dimensi onal Fourier transform of both sides. In detail, we multiply (2.2.24) by e 2 rri~-x and integrate over l; E JR.n. On the right side we recognize the n-dimensi onal Fourier transform of the heat kernel, corrected by the factor (4rrt)
=

1oo

(4rrt)(n-llf2e-4n:2rlxl> e-4n:2ry2 dt

(4rr)
+

1)/2)

Therefore we have the following explicit formula (2.2.25) Formula (2.2.25) shows that the functions x -+ P(x, y) define an approxima te identity fiR" P(x, y) dx = 1 and J;xl>o P(x, y) -+ 0 when

withy-+ 0, since clearly P(x, y) > 0, y-+ 0.

The method used to obtain (2.2.25) is a special case of Bochner's method of subordination . In this case, we see that the Poisson kernel is subordina te to the heat kernel, since we can obtain one as an integral transform of the other. In the next section we

108

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

will see that the same idea can be used to obtain the Riesz potential kernel associated with fractional powers of the Laplacian, where the exponential in (2.2.23) is replaced by a suitable monomial.

Fourier Transforms of Tempere~ Distributions

2.2.3

The space S of rapidly decreasing functions is a linear metric space, when we define the metric by

~

d~.~)= ~

m.k=O

2 -m-k

dmk

1+

.

dmk

sup

:=

lxlmiD"'¢(x)-

D"'~(x)j.

XEIR".I<>I=k'

dmk

A tempered distribution is a continuous linear functional L on the space S. The collection of all tempered distributions is denoted S'. The Fourier transform of a tempered distribution is defined by (2.2.26)

L(¢) = L(/P).

Clearly Lis again a tempered distribution and the mapping L -+ i is injective: if L(¢) = 0 for all ¢ E S, then the distribution L is identically zero. Convergence of tempered distributions is defined in the pointwise sense: Lk -+ L if and only if Lk(¢) -+ L(¢) for each¢ e S.

Exercise 2.2.46. Show that Sis a complete metric space with the metric d defined above.

Example 2.2.47. Any locally integrablefunctionf defines a tempered distribution by setting L(¢) =

{ f(x)¢(x) dx.

JIR•

A class of interesting examples is provided by the Riesz potentials, obtained as follows.

Example 2.2.48. Letf(x) = lxl-"' where 0
< n.

Then f is locally integrable and the Fourier transform can be computed by beginning with the heat kernel applied to¢ E S: (2.2.27)

We multiply each side of (2.2.27) by tk, integrate on [0, oo ], and apply Fubini. The inner integral on the left converges if and only if k<(n- 2)/2 while the inner integral on the

FOURIER TRANSFORMS ON THE LINE AND SPACE

109

right converges if and only if k > -1. In detail, the inner integral on the left side of (2.2.27) is transformed with s = 1 / 4t to obtain 00

1

2

l t kexp(-IYI /4t)dt = ---::--:---:( 47r t)"/2

nn/24k+ I

1

00

0

2 e -slyl s n/2-k-2ds

1

((n/2)- k - 2)!

7rn124k+ I

iyin-2k-2

while the inner integral on the right side of (2.2.27) is

which gives the result (2.2.28)

((n/2)- k- 2)! { n"/2

¢(x- y) d

JIR" iyln-2k-2

_

k!

{

y - n2k+2

A

e2lrit;·x

d

}JRn ¢(~)I~ j2k+2 ~

valid for all ¢ E S. Taking x = 0, the right side of (2.2.28) expresses the definition of the Fourier transform of the tempered distribution defined by the function~ ~ 1~ l- 2k- 2 . The left side shows that this is equivalent to integration with a constant multiple of the locally integrable function y ~ lyi 2k+ 2 -n. The identificatio n becomes complete when we set a = 2k + 2, and the result is paraphrased in the statement (2.2.29)

The Fourier transform of

I~ 1-a

is Cna ixia-n, where 0 < a < n.

Note that the case ot = 2 corresponds to the Newtonian potential kernel associated to the Laplace operator of !Rn when n ::=: 3. Then k = 0 and (2.2.28) takes the form (2.2.30)

((n/2) - 2)! 7l"

n/2

1

¢(x- y) IY jn-2 dy = IR"

1

2

7l"

1 IR"

A

e2:rrit;·x

¢(~) ~~ j2 d~.

Exercise 2.2.49. Show that the previous example can be generalized to complex numbers a satisfying 0 < Re(a) < n. Exercise 2.2.50. Let¢ E S(!Rn) where n > 3. Show that (2.2.30) is a solution of the equation ~u = -4n 2 ¢>, where~ denotes then-dimens ional Laplace operator.

2.2.4

*Characte rization of the Gaussian Density

We can use the Fourier transform to prove J. C. Maxwell's (1860) characteriza tion of the Gauss density, as follows.

Proposition 2.2.51. Suppose that n > 2 and that f(x), x E JRn is an integrable function which has the properties that there existfo,f1 , ••• ,fn so that f(xl, · · · , Xn) = f1 (XI) · · ·/,, (xn) = fo(.../xT Thenf(x) = Ae-Bixl 2 for some B > 0.

+ · · · + x;)

x

E

!Rn.

110

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. From Fubini 's theorem we see that J, E L 1 OR 1 ) for 1 ::S i ::S n. Taking polar coordinates, we see thatfi> E L 1 (JR+; r"~ 1 dr) and that the Fourier transform] can be expressed as a function g of the square of the Euclidean norm. Taking the Fourier transform, we see that

If j,(O) = 0 for some i, then we would have g(x) - 0 and we could take A = 0. Hence we can suppose that},(O) ¥ 0. Setting ~1 = 0 for j ¥ i shows that},(~) = c,g(~ 2 ) for some constant c, ¥ 0. Letting ~1 0 identifies the constant c 1 • • • c, = g(O) 1 ~n. Setting G(x) = g(x)jg(O) we obtain the functional equation

=

(2.2.31)

G(x

+

y)

=

G(x)G(y),

G(O)

=

I,

x--+ G(x) continuous.

Let 8 > 0 so that G(x) > 0 for 0 < x ::S 8. From this we can compute G on the rationals multiples of 8: G(8m/n) = G(8)"'1" and by continuity this formula extends to all real numbers in [0, 8] in the form G(x) = e- 8 ' , B := -8~ 1 log G(8). Now we can use the functional equation (2.2.31) to extend this to all x > 0. Since G is a bounded function, we must have B > 0, which completes the proof. •

In Chapter 5 we will see that this characterization of the Gaussian density is true in the wider context of probability measures on .!R.", not necessarily absolutely continuous with respect to Lebesgue measure.

Exercise 2.2.52. Suppose that G is a measurable and locally integrable function on lR and satisfies the functional equation G(x + y) = G(x)G(y) a. e. Prove that either G(x) = 0 or G(x) = e= for some a E .IR.. Hint: First show that G is a continuous function.

2.2.5

*Wiener's Density Theorem

The Fourier transform in one dimension can be effectively used to study the L 1 closure of the set of translates of a given L 1 function N

(2.2.32)

Lakf(x -xk) k=l

where ak are complex numbers andxk are real. A closely related set is formed by functions written as convolutions (2.2.33)

l

a(y)f(x- y) dy

where a E L 1 (IR). From Lebesgue's differentiation theorem it follows that any finite sum of the form (2.2.32) can be written as an L 1 limit of functions of the form (2.2.33); conversely, any convolution can be written as the limit of Riemann sums. Hence we see that the L 1 closure of (2.2.33) is identical to the L 1 closure of (2.2.32). Wiener's theorem characterizes this in terms of the Fourier transform. The following proof is adapted from Garding (1997).

FOURIER TRANSFORMS ON THE LINE AND SPACE

111

Theorem 2.2.53. Letf E L 1 (JR.). Then the L 1 closure o/(2.2.33) is the full space L 1 (JR.) if and only if the Fourier transform is never zero: f(~) =I= Ofor all~ E JR.. Proof. The necessity of the condition is immediate, since the Fourier transform of a * f is a(l;)](l;). If](~o) = 0, then the same is true for all convolutions and, by continuity, for all elements in the L 1 closure. Therefore the L 1 closure of {a *f: a E L 1 (1R)} is a proper subset of L 1 (IR). To prove the sufficiency, we first note that it's enough to prove that there exists a dense subset of L 1 (IR), all of whose elements can be written a *f for some a E L 1 . We let A 0 = {h Bo

=

L 1 (IR) :

E

h has compact support},

{h E Ao : his piecewise C 2 }.

Clearly A 0 is dense in L 1 (IR) since Fejer's theorem guarantees that we have the L 1 convergence h(x) = 1~

{M h(~) A

}_M

(

1-

~~~)

M

•

e27m/;

•

d~.

lf :

To proceed further, we introduce the notation A = f E L 1 (JR.)}, which consists of continuous functions vanishing at infinity, with the norm

llfiiA

:=

11/111·

We state and prove the following basic lemma. Lemma 2.2.54. For any f E L 1 (JR.) and g E B 0 , let G we have

lim

8~0

Proof.

]G8

lllf -f(O))GoiiA

= g,

G 0 (~)

=

G(~ 18). Then

= 0.

is the Fourier transform of t-+

i_f(s)8g(8(t- s)) ds,

whereas](O)G8 is the Fourier transform of t-+

8g(8t) (if(s) ds).

Therefore the required A norm is estimated as lllf- ](0))G8IIA

~i

( i lf(s)(g(t- 8s) - g(t))l ds) dt

where we have made the substitution t -+ 8t. The final integral tends to zero by the dominated convergence theorem. To complete the proof of the theorem, we let P be the piecewise linear function such that

P(~) = {~

ifl~l <

1

if I~ I > 2.

112

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Notice that P(~ 18) ::f. 0 implies that P(~ /28) = f(~) #- 0 to write

1, so that we can use the hypothesis

P8(~) . p(~ /8)f(~) p~~ / 8 ) /(~)

~

=

P(V8)

/(~) f(O) + P(~ /28)lf(~)

- f(O)).

From the lemma we see that for sufficiently sma118 > 0, the term P(~ /28)lf(~) -](0))/].(0) has norm less than 1/2, so that we can make the following convergent Taylor series expansion in the space A:

We have proved that P 8 = fQ 8 where Q 8 any translate of P 8 to obtain

for some Q8 E A. Note that L~N P(~ we can write

E

+ 3k)

A. In the same manner we can apply this to

= 1 for I~ I :5 3N

+

l. Hence for any h E Ao

which exhibits h as the convolution off and an L 1 function, which completes the proof of the theorem. •

2.3

FOURIER INVERSION IN ONE DIMENSION

In this section we give a self-contained treatment of convergence theorems for the Fourier integral in one dimension. Readers who have followed the treatment of Fourier series in Chapter 1 may wish to omit much of the current section, since many of the theorems are direct analogues of the corresponding theorems for Fourier series. An exception is the discussion of one-sided Fourier representations in Section 2.3.9, but this is not used in the sequej.

2.3.1

Dirichlet Kernel and Symmetric Partial Sums

The partial sum operator applied tof

E

L 1 (IR) is defined by

(2.3.1)

We now rewrite the integral defining the partial sum so that it makes no reference to the Fourier transform. This is called the explicit representation via the Dirichlet kernel.

FOURIER TRANSFORM S ON THE LINE AND SPACE

i: (/_: i: (/_:

113

In order to do this, we use Fubini to write

f:J(~)ezrr,l;x d~

SMf(x) =

= = =

! f

00

d~

e2rr1/;(x-y)dl; )j(y) dl;

sin 2n M(x- y) f(y) dy n(x- y)

-oo

=

e-2ml;}j(y ) dy) e2rril;x

sin 2n Mz f

oo

-oo

(x- z)

d

z.

1TZ

the required formula. The previous computati on is summariz ed by writing (2.3.2)

oo

sin 2nMz

-oo

JTZ

f

SMf(x) =

f(x- z) dz

or equivalent ly, since the kernel is an even function SMf(x) =

1

(X)

[f(x

0

+ z) + f(x-

z)]

sin 2n Mz

dz.

JTZ

The function z .._ (sin 2nMz)/n z is called the Dirichlet kernel and the integral operator is a convolutio n with the Dirichlet kernel. We recognize the Dirichlet kernel as the Fourier transform of the indicator function of the interval [- M, M]. As a first applicatio n, we use the Gaussian identity (2.2.9) to compute the (improper ) integral of the Dirichlet kernel. Applying (2.2.9) with x = 0 andf = 1r-I.IJ

11. m

f

t-+0

00

_

00

_si_n_2_n--'-~ _ 4 rr 2 1.; 2 e d~ = 1, JT~

or equivalen tly by changing variables to

f

.m 11

N-+oo

00

_ 00

z=

~

.../i, N

= 1j

_si_·n_2_n_N_z _ 4 rr 2 z2 e

JTZ

.../i we have

dz = 1.

This can be applied first to compute

f

1

-1

sin 2n Nz _ rr2z2 ----e 4 dz= JTZ

(f

oo

{

-oo-

Jlz!~l

)

sin 2rr Nz _ 4 rr2z2 d -1T-Z e

z.

The first integral tends to 1, while the second integral tends to zero, by the RiemannLebesgue lemma. Now

f

1

-1

sin 2rr Nz d z = rrz

f

1

-1

sin 2rr Nz _ rr2.2 (1- e 4 ~ ) dz rrz

+

f

1

sin 2rr Nz _ 4 rr2z2 e dz. _1 rrz

The first integral tends to zero by the Riemann- Lebesgue lemma and the final integral tends to 1, by the previous step. We have proved that 1 !ZNrr sin t lim - - dt =

N-+oo 1T

-2Nrr

t

lim N-.oo

f

1

_ 1

sin 2n Nz rr z

dz = 1.

114

INTRODUCTION TO HJURfER ANALYSiS AND WAVELETS

This is the famous sine integral, which is often computed from Cauchy's theorem on complex integration. In Chapter 1 this was computed from the properties of Fourier series. Now we have redone this using the Riemann-Le besgue lemma and the explicit Gaussian example. It is customary to write

21x

Si(x) = -

TC

o

sin t

- - dt,

t

so that limx->oo Si(x) = 1, lim_.->-= Si(x) = -1.

Exercise 2.3.1. Prove the inequality 11- Si(x)l Hint: Integrate-by-p arts

2.3.2

J,M (sin tft) dt and let M

< (4/rcx)fora llx > 0.

~ oo.

Example of the Indicator Function

We now consider in detail the case of the indicator functionf(x) = 1(a. b) (x). The Fourier transform is

for~ sum

=f: 0 andf(O) =

(b- a), by definition. Now we consider the nonsymrnetr ic partial

The first term is written

while the second term has an identical structure. When we take the real and imaginary part, we see that the real part may be written in terms of the sine integral Si(x) = (2/rc) (sin t/t) dt, hence convergent. But the imaginary part is written in terms of integrals involving J0M [cos ~(x- b)- cos ~(x- a)]/~ d~, which is convergent if x-# a, x-# b, but otherwise diverges logarithmica lly. Therefore the nonsymrnetr ic partial sum SM,Nf does not converge in general. Put otherwise, the improper Riemann integral will not suffice for the Fourier inversion of this function. This apparently anomalous behavior may be attributed to the generality of complex notation. Indeed, if we had begun with the basic trigonometri c form of the Fourier integral (2.1.8) this would not occur, since the correspondin g complex form will necessarily be the symmetric partial sum.

J;

Exercise 2.3.2. Showthatfor theaboveexa mple, So.Nf(a) ,...,_ ClogN andidentify the constant C.

FOURIER TRANSFORMS ON THE LINE AND SPACE

115

We now show explicitly that the symmetric partial sums converge. SMf(x) = !M f(l;)e2rrtl;r

d~

-M

= iM [sin =

2:rr~(x- b)- sin 2:rr~(x- a)]/n~ d~

t£Si(2Mrr(b- x)) - Si(2Mrr(a- x))].

It is immediate that if a < x < b this converges to 1, while if x < a or x > b it converges to zero. At the endpoints x = a, x = b it converges to ~. Furthermore these

approximating functions are uniformly bounded by 3.

Exercise 2.3.3. Check these statements. 2.3.3

Gibbs-Wilbra ham Phenomenon

The Fourier inversion of the indicator function provides the simplest occurence of the Gibbs- Wilbraham phenomenon. This is the detailed statement of nonuniform convergence that is present in the Fourier analysis of discontinuous functions. Indeed, if we had uniform convergence, then the above sequence of continuous functions would have a continuous limit. But the indicator fails to be continuous at its endpoints. In order to see this in more detail, we take the special case a = 0, b = 1. Applying the previous discussion, we see that SMf(x) = HSi(2M:rr(1 - x))

+ Si(2M:rrx)].

For any fixed x E (0, 1) this converges to 1, when M-H:JO. But if we take x = 1j2M ---+ 0, then fM(I /2M) ---+ (1/2)[1 + Si(:rr)], which is now shown to be larger than 1. Indeed Si(:rr) =

2 JT

rrr

lo

sinx 2 ~dx = :rr

=

~

rrr ( 1 -

lo

(:rr1l' 2

x2 3!

+

4

x

6

)

5!- 7!

dx+ . ..

~~ + ~- 35~~so) + · · · 1l' 4

1l' 6

300

17,640

=2-+-9 = 2 - 1.11

x

+ 0.33

- 0.04

+···

+ ...

= 1.18 to two decimal places

so that to two decimal places, limM SMf(l/2M) = 1.09, demonstrating the Gibbs overshoot.

2.3.4

Dini Convergence Theorem

Returning to the theory, we now develop a basic convergence theorem.

116

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Theorem 2.3.4. Suppose that f(x), -ex:>< x <ex:> is a complex-valued integrable function that satisfies a Dini condition at x: for someS E 0,

1 8

(2.3.3)

f_(_x_+_t_)_+_J_ _(x_ _ t)_ _2_S_I dt < ex:>.

--='

1

Then

!M

lim M-HXJ

f(f;)e2Jrii;x

-M

d~

= S.

(It is not asserted that S =f(x)). Proof. From the Dirichlet kernel representation, we have the Fourier partial sum SMf(x)

!

=

M

-M

~

f(t;)e 27"t;x d~

foe sin 2:rr Mz f(x +

=

-=

:rrz

z) dz.

Having proved Fourier inversion for the function e-rrx 2 , we can replacef(x) byf(x) -se-rrx 2 • The new choice off is also in L 1 and satisfies the Dini condition with S = 0. The function z -+ [f(x + z) + j(x - z}]jz is integrable, since the Dini condition takes care of ~z/:sa whereas

1

11

l:l>li

lf<x+z)+f(x-z )i - - - - - - - - dz ::5 z 8

lf(x

+ z) + f(x

- z)l dz < oo.

1:1>8

Using the Riemann-Lebesgu e lemma, it follows that the SMf(x)

-+

0, as required.

•

Corollary 2 .3.5. Suppose that f satisfies a local Holder condition with exponent a> 0: IY

-xl

< 8.

Then Fourier inversion holds with S = f(x). Proof. Taking S = lf(x

f

(x ),

+ t) + j(x- t)

But the integral

we have for 0 < t < 8 -

2/(x)l

=

!(f(x

+ t ) - f(x)) +

(f(x- t ) - f(x})l ::5 2Ct 01 •

J; ~-I dt is convergent, hence the Dini condition is satisfied.

Corollary 2.3.6. Suppose that f has right and left limits f(x a one-sided local Holder condition with exponent a > 0:

if(y)- f(x

+ 0)1

< Clx- Yla,

0) and satisfies

x
lf(y)- f(x- 0)1 < Clx- yla,

Then Fourier inversion holds with S = [f(x

±

•

X-

0 < y <X.

+ 0) + f(x-

0)]/2.

Proof. We have for 0 < t < 8

lf(x

+

t)

+ f(x-

But the integral

t) -

251

=

l(f(x

+

t) - f(x

+

0))

+

(f(x- t) - j(x- 0))1 ::5 2Ct01 •

J; t"- dt is convergent, hence the Dini condition is satisfied. 1

•

FOURIER TRANSFORMS ON THE LINE AND SPACE

117

The theorem of Dirichlet-Jo rdan also has a counterpart for Fourier transforms.

Theorem 2.3.7. Suppose that f Then limM SMf(x) = Y<x + 0) Proof. Letting F(u) = 4(f(x

L 1 (lR) is offinite total variation on the real line. + !f<x - 0). E

+ u) + j(x- u)], we have

1

00

SM.f(x) =

F(u) d Si(2.7r Mu) du.

For any K > 0, the contribution to the integral from u :=::: K is bQ_unded by 11111• I K, which can be made small by taking K sufficiently large. On the interval (0, K) _we can integrate by parts: {

F(u) d Si(2.7rMu) = F(K- 0) Si(2rrMK) -

J(O,K)

{

Si(2rrMu) dF(u).

J(O,K)

The integrand is bounded and tends to 1, so that we can apply dominated convergence to conclude lim { M

F(u) d Si(2rrMu)

J(O,K)

= F(K- 0)- [F(K- 0)- F(O + 0)] = F(O + 0),

completing the proof.

•

Exercise 2.3.8. Prove that iff is of finite total variation, then the partial sums SM.{ are uniformly bounded: supxeiR.M>O ISMf(x)l < oo.

2.3.4.1

Extension to Fourier's single integral

We have proved the convergence of the Fourier inversion for functions in L 1 (lR), which satisfy a Dini condition or have finite total variation. The operator SM.{ can be extended to a wider class of functions, if we note that the formula (2.3.2) is well-defined under the sole condition that fiR lf(x)l/(1 + lxl) dx < oo. This extended operator is called Fourier's single integral by Zygmund (1959). If we use the extended definition of the operator f --. SM.{ in formula (2.3.2), then we can extend each of the above theorems. We leave the details as exercises.

Exerc:ise 2.3.9. Suppose that fiR lf(x)l/(1 +lxl) dx < oo andthatf satisfies a Dini condition at x. Prove that limM SM,{(x) = S. Exerc:ise 2.3.10. Suppose that fiR lf(x)l/(1+ lxD dx < ooandthatf isofbounded variation in a neighborhoo dofx. Prove thatlimM SMJ(x) = ~f(x+O)+ ~f(x-0).

2.3.5

Smoothing Operation s in 1R. 1-Averagin g and Summabili ty

The problem of pointwise convergence of Fourier series and integrals is beset with numerous pathologies, of which we recall two, in the context of Fourier series: • There exists a continuous function whose Fourier series diverges at a point. The first such example was found by du Bois-Reymo nd and later simplified by Fejer.

118

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

By superposing these at different points, one can construct a continuous function whose Fourier series diverges at an infinite set of points. • There exists an integrable function whose Fourier series diverges at every point. The first such example was found by Kolmogorov, thereby answering in the strongest negative sense the possibility of a general theorem on pointwise Fourier inversion for integrable functions. These examples, which can be replicated in the context of Fourier transforms, suggest the difficulty of finding general sufficient conditions for convergence. In the other direction, it is impossible in general to obtain a necessary condition for convergence: if f is an odd function[/( -x) = -f(x)]. its Fourier transform reduces to a sine transform: F(~) = J f(x) sin 2rr~x dx, which is also an odd function; hence SMf(O) = 0 identically and limM SMf(O) = 0 = f(O). Thus pointwise Fourier inversion holds at x = 0 without any further regularity conditions. In a positive direction, it was proved by Carleson (1966) that for an L 2 function, the Fourier series converges at almost every point. This was later generalized by Hunt (1968) to all of LP if p > 1. The Carleson-Hunt result is deep and difficult, beyond the scope of this work. Instead we shall be content with theorems that replace pointwise convergence by a weaker notion. If the sequence of numbers SM.[ (x) fail to converge, it is natural to form averages and hope that the averages behave better than the original sequence of partial sums. There are two possible ways to average:

J:

• Average with respect to x: for example, form 1/(b- a) f(x) dx. • Average with respect toM: for example, the arithmetic mean (1/M) (Fejer mean). Another choice is the Abel mean € f 000 e-EmSmf(x) dm.

2.3.6

J0M Smf(x) dm.

Averaging and Weak Convergence

These smoothing operations lead immediately to general convergence theorems without additional smoothness conditions. Theorem 2 .3.11 . Suppose that f is any integrable function on lll. Then for every a
lim

{b

M-+oo Ja

SMf(x) dx =

1b

f(x) dx.

a

Proof. To see this, we first recall that the Dirichlet kernel is even, hence the operator I --+ SMf is self-adjoint, so that we can write

lb

SMJ(x) dx = ll
Since l
•

FOURIER TRANSFORMS ON THE LINE AND SPACE

119

This type of averaging is natural in applications, where f may represent a density function of mass, charge or probability. We are interested only in the mass/charge/probability of an interval, which is defined by the integral f(x) dx. The previous theorem states for any integrable function f, we can always recover the mass/charge/probability of an interval as the limit of the Fourier partial sums.

I:

Exercise 2.3.12. Suppose that f-l is a .finite Borel measure on the rea/line. Modify the above proof to show thatlimM SMJ-t(l;) dl; = f.-l((a, b))+ !J-t({a}) + !~-t({b}) 2 where SM~-t(l;) = J':!M jl(l;)e rril;x dx.

I:

We can use the above arguments to produce a continuous function vanishing at infinity, which is not the Fourier transform of any integrable function. 1 Let (2.3.4)

F(i;) =

(1

+

II; l)l!g (2

+

II; I)

Suppose that F = j for some f E L 1 (IR). Let g = 1[O. 11 be the indicator function of the unit interval, with g(l;) = (1- e- 2rril;)/2nil; for I; ::j: 0. Since g is of bounded variation, the Fourier partial sums SMg converge boundedly tog and we can write

r fg =lim r f SMg =lim r gSMf =lim 1M f(-l;)g(l;)dl;. M JR M lrrt M -M

lrrt

But a direct calculation shows that the real part of the last integral is

1 - COS 2.rr I; -(1-+-11;-1,.--),--lo_g_(-=-2....:c+_ll;-l) d I;'

1

M

-M

which diverges when M

2.3.7

~ <Xl.

Cesaro Summability

We now turn to the question of Cesaro summability. To study this, we rewrite the Fejer mean in terms of the original/, as follows:

_.!._ {M S,J(x) dm = _.!._ {M (1M f(l;)e2rril;x dl;) dm M

Jo

Jo

M

= =

~ 1M (/_'~ (/_: e 2 rril;(<-y)f(y) dy) dl;) dm _!_ 1= M

-oo 00

=

1

Igari, 1996,p. 172

-M

1

-oo

( {M sin 2mrr(x- y) dm)f(y) lo rr(x- y)

1 - cos(2M.rr(x- y)) 2M.rr2(x- y)2 f(y) dy.

120

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The Fejer kernel is defined by KM (0) = M and 1 - cos (2Mrrx) KM(x) = - 2Mrr 2 x 2

(2.3.5)

,

.,\ =I= 0.

It has the properties of an approximate identity, expressed by • KM(x) :::_ 0, J~OQKM(x) dx = 1. • For any 8 > 0, ~x)>o KM(x) dx -+ 0, when M-+ oo. The nonnegativity is obvious. The normalization can be found from Fourier inversion, as follows: The Fourier transform of (1 - 1.; I) lro.n(l.; I) is (1 -cos 2rry)j2rr 2 y 2 , which is an L 1 function. Therefore we can apply Fourier inversion at y = 0 to conclude that jiR(l -cos 2rry)/2rr 2 y 2 dy = 1, which transforms into fri{. KM(x) dx = 1 when we let y=Mx. To estimate the integral for lxl > 8, we replace the sine by 1, to obtain

1

)x)>8

-1

1 KM(X) dx < Mrr 2

lxi>.S

2 dx2 = --+ 0 x M8rr 2

whenM--+ oo. In order to minimize the new notation, we write

It will always be clear from context whether we are operating on a function or simply considering the kernel. Theorem 2.3.13. The Fejer means have the following properties:

• Iff is integrable on lR and continuous at x, then the Fejer means converge to f(x): KMf(x) --+ f(x) when M--+ oo. • Iff is integrable on lR then the Fejer means converge in L 1 : IlK~- fll 1 --+ 0, whenM--+ oo. • Iff is bounded and uniformly continuous on JR, then the Fejer means converge uniformlytof: supxEIR IKmf(x) -f(x)l-+ OwhenM--+ oo. Proof. Since

JJR K,.,

= 1, we can write

KM.f(x)- f(x) = !K,.,f(x) - f(x)l

:=:

J:

K,.,(y)[f(x

+ y)- f(x)] dy,

(1 + 1 ) lyld

IYI:!~

KM
+ y)- f(x)] dy.

Iff is continuous at x, then given E > 0 we can choose 8 > 0 so that if(x + y)- f(x)\ < E for IY I < 8. Therefore the first integral is bounded by E. On the other hand, the second term is less than 2 supJR If I ~vi>~ K,., (y) dy, which tends to zero, proving the first statement.

FOURIER TRANSFORM S ON THE LINE AND SPACE

121

To estimate the L 1 norm, we write

f:

JKM.f(x)- f(x)J

dx :::0 = =

f: (/_: f: f:

KM(y)Jf(x

+

y) -

KM(y) (/_: lf(x

+

y)- /(x)J

f(x)J dy)

dx

dx) dy

KM(y)JJfv- /IJ1 dy.

But we can apply the reasoning of the previous step withx = 0, and/ replaced by 11/y -JIJ,, which is bounded by 211/11 1 and which is continuous at y = 0. Now if f is bounded and uniformly continuous , we can choose 8 > 0 so that lf(x + y)- f(x)l < E simultaneo usly for all x, when Jyl < 8. Thus

JKMf(x) -

j(x)J :::0

E

+ 2 SUp J/Jl IR

Hence lim supM supxeiR lKMf(x)- f(x)l :::0

E

for any

KM(y) dy.

IYI>8

E,

which completes the proof.

•

We used above, without proof, the fact that 11/y -flit ---+ 0 when y ---+ 0. This can be proved in the same spirit as the Riemann- Lebesgue lemma. First prove it for indicator functions of an interval (a, b), then extend to finite linear combinati ons, and then to bounded measurabl e functions with compact support. But any Lt function can be approxima ted in L 1 by a bounded measurabl e function with compact support.

Exercise 2.3.14. Carry out the details of the proof that llh· -flit

---+

0.

2.3.7.1 Approxim ation propertie s of the Fejer kernel As with Fourier series, we can find a universal bound for the accuracy of the Fejer approxima tion, as follows.

Theorem 2.3.15. Letf E Lt(JR) have the property that IIKM/ -flit= o(I/M), M---+ oo. Thenf = 0 a.e. Proof. The Fejer means are represented as

(2.3.6)

(2.3.7)

=

1 IR

sin 2 rrMy 7r

2M2

y

f(x- y)dy.

122

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

In particular KMf E L 1 (1R) and its Fourier transform is (1- 1~1/M)}(~)l[-M.Ml(~). Hence for any M > I~ I we have ( 1-

i.

'!) )J<~) =

e-2rrr<x(KM /)(x) dx

lil.j(~)

=

1~](~)1

:5 M .{ lf(x)- (KMf)(x)l dx

{ e-2rrrt;x[f(x ) - (KMf)(x)] dx

fiR

M

= Mllf- KMfllt·

Letting M -+ ex:> gives the conclusion that](~) = 0 for ~ =I= 0. But function, hence](~) 0, which implies thatf = 0 a.e.

=

J is a

continuous •

Iff has some additional regularity properties , one may obtain the first term in the asymptoti c expansion of KMf when M ---+ oo.

Exercise 2.3.16. Suppose thatf

E

L 1 (1R) and that]

E

L 1 (1R), ~J

E

L 1 (1R). Then

(2.3.8) Hint: Begin with the Fourier representat ion (2.3.6) of KMf, noting thatj(x) = and estimate each of the integral terms J;t;I>M and J;
2.3.8

JJR e2 rrit;xj(~) d~

Bernstei n's Inequali ty

The Fejer means can be used to give a proof of an important inequality in approxima tion theory, originally due to Serge N. Bernstein (1912). Suppose that a function is represente d in the form (2.3.9)

f(x) =

1M

e2rritx J-L(dt)

-M

where 1-L is a finite measure supported by the interval [ -M, M]. This includes a finite trigonome tric sum when we specialize f-L to be a discrete measure on an arithmetic sequence with 27T t = 0, ± 1, ±2, .... In general,J is an infinitely differentia ble function whose derivative s are bounded. Bernstein 's inequality gives an upper bound for these derivative s in terms of the upper bound off.

Theorem 2.3.17. Under the above conditions , we have for all x E lR (2.3.10)

lf'(x)l

< 4nM sup ~EIR

lf(x)l.

FOURIER TRANSFORM S ON THE LINE AND SPACE

123

Proof. It is no loss of generality to prove the inequality at x 0 = 0, since we can change the measure 1-t by replacing it by e 2 rrllxo ~-t(dr). We begin with the Fejer kernel I -cos 2Mrrx =1M (M- iti)e2rrm dt := 1"" D..M(t)e2rrttx dt

2rr 2x 2

-M

-oo

where D..M(t) is the triangular function defined by L:l..M(t) = M- iti if ltl < M and zero otherwise. From this it follows that = I -cos 2M:rrx ------e2 "rnr dx = D..M(t) oo 2:rr 2 x2

!

00

1 _

sin 2Mrrx

00

l - cos2M:rrx ~ e-rrrxr dx 2:rr 2 x 2

= --:1 [D..M(t + 21

M)- D..M(t- M)].

The right side is equal to -ron the interval [ -M, M] . Integrating both sides with respect to the measure J.l(dt) on the interval [ -M, M] and applying Fubini, we obtain from (2.3.9)

" ' !

'""'

sin2M:rrx

l-cos2Mr rx j(x)dx 2 :rr 2 X 2

11M

= -. 2 I

-fJ.l (dt)

-M

= f'(O) -4 . 7r

Therefore

1/'(0)1 4:rr

::5

-~~~
1"" 1- cos2M:rrx dx _ -oc 2rr2x2 - M

-~~too lf(x)i,

•

completing the proof.

By applying this repeatedly , we obtain estimates for the higher derivative s. Corollary 2.3.18. Under the above conditions , we have for all x k = 1, 2, ...

E

IR and any

lfkl(x)l < (4rrMl sup lf(x)l. xElR

Bernstein 's inequality can be used to characteri ze the smoothne ss of functions on the real line in terms of the speed of convergen ce of their approxima tion by Fourier integrals on finite intervals. In complex analysis, it is shown that these approxima nts are entire functions of exponenti al type. The following propositio n is an immediate corollary of Bernstein 's inequality . Propositi on 2.3.19. Suppose that f E C(IR) is a bounded continuou s function with the property that there exist approxima nts of the form

(2.3.11) fr(x) =

{

}IH~J

e2rrix~ 1-LT (d~),

11/r- flloo = o

with

Thenf E Ck{JR). Proof. For any N

E

z+

we can write ex.

f =f2N

+ Lif:v+' -/21 ). J=N

(TL

1 ),

T-+ oo.

124

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Clearly j 2 N is differentiable to any order. Now note that

= o ( (~

Y+)

+

o ( (~)

k+)

=a((~)*+). Butfv+l - f 21 is of the form (2.3.9) with M = 2'+ 1 • Therefore by Bernstein's inequality, we have for any n 1({21 +1 - f21 )(x)l :S [4rr21 + 1 rsuplfv+I(X) -fv(x)l <EIR

which is the general term of a convergent geometric series provided that n :S k. Hence the differentiated series L_1 ifv+l - f 21 ) are convergent for n = 1, 2, ... , k, which proves that the limit function/ is k-times differentiable. •

In closing, we remark that in the best known version of Bernstein's inequality, the constant 4rr M is replaced by 2rr M and that this is sharp. Indeed, by considering the examplef(x) = e 2rriMx, we see that lf'(x)l = 2rrMif(x)l. For details, consult Zygmund (1959), vol. 2, p. 276.

2.3.9

*One-Sided Fourier Integral Representation

Sometimes we have to deal with functions that are defined on the halfline 0 < x < oo. We can obtain several inequivalent representations by trigonometric integrals by extending the function to the entire real line -oo < x < oo in different ways.

2.3.9.1 Fourier cosine transform We can extend/ as an even function by settingfeven(x) =f(x) for x:::: 0 andfeven(x) = f( -x) for x < 0. Iff E L 1 (0, oo), thenfeven E L 1 (lR). In order to have a symmetrical theory we define the Fourier cosine transform as

Fe(~)

1

00

=

f(x)

cos(rr~x/2) dx.

The Fourier transform offeven(x) is computed as :Flfeven)(n =

J:f~ven(x)e-Zrrit;x dx =

21= f(x)

cos(2rr~x) dx = 2Fc(4~).

FOURIER TRANSFO RMS ON THE LINE AND SPACE

125

If in additio n/ satisfie s a Dini conditi on at x, with S = f(x), then we have the Fourier inversi on as the improp er integra l f(x) = /_: .Fifeven)(/;)e 2 rri~.x dl; = 2 =

~o= 2Fc(41; ) cos(2.rr l;x) dl;

~o= Fc(v) cos(.rrv x/2) dv,

X>

0.

The partial sums of the Fourier cosine transfo rm satisfy (SM/even)' (0) = 0, which suggest s the relation to the "bound ary conditi on" f' (0) = 0. 2.3.9.2

Fourie r sine transfo rm

We can extend / as an odd functio n by setting f00ct(X) = f(x) for x > 0 andfooc t(X) = -/( -x) for x < 0. If/ E L 1 (0, oo), thenfooct E L 1 (IR). In order to have a symme trical theory, we defi"ne the Fourier sine transfo rm

1

00

Fs(l;) =

f(x) sin(.rrl; x/2) dx.

The Fourier transfo rm offooct(x) is an odd functio n, written .Fifooct) = 2i ~o= f(x) sin(2.rr l;x) dx = 2iFs(41 ;). Iff satisfie s a Dini conditi on at S = f(x), then we have the Fourier inversio n in the form

1

00

f(x) =

Fs(l;) sin(.rr/; x/2) dl;,

X>

0.

Exercis e 2.3.20 . Check this directly from the Fourier inversio n theorem . The partial sums of the Fourier sine transfo rm satisfy (SMfooctHO) = 0, which suggest s the relation to the "bound ary conditi on" /(0) = 0. 2.3.9.3 Gener alized h-trans form It is also natural to conside r Fourier integra l represe ntation for functio ns that satisfy

a bounda ry conditi on of the form

/'(0) = hf(O). The case h = 0 corresp onds to the cosine transfo rm, while the limiting case h -+ oo corresp onds to the sine transfo rm. In order to motivat e the proper integra l transfo rm, we look for the combin ations of sin l;x, cos l;x that satisfy the bounda ry conditi ons. It is immed iately verified that the functio nf(x) = I; cos I; x + h sin l;x satisfie s the bounda ry conditi on. This functio n also has the propert y that hf - f' = (/; 2 + h 2 ) sin l;x, an odd functio n. This immed iately suggest s a new recipe for extensi on of an arbitrar y f to the half line x < 0, namely

126

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

to require thatj'(x) - hf(x) be an odd function. This leads to a first order differential equation, which is solved in detail by writing ](x) = f( -x) - 2h fo-x eh
(2.3.12)

X<

0,

while](x) =f(x) for x > 0. To proceed further we consider separately two cases. Case 1. h > 0: In this case we verify directly that iff E L 1 (0, <Xl) then] E L 1 (1R). The first term of (2.3.12) is clearly integrable, while for the second term we have

f_~ l](x)l dx

< 2h

= 2h =

J:

(fo-x eh
~a= IJ(y)l

21=

(!:·

lf(y)l dy <

eh<x+y)dx) dy

<Xl.

Therefore j is integrable. To compute the Fourier transform, we write -CX)

<

X

<

<Xl.

The Fourier transform of the convolution is the product of the Fourier transforms while the Fourier transform commutes with the reflection x ~ -x. For the individual terms, we have

:F(j ·

1(0.oo))

:F( e -h· 1
=~a= j(x)e- 2 rril;x dx

=

Fc(4~)- iF5 (4~),

1 -~a= e -hx e -2rril;x dx -_ - - - -

o

h

+

2;ri~

Therefore

This is the h-transform of the given functionj(x), 0 < x <

<Xl:

(2.3.13) The numerator is an odd function. In case h = 0 the h transform is twice the Fourier cosine transform, whereas in the case h ~ <Xl it reduces to (- 2i) x the Fourier sine

FOURIER TRANSFORMS ON THE LINE AND SPACE

127

transform. The Fourier inversion formula gives the improper integral (2.3.14)

f(x) =

i:

= -2i

e2rri~x Fh(~) d~

f

00

h sin(2n~x; + 2n; ~os(2n~x) (hFsC 4 ~) h + 4n ~

_ 00

= 4 (

lo

=

[

e2rr u; x

oo

00

. h - 2n ~~

h sin(nvxj2; h

) 0

(hF5 (4~)

+

2n~Fc(4~)) d~

+ 2 n~FcC 4 n) d~

+ (n;/ 2) cos(nvxj2) (hFs(v) + + (n v 2 /4)

In this form we see more clearly the limiting cases h summarized as follows.

~

0 and h

(nvj 2 )Fc(v)) dv. ~

oo. The results are

L 1 (0, oo) with h > 0. Then], defined by (2.3.12) is integrable on JR. Iff satisfies a Dini condition at x, then f may be recovered from its h-transform, defined by formula (2.3.13) and (2.3.14 ).

Proposition 2.3.21. Suppose that f

E

Case 2. h < 0 : In this case we have the additional complication of a nonzero integrable function whose h transform is identically zero. The functionf(x) = ehx is directly computed to have Fh(~) = 0, hencef cannot be recovered from its h-transform. This is the only obstruction, however.

Proposition 2.3.22. Suppose thatf

1

E

L 1 (0, oo) with

00

(2.3.15)

f(x)ehx dx = 0.

Then] defined by (2.3.12) is in L 1 (IR) and the Fourier inversion holds at every point where the Dini condition is satisfied. Proof. If we combine (2.3.15) with (2.3.12), we can write ](x) = /( -x) - 2h

/_~ eh(y+x>J(y) dy.

Now we can estimate the L 1 norm as before: lh /_:

(1~ eh(y+x>IJ(y)ldy) tb:l

=lex> (1:

lhleh<x+y) tb:)

1/(y)l dy

= 1CX>[1-eh_v]if(y)ldy
to].

•

The new behavior for h < 0 is related to the existence of a point spectrum in the Laplace operator with these boundary conditions. The functionf(x) = ehx satisfies the differential equationf" = h 2f and also satisfies the boundary conditionf'(O) = hf(O).

128

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

If h > 0 this function is not in L 1 and hence does not figure in the Fourier analysis. But when h < 0 it is integrable and must be dealt with.

In case/ does not satisfy the orthogonality condition, we simply subtract a multiple of elu and then apply Proposition 2.3.22. The functionf(x) + helu f 00 f(y)ehYdy will 0 satisfy the orthogonality condition, leading to the representation

1

00

C =

f(y)hehy dy.

This illustrates the role of the point spectrum in the Fourier analysis.

L2 THEORY IN lRn

2.4

The Fourier transform is well-adapted to the space of square-integrabl e functions, denoted L 2 (IRn). Since L 1 and L 2 are not properly contained in one another, we first restrict to the common dense subspace S and extend by continuity.

2.4.1

Plancherel's Theorem

The Plancherel theorem serves as a replacement for the Riesz-Fischer theorem that appeared in the L 2 theory of Fourier series in Chapter 1. In contrast with the theory of L 2 (1I'), in the present context we have an isometric bijective correspondence on the space L 2 (1Rn). The precise statement follows. Theorem 2.4.1. Plancherel: The Fourier transform can be extended to the entire space L 2 (1Rn) so that the mapf---+ :Ff preserves the L 2 norm. Furthermore the extended mapping is 1:1 onto all ofL 2 ('R..n).

The key to the proof is to establish the isometry of the Fourier transform on the spaceS, as follows. Theorem 2.4.2.

Parseval: For any f

E

S, we have

Proof. This depends on three simple facts, valid for u, v

(2.4.2) (2.4.3) (2.4.4)

{ uv =

'R"

{

E

uv

}f(n

~(x) =

u( -x) := Ru(x)

~(x)

~(-x).

=

S:

Formula (2.4.2) is a restatement of the Fourier reciprocity lemma, (2.4.3) is an expression of Fourier inversion for Schwartz functions, and (2.4.4) is from the definition of the Fourier

FOURIER TRANSFORMS ON THE LINE AND SPACE

129

transform. Letting u =f, v =],we have fJ =1 and

1 tl 1 uv 1 uv 1 u~. lR"

which was to be proved.

=

=

JR"

=

IR.n

IR.n

• 2

Proof of Plancherel's Theorem. By Proposition 2.2.42, any f E L (1R.") can be approximated in the L 2 norm by a sequence jj E S. Having done this, we apply Parseval to obtain

Ji

By hypothesis the left side tends to zero when j, k - oo, hence is a Cauchy sequence. By the completeness of the space L 2 , there is a well-defined limit :F(f) in the L 2 norm so that IIF(f)- 112 - 0, which defines the required extension/- :Ff. In particular,

Ji

II:Ffllz =lim llhll2 =lim ll/;ll2 = 11/llz. J n

Clearly this definition is independent of the approximating sequence, since if gi is another approximating sequence, another application of Parseval shows that llgi - 11 2 - 0. To prove the onto property, we first note the properties of the reflection operator 'R, defined by 'Rf(x) =f( -x). Applied to the Fourier transform, we have for any f E S,

Ji

'Rf(l;)

= {

Jan

e2rrix·lfJ(x) dx

= {

}Rn

e-2rri.x

Clearly II'R/IIz = 11/llz for all/ E L 2 (1R"). Now if 1/f E L 2 (1R"), we approximate by 1/fj 1/fj = 'R ..jrj =

where l/Jj = 'R{/Fj· Whenj- oo we have l/Jj -

E

lfJ( -x) dx

= ('Rf)(~).

S. From (2.4.3)

('R (/r,) = ~j 'R(:Fl/f), ~j

--+ :F('R:Fl/f). Hence

1/f = lim 1/fj = lim ~j = :F('RFt/F), J

J

•

which proves that the operator :F is onto all of L 2 (1R").

Exercise 2.4.3. Prove that the Fourier-Planche re/ transform on L 2 (Rn) satisfies the properties of linearity, translation, and phase factor that were proved for the Fourier transform on L 1 (lRn), described in Proposition 2.2.1.

2.4.2

*Bernstein's Theorem for Fourier Transforms

In Chapter 1 we proved that any HOlder continuous function with exponent ot > ~ has an absolutely convergent Fourier series. Here is the corresponding result for Fourier transforms.

Proposition 2.4.4. Suppose that f

L

E

L 2 (lR) satisfies

lf(x- h) - f(x)l 2 dx

fior some C

I

< C 2 h 2a A

> 0, ot > 2 and all h sufficiently small. Thenf = .:Ff E L 1 (lR).

130

INTRODUcriON TO FOURIER ANALYSIS AND WAVELETS

Proof. The Fourier-Planc herel transform of the translatefh satisfies}i,(~) = e-Zrrihlf}(n so that from the Parseval relation, we have

Taking h = (1/6)2-k, we estimate theL 1 norm of] for I~ I E [2k, 2k+l] by Cauchy-Schw arz, noting that in this range le- 2 rrrl;h - 11 :;:: 1: ·

(1ksi<Js2<+'

tf(~)l d~)

2 ::::

2

1

k+

1ksJ
:::: 2k+l {

tf
2

d~

le-2rril;h -

J2ksJiils2k+ 1

112 tf(~) 12 d~

:::: 2k+l ( c2 j36)2-2ka

=

(C 2/18)2k.

Therefore if a > ~

and the proof is complete.

•

In parallel with the theory of Fourier series, one can characterize the L z smoothness of a function in terms of the speed of L z convergence of the Fourier transform. These are listed in the following exercises.

Exercise 2.4.5. Suppose that for some 0 < a < 1 and some C > 0, we have 11/- fh liz < Cha for all h > 0. Prove that J;~i>M tf(~)l 2 d~ < CtM-2a for all

M > 0 and some C 1 > 0.

Hint: Mimick the proof of Theorem 1 .3 .3.

Exercise 2.4.6. Suppose that for some 0 < a < 1 and some C 1 > 0, we have J;~i>M tf(~)lzd~ ::S CtM- 2a for all M > 0. Prove that II/ -Ji.llz < Cha for all

h > 0 and some C > 0.

Hint: Mimick the proof of Theorem 1.3.3.

Exercise 2.4.7. Suppose thatforsome C 1 > 0 we have J;~I>M tf(~)lz d~ < C,M-z for all M > 0. Prove that II f - /h liz < Ch log ( 1I h) for 0 < h < c > 0.

!,for some constant

Exercise 2.4.8. ASuppose that for some k E z+ and for some 0 0 we have J;~I>M lf(l;)ed~ ::5 CtM- 2k-Za. Prove thatf, ... .t are absolutely continuous withf' E LZ(JR.), ... ,J E LZ(JR.) and that 11/,(k) - t liz < Cha for all h > O,forsome C > 0.

FOURIER TRANSFORMS ON THE LINE AND SPACE

2.4.3

131

The Uncertainty Principle

The L 2 theory of the Fourier transform can be used to discuss the Heisenberg uncertainty principle, as follows. We restrict attention to the case of one dimension. Iff E L 2 (lR), a quantitative measure of the spread about x = 0 is given by the dispersion about zero and defined by the formula J~ x 2 if(x) 12 dx

J~

Do(/) =

if(x) 12 dx .

This is defined whenever the relevant integrals are finite. The name is justified by Chebyshev's inequality, namely for any M > 0 J;.,.I>M lf(x)l 2 dx 1 f~oo x 2 if(x)i 2 dx D 0 (j) < = . 00 J_ l/(x)l 2 dx - M 2 J~= if(x)l 2 dx M 2 00

The fraction of the L 2 norm due to lx I > M is controlled by Do(/).

Exercise 2 .4. 9. Suppose that f is a Gaussian density function: j(x) =

Check that D 0 (j)

=

1

v'4JTt

e

-x 2 /4t

t and D 0 lf)

=

with

,

l/(16rr 2 t).

In this example the product is Do(f)Dolf) = 1/16rr 2 • This can be paraphrased as the statement that if a Gaussian density function is highly concentrated about its midpoint, then the Fourier transform will be widely spread about its midpoint. A remarkable statement holds in the general case, where we have the following inequality:

Proposition 2.4.10. Uncertainty principle: Letf

E L 2 (1R),

be a complex-valued function that is absolutely continuous andfor which x f E L 2 (1R) andf' E L 2 (1R). Then we have the inequality

(2.4.5)

1 rr 16 2

~

Do(f)Doif) >

.

Equality holds if and only iff is a Gaussian density function centered at x = 0; 2 2 in detail,f(x) = Cte-x fa ,f(J;) = C2e-rr a ~ for suitable constants Ct. C2, a 2. 2

2

2

Remark. The term "uncertainty principle" comes from the interpretation that we cannot localize both f (x) and j (~) in their respective spaces. Iff (x) is localized about x = 0, then Do(/) will be small; the uncertainty principle then asserts that D 0 lf) will be correspondingly large, indicating a lack of localization about ~ = 0. Proof. For notational clarity, write F =]. Unless otherwise noted, all integrals are taken over the entire real line. First we apply the Cauchy-Schwarz inequality to 1/(x)l = + x 2 )1/(x)l x JI/(1 + x 2 ) to conclude that/ E L 1 , similarly F =] E L 1 , hence they are Fourier transforms of one another, and in particular bothf and F are equal a.e. to continuous functions vanishing at infinity. In addition, F' exists as an L 2 function, since its

JO

132

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Fourier transform is assumed to be L 2 . Both the numerator and denominator of the expressions defining Do(/) and D 0 (F) will be transformed by Parseval's theorem. !n this way one is led to examine a corresponding integral involving F'(~). Writing F = f, we write the real part of the integral of~ F F' in two different ways. On the one hand,

2Re

f ~FF'd~

=

f ~(F'F+FF')dl; f ~(FF)'d~ f =

=

2

l;(IF1 )'dl; = -

f IF1 d~ 2

where we have integrated by parts in the last step. To justify discarding the term at the limits, note that since all of the integrals are absolutely convergent, they are also convergent improper Riemann integrals, so that MIF(M) 12 and NIF( -N)I 2 tend to limits when either M, N -+ oo in any order. If either limit is nonzero, then F(~) ""' const/1~1 1 1 2 , which contradicts the L 2 integrability of F. Now on the other hand, (2.4.6) -Re

J~FF' d~ ~ I/ ~F(~)F'(~) d~ I~ (J I~F(~)I2 d~)

1/2 ( /

IF'(~)I2 d~) 1/2

where we have applied the Schwarz inequality to the functions F'(~) and l;F(l;). Now we apply Parseval's theorem twice, recalling that the Fourier transform of xf(x) is F' (~) /( -27r i):

f IF(~)I d~ = f 2

lf(x)l 2 dx,

f

2

IF'(/;)1 dl; = 47r

2

f

2 2 x 1f(x)l dx.

Squaring both sides of (2.4.6) and making these substitutions gives the desired result, in the form (1/l67r 2 ) f 1!1 2 f IFI 2 ~ f lxfl 2 f I~FI 2 • In case equality occurs in (2.4.6), we obtain two conditions: (i) Schwarz's inequality implies that F'(l;) and l;F(~) must be proportional a.e. thus F must satisfy a.e. the differential equation F'(~) = -A~F(~) for some complex constant A; (ii) the imaginary part of J ~ F F' d~ must be zero. From (i) it follows that the derivative of the function G(~) = F(~)eA< 1 2 is zero a.e., hence G(~) = C a.e. for some complex constant C, which 2 proves that F(~) = ce-A< 12 a.e. But we noted above that F is continuous, hence the equality holds everywhere. This function will yield a finite value of D 0 (F) if and only if ReA > 0. To show that ImA = 0, we write A=()(+ ifJ and compute F'(~) = -CA~ e-A/; 2 12 , F(~) = ce-A/; 2 12: 2

f l;F(~)F'
ICI 2 A

f

1; 2e-a<

2

d~.

The imaginary part of the integral is zero if and only if fJ = ImA proved. The constants can be identified by setting A = 21r 2 u 2 .

=

0, which was to be •

In order to have a more flexible form of the uncertainty principle, we define the dispersion about a of a complex valued function/ by D

_ J~00 (X- a) 2 1f(x)l 2 dx a(f)f~oolf(x)l 2 dx

This can be reduced to the above case by defining fa.b(X) = e2rribxf(x- a),

FOURIER TRANSFORMS ON THE LINE AND SPACE

133

It is immediately verified that the Fourier transform of fa.b is e 2 rciab F -b,a and that DaCf) = Dolfa.b). Db(F) = Do(F -b,a). Applying the uncertainty principle we see that Da(f)Db(F) > 1/16rr 2 with equality if and only ifj(x) = Ce 2 rcibxe-<x-a) 2 / 2 2 some complex number C. (1'

Exercise 2.4.11. Show that the uncertainty principle can be generalized to lRn in the form n2

D3(f)D3(}) >

16rr 2

with equality if and only if f(x) = ce-lxl /a , F(~) constants C 1 , C 2 , a 2 • Here we use the notation 2

/Y).(j) = 0

2

C2e-rc 1~1 a for suitable 2

2

2

fn~." lxl 1f(x)l dx jlR" lf(x) l2 dx 2

2

Hint: Apply the proof of Proposition 2.4.1 0 in each coordinate and then apply the Cauchy-Schw arz inequality.

2.4.3. 1

Uncertainty principle on the circle

Heisenberg' s inequality has no direct analogue for Fourier series on the circle. This is related to the fact that there is no direct counterpart of the Gaussian density on the circle. The following modified form of Heisenberg' s inequality was discovered by Grunbaum (1990). Proposition 2.4.12. Suppose that 0 =1= f E L 2 (1I") is absolutely continuous with f' E L 2 (1I") andf(xo) = Ofor some xo E 1I". Then

lnl 2 1f(n) 12 LneZ lf(n)l 2

LneZ

xo) 2 1f(x) 12 dx J11' lf(x)l 2 dx

j1l'(x -

Proof. By changing x to x - x 0 , we may suppose that x 0 Then

2 Re (/_:

1

>-

4

= rr, so thatf(rr) =f( -rr) = 0.

x]f') = /_: x(]f' + ]j)

f:

=

x(l/1

= - /_:

2 )'

1/1 2

so that we can apply the Cauchy-Schw arz inequality to obtain

-I

2

1"'

1/1 2

=

-Re

-1T

1"'

xif' ~

-1T

Now square both sides and use the Parseval identity for f andf' to obtain

~ 11/1 2 11'

L

neZ

lf(n)l 2

~

{

JT

x 2 1/1 2

L

neZ

n2 1f(n)l 2 ,

134

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

which proves the inequality with ~· If equality holds, then we must have f' = Ax/ for some A and alsof(rr) = 0, which implies that/ is identically zero, a contradiction. •

2.4.4

Spectral Analysis of the Fourier Transform

In this section we show that the Fourier transform on JR. 1 has a complete orthonormal system of eigenfunctions. To get started, we note that the Gaussian density function with a 2 = l/27l' is its own Fourier transform, since

If we differentiate both the sides of this identity, we see that the Fourier transform of xe-rrx 2 is -il;e-rrt; 2 •

2.4.4. 1

Hermite polynomials

To proceed more generally, we introduce the generating function (2.4.7)

rx

e -

r 2 ;2

CX)

=

~

~ k=O

k

t -Hk(x) k!

= Ho(x) + tH1 (x) +

2

t -Hz(x) 2

+ ···

This power series converges for all t, real and complex~ the coefficients Hk(x) are the Hermite polynomials. Since the generating function is a Taylor series in the variable t, the coefficients can be obtained by successive diffferentiation as (2.4.8)

k = 0, 1, 2, ...

Equivalently, we can write

The first few are written as follows: Ho(x) = 1

H1(x) =x Hz(x) =x 2 - 1 H3(x) = x 3 - 3x H4(x) = x 4

-

6x 2

+ 3.

From the generating function, it follows that Hzk-t (0) = 0 and that H 2 k (0) = (2k)! 1k !2k fork= 1, 2, ....

FOURIER TRANSFORMS ON THE LINE AND SPACE

Lemma 2.4.13. H~(x) =

kHk-1

135

(x).

Proof. From (2.4.8), we write the derivatives and use Leibnitz's rule to write H;(x)

(!) k (tetx-t2/2)1r=O = t (!) k (etx-r2;z)lr=O + k (!) k-1 (erx-r2;z)lr=O =

•

= kHk-1 (x).

Lemma 2.4.14. For x > 0

[oo e_Y2 Proof. Let y = x

/2 dy <

j!fe-x2

/2.

+ t. Then loo e_Y2!2 dy = e-x2;z1oo e-'xe-r2;z dt .:5 e-x2;z1oo e-r2;zdt

=

i: e-x

~e-x2;z.

•

Lemma 2.4.15. For 0 < a < 2.../2Tii

Ak :=

212

1Hk(ax)l dx < 3hrr2kk!.

Proof. We have for x > 0 Hk(ax) = Hk(O)

+ ak 1x Hk-1 (ay) dy,

1Hk(a.x)l.5 IHk(O)I +ak 1x iHk-1(ay)ldy.

Therefore 00

1

e-x

212

1Hk(ax)l

dx

.:5

~IHk(O)I + ak 1

.:5

~IHk(O)I +ak~ 1oo 1Hk-1(ay)le- h2 dy.

00

00

IHk-1(ay)l ( /

e-x

212

dx) dy

We perform the corresponding computation for x < 0. Combining the two, we see that the sequence At satisfies the system of inequalities. k ~ 1

136

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

which telescopes to

•

which was to be proved.

Hint: Use mathematical induction, applied to 2.4.4.2

H~

=

k.Hk-l·

Eigenfunctions of the Fourier transform

With this preparation: we can now list the eigenfunctions of the Fourier transform.

Proposition 2.4.17. (2.4.9)

Proof. To prove (2.4. 9), we can use the generating function (2.4. 7) with t real and x replaced by x../2 to write

We apply Lemma 2.4.15 with a =

../2, to see that the series

~ ~~ /_: e-x 12 1Hk (xJ2) I dx 2

converges for ltl < 1/2. Hence we can integrate term-by-term to find that

This integral can be evaluated by completing the square in the exponent and making the substitution y = x- t../2 to obtain e'2/zj"" e_Y212e-i~Cy+r-.12> dy =

.../2iie-t;2!2e-·~r../2er2Jz_

-oo

When we compare this with the original generating function, we see that the only difference is the replacement oft by -it. But the series defining the generating function converges for all complex t, from which we conclude that for ltl < ~.

•

FOURIER TRANSFORMS ON THE LINE AND SPACE

137

If we make the substitution x = y,./2ii, ~ = v,./2ii, this can be written in terms of the usual notations as (2.4.10) k = 0, 1, 2, ....

One can reinterpret the above result as providing a basis of functions in which the Fourier transform has a simple structure. For example, if a function is written as a finite sum:

then the Fourier transform is N

F(~) = 'L,C-i)kake-JT~ Hk(2~../ii). 2

k=O

2.4.4.3

Orthogonality properties

The orthogonality properties of the Hermite functions are obtained from a second-order differential equation which will be proved. Computing as above, we find

H;(x)-

H~(x) =

2 (!)k (terx-r ; 2)lr=O

H;(x) =

(!)

k

(t2erx-r2/2)1r=O

xH~(x) =

(!)k (xterx-r 212 )1r=O

xH~(x) =

(!)

2 k (t(t- x)erx-r ; 2)1r=O

2 = (!!__) (t!!__(erx-r /2)) dt

dt

I

t=O

We now prove the orthogonality of these functions with respect to the measure with 2 density e-x /Z. To do this we introduce the differential operator

138

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

Thus iff, g are polynomia ls, we can integrate by parts as follows:

L

g(x)Lf(x) e-x 212 dx =

= =

L -l

2 g(x)[f' e-x 12 ]' dx

+

g'(x)[f'e-x 2/2]' dx

Lf(x)[g' e-x2/2]' dx

= Lf(x)Lg( x)e-x212 dx.

Applying this withf = Hm. g = Hn, we see that (n- m) Lf(x)g(x )e-x212 dx = 0,

which proves the orthogona lity. To obtain the normaliza tion, we write L

Hn(X) 2e-x 212 d.x = (-l)n L

= ( -l)n-l

Hn(x)D~(e-x 2 1 2 )dx

L

2 DxHn (x)D~-I (e-x 12) dx

= n(-l)n-1 L = n

l

Hn-!(x)D~-!(e-x2/2)

Hn-! (x)2e-x2/2 dx.

Proceedin g inductivel y, we see that fiR Hn(x) 2 e-x212 dx = n! fiR e-x212 = n!v'21r. The orthogona lity properties may be concisely written k =j k#j.

(2.4.11) We also introduce the normalize d Hermite functions ) -x2/4 (x ) -- (27r )-!/4Hk(X hk rr:~e ,

vk!

k = 0, 1, 2 ...

which satisfy k =J k #J

2.4.4.4

Complete ness

Finally we discuss the question of completen ess of the Hermite functions. We want to show that the closed linear span of finite linear combinati ons of the Hermite functions is the entire space L 2 (lR). If not, there would exist/ E L 2 (!R) which is orthogona l to all of the eigenfunc tions: f!Rf(x)hk( x) dx = 0 fork = 0, 1, 2, .... Since the Hermite

FOURIER TRANSFORMS ON THE LINE AND SPACE

139

polynomial = x" + lower-order terms, we conclude that JJRf(x)x"e-x 2 14 dx = 0 for n = 0, 1, 2, .... Now we can compute the Fourier transform off(x)e-x 2 14 by integrating term-by-term :

where the interchange of sum and integral is justified by noting that the modulus of the integrand is bounded by

which is an integrable function, for any~- Hence the Fourier transform ofj(x)e-x2 f 4 is zero, therefore f (x) = 0 a.e. This immediately shows that any L 2 function has an L 2 convergent Hermite series. Indeed, we define the Fourier-Herm ite coefficients off E L 2 (1R) by

Any finite linear combination fN = L::=o akhk is orthogonal tof- /N, thus

11/11 2

=II/- !Nil~+

II/NII~ 2: II/NII~ =

N

L lckl k=O

2

whichprove sBessel'sine quality: L:~o lckl 2 < 11/11~- Inparticular ] := L:~oakhdsan L 2 convergent series and the difference /- j is orthogonal to hk(x), fork = 0, 1, 2, ... , 00 hence by the above argument/ - f- = 0 a.e. thus/= Lk=O akhk. in the sense of L 2 .

2.5

SPHERICA L FOURIER INVERSIO N IN IR.n

Bochner ( 1931) studied the pointwise convergence of the spherical partial sums of the Fourier integral in Euclidean space. The purpose of this work is to determine the minimal smoothness assumptions necessary for pointwise Fourier inversion in lR". In this section we will give an up-to-date treatment of this material, based on Pinsky ( 1994) and Pinsky and Taylor ( 1997).

2.5.1

Bochner's Approach

The spherical partial sums are defined by (2.5.1)

140

INTRODUCTION TO FOURIER ANALYS~S AND WAVELETS

This integral may be rewritten as an integral transform onf by applying Fubini as follows: SMJ(x) =

1 ({ IH::N

)IR"

f(y)e-2rriy·l; dy)

= { f(y) ( { = =

e-2rri(x-y)·l;

}JR.n

jii;I5M

f

D'M(x- y)f(y) dy

f

D'M(z)f(x- z) dz

}JR.n

./.IR"

e2rrix·l;

dl;

dl;) dy

where the n-dimensional spherical Dirichlet kernel is defined by (2.5.2)

f

D'f.t (z) =

e-2rriz·l; dt; = D'f.t (lzl)

JII;/5M

where we abuse the notation and identify a radial function on lRn with a function on the positive real line. Noting that D'f.t is a radial function, we may further reduce SM.f in terms of the spherical mean value, defined by an integral over the surface of the unit sphere: (2.5.3)

fx(r) := -1Wn-1

1

sn-1

f(x

+ rw) dSw

resulting in (2.5.4) Lemma 2.5.1. The spherical Dirichlet kernel may be computed in terms ofBessel functions according to

(2.5.5) Proof. Taking a system of spherical polar coordinates with ~ 1 = f.J.. cos(}, we have D'J.t(r) = 9n

1M 1.,.

e2.,.irucos9f.J..n-l(sin9)"-2df.J ..d9.

From equation (2.6.6) in the appendix to this chapter, the(} integral is recognized as the Bessel function (rJ.J..)< 2-n>I 2J
The useful properties of dimension constant.

D'f.t

are summarized as follows, where Cn denotes a

FOURIER TRANSFORMS ON THE LINE AND SPACE

141

Proposition 2.5.2. (i) D1t(O) = (ii)

(TC M2)nf2 (n/ )! ,

n > 1

2

D1t(r) = CnM(n-l)/ 2 (cos(2nMr- 8n)

+ 0 (it)),

n 2:: 1, r > 0, M ~ ex>

CMn

(iii) ID1t(r)l < (1

+ ~r)
n > 1, r > 0 n > 3, r > 0.

Example 2.5.3. Jfn = 1 we have the one-dimensional Dirichlet kernel sin(2n Mr)

1 DM(r) =

TCr

.

Example 2.5.4. Jfn = 2 we have the two-dimensional Dirichlet kernel 2 DM(r) =

MJ, (2n Mr) r

.

The higher-dimensional Dirichlet kernels can be obtained by differentiation, using the recurrence formula (iv) from Proposition 2.5.2.

+ 1 and the spherical mean r ~ fx(r) is absolutely continuous, together with its derivatives of order k - 1 and that f 000 ,J-llfxV) I dr < ex> for j = 1, ... , k. Then the spherical partial sum converges: limM SMf(x) =]x(O + 0).

Theorem

2.~.5. Suppose that f E L 1 (Rn) where n = 2k

coo

Proof. Let T/M(r) be a function with 0:::: T/M(r) :::: I, T/M(r) = I for r:::: M, T/M(r) = 0 for r > M + I, and I7Jt'(r)l :::: Cj for j = 1, ... , k. We write (2.5.4) as SM.f(x) IM(x) + IIM(x) where

1 1JM(r)fx(r)~(r)r"1 00

IM(X) =

Wn-1

=

Wn-1

1 dr

00

IIM(x)

'f'/M(r))jx(r)IYJ.,(r)r"- 1 dr.

(1 -

The second term can be estimated by IIM(X) ::::

Loo lfx(r)~(r)lr"- 1 dr {oo M"

Wn-1

:::; C,

JM

lfx(r)i (I

+ Mr)
: :; ~ Loo r"- lfx(r)l dr = o (~) 1

00

where we have used the fact that j 0 r"- 1 lfx(r)l dr < <X>. The term IM(x) is estimated by repeated integration-by-parts, where we exploit the fact that the integrand has compact

142

INTRODU CTION TO FOURIER ANALYSI S AND WAVELET S

support to write

1

00

1

00

-

YJM(r)fx (r)D'J.t( r)r"- 1 dr =

o

00

=

1

-1 d 17M(r)r" - 1fx(r) - d D'J.t- 2 (r)dr 2 rrr r

d - [r"- 2 '7M(r)fx( r)]D'J.t- 2 (r)dr.

dr

We repeat the partial integrat ion k times to obtain IM(x)

=

Wn-1 --k

(2rr)

1

00

r

(

0

1 d

--d r r

)k

-

[r"- 2 '7M(r)fx (r)]

sin(2rrM r) rrr

dr.

Writing out the derivativ e by Leibnitz , we find that

r(~:rr [r"- 2YJM(r)fx(r)] =

t;C

1kr' (:r)J [YJM(r)] ..(r)]

for suitable constan ts c,k. For each) ~ 1 we have (djdr)'[' 7M(r)fx (r)]--+ (djdr)j] x(r) in L 1 (0, oo) when M --+ oo. Therefo re by the one-dim ensiona l Rieman n-Lebes gue lemma we have for M --+ oo 00

1 0

sin(2rrM r) - - - - r '. rrr

(d)' -

dr

-

['7M(r)fx (r)J dr--+ 0

j = 1, ... 'k.

For j = 0 we have the one-dim ensiona l Fourier inversio n of the absolute ly continuo us function '1M(r)].. (r) at r = 0, which gives

;)k Coifx(O + 0). 2

w, .J~oo IM(x) = (

1

-

The constan t is identifie d by choosin g a function for which we have already proved the Fourier inversio n, e.g.j(x) = e-,.lxl 2 • •

The sharpne ss of the conditi ons is reveale d by the followi ng basic exampl e in three dimens ions.

Examp le 2.5.6. Let n otherwi se.

3 and let j(x) -

1 for 0 < lxl < a and f(x) -

0

To comput e the spheric al mean value, we note that]x( r) is the fraction al area of the sphere S(x; r) which is contain ed in the ball B(O; a). This is zero if r > a + lxl and is one if r < a - lxl. Otherw ise it is comput ed as fx(r) = -1 47l'

1

{w:lx+rwl :5a}

4rlxl

dw

FOURIER TRANSFORM S ON THE LINE AND SPACE

143

Therefore if 0 < lxl < a, the spherical mean value]x(r) is ]x(r) = {

~2

if 0 < r < a - lx I lxl)2

(r -

-

if a - lx I < r < a

4rlxl 0

+

lx I

if r >a+ lxl.

The function r ---+ ]x(r) is Lipschitz continuou s, in particular absolutely continuou s, hence Theorem 2.5.5 applies to prove convergen ce at x. However if x = 0 we have ]x(r) = 1 for r < a and zero otherwise , a discontinu ous function. The Fourier inversion fails in a very simple way in this case, since we can use (2.5.4) to write SMf(O) = 4.rr

faa D~(r)r 2 dr a

= -2

1 0

d r-D1(r) dr dr

= -2aDl,(a )

+

zfoa D,{,(r) dr

2 = - - sin(2M.rra ) .rr

+2

1a

sin(2.rrMr ) dr . o .rr r

The second term tends to 1 when M ---+ oo, whereas the first term oscillates between ±2/.rr; in detail 2

lim inf SMf(O) = I - - , M

7r

limsupSM f(O) = 1 + M

~7r

This example provides a concrete illustratio n of the nonlocal dependen ce of Fourier inversion in three dimension s. The function is smooth in a neighborh ood of x = 0 but has a jump at Jxl = a. The jump effects the impossibi lity of Fourier inversion at x = 0. Figure 2.5.1 gives the profile of the spherical partial sum for this example. Kahane ( 1995) generalize d this example to the setting of a bounded region in JR 3 bounded by a smooth surface. If the surface is analytic and if the spherical Fourier inversion fails at a single point, then the surface must be a sphere and the point must be the center. For more general smooth surfaces, one may have divergenc e of the spherical Fourier partial sum at any preassigne d finite set of points. This example can be modified to provide a concrete example of nonlocaliz ation.

Example 2.5.7. Let f(x) = I for 0 < a < lxl < band zero otherwise. Then by applying the previous example twice and subtractin g, we have 2 SMf(O) = -[sin(2M .rra)- sin(2M.rrb )] + o(l) . .rr Clearlyf( x) = Ofor lxl
Exercise 2.5.8. Let n = 2 and let f(x) = 1 for 0 < lxl < a and f(x) = 0 otherwise. Prove that if x =I= 0, then r ---+ ]x
1·

144

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

1 + .2. 7t

0.2

0.4

0.6

1.2

0.8

1.4

FIGURE 2.5.1 The spherical partial sum of the indicator function of the unit ball in JR 3 with M

= 99j2rr.

Theorem 2.5.5 can also be formulated in the case of even dimensions. The basic case of two dimensions is dealt with as follows.

Proposition 2.5.9. Suppose thatf E L 1 (1R 2 ) and that the spherical mean value r---+ jx(r) is absolutely continuous with f 000 t1;(r)l dr <ex>. Then limM SMf(x) ::::; h(O

+ 0).

Proof. Appealing to (2.5.4) and (2.5.5) and the identity (dfdr)J0 = -J" we have

1

00

SM(x) = 2rr

MJ 1 (2Mrrr)]x(r) dr

00

=-

1 0

=fx(O)

d

]x(r)-J0 (2rrMr)dr dr [""

+ Jo

. d Jo(2rrMr) d/Ar)dr.

The final integral tends to zero by the dominated convergence theorem.

•

Exercise 2.5.1 0. S/tow that the conclusion of Proposition 2.5.9 holds true iff has compact support and the spherical mean r ---+ ]x(r) is assumed to be piecewise absolutely continuous. In particular for f = 1[0.aJ(Ixl), we have spherical Fourier inversion at x = 0. Now we can formulate Theorem 2.5.5 in even dimensions.

Theorem 2.5.11. Suppose that n = 2k and the spherical mean r ---+ ]x(r) is absolutely continuous, together with its derivatives of order j = 1, ... , k - 1 and

FOURIER TRANSFOR MS ON THE LINE AND SPACE

145

that roo ,J-il]x
Exercise 2.5 .12. Complet e the details of the proof of Theorem 2.5.11, followin g

the correspo nding reductio n in the odd-dim ensional case.

2.5.2

Piecew ise Smooth Viewpo int As in the case of one-dim ensional Fourier inversion , it is not possible to obtain any necessar y conditio ns for spherica l Fourier inversio n in lRn. Howeve r we can isolate a class of function s for which we can obtain some simple necessar y and sufficien t conditio ns for converg ence of SMf(x), M---+ oo. The original referenc e is Pinsky (1994). Definiti on 2.5.13. f

L 1 (lRn) is piecewis e smooth of degree k with respect to x E lRn if there exists a subdivis ion 0 = ao < a1 < · · · < aK such that r ---+ fx(r) is absolute ly continuo us on each subinter val, together with its derivativ es of order k - 1 and that J 00 rj-l l]x- JY)
+ 1 and that f E L 1 (lRn) is piecewis e smooth of degree k with respecn o x E lRn. Then limM SMf(x) exists if and only if r ---+ ]x(r) is of class ck-l; in detail o]Y) (a;) = 0 for 1 < i ::: K, 0 ::: j < k - 1.

Theorem 2.5.14. Suppose that n = 2k

Proof. The computat ions in the previous section can be repeated in this context on each subinterv al (a,_ 1 , a,). Each time we integrate by parts on (a;_ , a;), we obtain a contribut ion 1 from the endpoints . When these are" summed, .the resultant contribut ion can be expressed in terms of the jumps. In detail, we have, k

(2.5.6)

/Mf(x) = Wn-l

K

~ ~ ( 2~) 00

Wn-l

{

+ (2 7r)k Jo

(

r

j

1 d

2

a7- JTJM(a;)IJZ,- 21 (a;)8lxU-I >(a;)

-;. dr

)k ( n- 2

r

-

TJM(r)fx(r )

)

sin(27rM r) 1rr

dr.

!f each of the jump terms is zero, then we obtain the desired converge nce: limM SMf(x) = fx(O + 0). Converse ly, suppose that limM SMf(x) exists. Then it must be equal to]x(O + 0), by Gaussian summabil ity. The final integral also converge s to]x(O+O ), by one-dime nsional Fourier inversion . Therefore the sum involving the jump terms must converge to zero. If we now divide by M
•

lim "'8fx(a; ) cos(Ma, -On) = 0. M-+oo~ i=l

To complet e the proof, we state and prove a lemma on finite trigonom etric sums.

Lemma 2.5.15. If {C;}~ 1 are complex numbers and (} limx->oo L~ 1 C; cos(xa; - (}) = 0, then C; = 0.

E lR

is such that

146

INTROL "CTION TO FOURIER ANALYSIS AND WAVELETS

Proof. Multiply the sum by cos(xa1 -(}),integrate over [0, M] and divide by M. Each term in the sum tends to z
4Cj.

K

'f...J

\

lim '""'- 8fx0 ' ~ ... ) cos(Ma, - (}n_z) = 0.

M-+oo~ l=l

-

'!

Applying the lemma once more we see that o]}nx(a;) inductively proves the result.

=

0 fori

=

I, ... , K. Continuin~ •

The results of this section can be reformulated in terms of the smoothness index, which is defined as follows: If the spherical mean value r -+ fx (r) is discontinuous, we set J(f; x) = -1. Otherwise r-+ jx(r) is continuous with a certain number of continuous derivatives, denoted J (f; x). The convergence theorem for piecewise smooth functions can be rephrased as follows, where [ ] denotes the integral part.

Theorem 2.5.16. Suppose that f E L 1 (lR.n) is piecewise smooth with respect to x E Rn with smoothness index J(f; x). Then the spherical partial sum SM.f(x) converges when M-+ oo if and only if J(f; x) > [(n- 3)/2], in which case the limit is]x(O + 0). If J(f; x) < [(n- 3)/2], then we have -00< lim inf M-k(SMf(x) -}x(O + 0))< lim supM-k(SM.f(x) -}x(O + O))
M

where k = (n- 5- 2Jlf; x))/2 > 0

2.5.3

Relations with the Wave Equation

We have already seen the close relation between Fourier analysis and the partial differential equation of heat flow, whose steady-state limit is the Laplace equation. In each case the solution is defined by integration of an approximate identity applied to the initial-boundary data, as we have seen in detail. When we come to the wave equation the situation is different, since the solution is no longer expressed as an integral with respect to a positive kernel, but rather a Schwartz distribution on the surface or interior of a sphere. To see this in detail, consider the initial-value problem for the wave equation

(2.5.7)

a2 u -a2 t

(2.5.8)

L n

=

f).u

:=

u(x; 0) =f(x),

a2 u

--2 i=l axi

au

-(x; 0) = 0

at

where/ E Sis a rapidly decreasing function. This initial-value problem can be solved in terms of the Fourier transform] by the formula

(2.5.9)

u(x; t) =

{

]JR.

cos(2rrti~l)j(~)e 21ri~·x d~.

FOURIER TRANSFORMS ON THE LINE All..!5 SPACE

147

It is immediately verified that u solves the wave equation with the given initial conditions, in the sense that lim 1 ~o u(x; t) = f(x) and lim 1 ~o au;at(A:·t) = 0. A corresponding formula can also be developed for the more g~eral initia' ~onditions, which is left as an • • > exercise. · ~ In order to proceed further, we write (2 :. 'J) as ,, .

u(x; ,., ................

~) =

1 c~~(2rrtJ.L)S'(J.L) 00

dJ.L

4

where

e2rril;·xj(~) d~'

S(J.L) = SJ(x) := {

S'(J.L) =

11/;I<JL

{

e2rril;·xj(~)wJL(d~),

11/;I=JL

where wJL (d~) is the surface measure on the indicated sphere. Since/ E Sit follows that j E Sand that both J.L -+ S'(J.L) and t-+ u(x, t) are rapidly decreasing functions. Hence we can apply one-dimensional Fourier inversion to obtain

1

00

S'(J.L)

=

cos(2rrtJ.L)u(x; t) dt.

Integrating once more on [0, M] and applying Fubini, we obtain the following proposition. Proposition 2.5.17. Suppose that f E S and that u(x; t) is the solution of the initial-value problem (2.5.7) and (2.5.8)/or the wave equation. Then the Fourier partial sum can be retrieved through the formula 00

(2.5.10)

SMJ(x) =

1 0

sin(2rr Mt) ]"( t

u(x; t) dt.

Formula (2.5.1 0) was developed and applied by Pinsky and Taylor ( 1997) to study pointwise Fourier inversion oniEuclidean space and other classical spaces on which the wave equation has a known solution. Proposition 2.5.17 will now be used to find an explicit representation for u(x; t), not involving the Fourier transform. To do this, recall the representation of SMf(x) in terms of the Dirichlet kernel

1

00

SMf(x) = Wn-l

D'M(r)]x(r)rn-l dr.

Since/ E S. the function r -+ ]Ar) is also smooth and rapidly decreasing, so that we can integrate-by-parts and use the properties of the Dirichlet kernel to write D'M(r) _ -(Ij2rrr)(8j8r)D'M- 2(r) and obtain if n = 2k + 1 Wn-l { SMJ(x) = (2n-)k

1o

00

sin 2rrMr ( a rr r r rar

)k [rn-2-fx(r)] dr.

But Proposition 2.5.17 also provides a representation in terms of the same onedimensional Dirichlet kernel. Therefore we obtain the following proposition.

148

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proposition 2.5.18. The solution of the initial-value problem (2.5. 7) and (2.5.8) is given by the explicit formula (2.5.11)

u(x; t) =

Wn-I k

(27r)

t

0 ( tot )

k

-

[tn- 2fx(t)].

Example 2.5.19. lfn = 3, then the solution is u(x; t) = (d/dt)[t]x(t)] = t];(t) + jx(t). The second term corresponds to a measure on the sphere {y: ly-xl = t}, while the first term corresponds to a (dipole) distribution, which is the first derivative of a measure-in the sense of Schwartz distributions. The representation formula (2.5 .11) can be used to exhibit the finite speed of propagation of the wave equation, as contr~ted with the "instantaneous speed of propagation" of the heat equation.

Proposition 2.5.20. Suppose that f(y) = 0 in a ball of radius a centered at x. Then u(x; t) = 0 fort < a. Proof. If f derivatives.

=0

in the ball,

then

]x
for t
likewise for all higher time •

The odd-dimensiona l wave equation also exhibits Huygens' principle, stated as follows.

Proposition 2.5.21. Suppose thatf is supported in a ball ofradius R centered at x. Then u(x; t) = 0 fort > R. Proof. In this case the surface integial]x(t) = 0 fort > R, likewise for the higher time derivatives. •

The wave equation in even-dimensiona l space can be soJved by the method of descent, which we illustrate in the three-dimension al case. Iff e S(1R2 ). we define F(x, y, z) = f(x, y). If we parametrize the upper and lower halves of the sphere by writing w3 = ±.Jt 2 - wf - wi. the surface integral over the sphere of radius t in three dimensions is transformed into the two-dimensiona l integral F x.y,O (t) _-

1

--

21Tt

1

lwi
f(x

+WI,

Y

+

W2)

.Jt2- wi- w~

d

WI

d

W2.

From this formula we see that the finite speed of propagation still holds: iff = 0 in a disk of radius a centered at x e 1R2 , then u(x; t) = 0 fort < a. However Huygens' principle is not valid in two dimensions, since iff is supported in a ball of radius R, the solution u(x; t) is written as an integral over the interior of the disc of radius t about x, hence will be nonzero for all t. · Figure 2.5 .2 illustrates Huygens' principle; f = 0 outside the sphere of radius R about (0, 0, 0). If ct > R + lxl. thenf = 0 on the surface of outer sphere (labelled Ill) where the integration is performed.

-

FOURIER TRANSFORMS ON THE LINE AND SPACE

149

FIGURE 2.5.2 Illustrating Huygen's Principle. FromM. Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications. Used by permission of The McGraw-Hill Companies.

2.5.3.1 1he method of Brandolini and Colzani The wave equation can be effectively used to study pointwise Fourier inversion and localization. In this section we treat the case of the Fourier integral. Brandolini and Colzani (1999) have also treated the corresponding problem for multiple Fourier series. We begin with the spherical partial sum operator SMf(x)

= f

ji~I~M

=

f

lrrt·

f(~) e 2 1ri~-.x d~

lro.MJ (I~ l)f(~) e 21ri~-.x d.;.

Let 1/1" E eN (lR) for some N > n/2 be a nonnegative even function with frrt 1{1" = 1 and lfr(t) = 0 for ltl > €, for some € > 0. For example we can take an iterated Fejer kernel (2.5.12)

1/J"(t) = CN8 ( 1

-COS

2:rr8t)N

(:rr8t)2

where eN is a positive constant. Explicit computation shows that {/J-(t) = 0 for ltl > N8, so that we can take 8 = € IN. Without loss of generality, we take 8 = 1 in what follows.

150

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Now lro.MJ is the restriction to the positive real axis of the even function 1r-M.MJ- Writing l[-M,M] = 1[-M,M] * 1/F + 0[-M,M) - 1[-M,M) * 1/F), we have SMf(x) = SlJ(x)

+ S1f(x)

where

r

S1f(x) : =

}JRn

lr-M.M]

0[-M.M) -

* 1/F)(I~ l)](~)e2Jri~-x d~.

The success of the method depends on two fundamental points: (i) S}.,J(x) depends only on the values off in a ball of radius E about x. (ii) S1f(x) is essentially bounded by a constant multiple of fM:siH:sM+l lf(~)l d~. To see this, first write the convolution Or-M.MJ

* 1/F)(t)

=

l_

2

e rristj[-M.MJ(s)..,lr( s) ds 00

=2 SlJ(x) = 2

1

cos(2rrst)

~

1/F(s)ds

Jl"S

foo lfr(s) sin 2 rr Ms ( { ](I.; l)e 2rri~-x cos(2rr sl.; I) Jo rrs Jan 00

= 2

sin 2rrMs

1

1/F(s)

sin2rrMs

0

d.;)

ds

u(s, x) ds

Jl"S

where u is the solution of the wave equation uu = du with u(O, x) = j(x), u 1 (0, x) = 0. Indeed, this equivalence has been demonstrated for the class S, and the solution formula (2.5 .11) easily extends to f E L 1 + L 2 . This formula shows two important properties.

=

=

• Ifj 0 in a ball of radius E about x, then S1f(x) 0. • If s ---+ u(s, x) satisfies a Dini condition at s = 0, then limM S1f(x)

= j(x).

To estimate S1f(x), we note that lr-M.MJ * 1/F(t) = 1,~: lfr(y) dy. Assuming the iterated Fejer kernel (2.5.12) with 8 = 1, we see that forM> 1

1

/ti>M

11/F(t)j dt <

l.

M

CN ZN-l,

M> 1

1/F(t) dt = 1

which can be combined into the overall bound

1

/t/>M

.l.(t) dt < o/

-

(1

eN

+ M)2N-l,

0 < M < oo.

FOURIER TRANSFORM S ON THE LINE AND SPACE

We use this to estimate the difference Or-M.MJ

* 1/f)(t)

151

- lr-M.MJ(t) separately for ltl >

M and ltl < M. In the first case 1[-M.MJ(t) = 0 and we have

1[-M.M]

* 1/f(t)

=

f

t+M

1/f(y) dy <

t-M

!= t-M

1/f(y) dy < (1

+

CN I t - MI)2N-I ,

while in the second case 1r-M.MJ(t) = I so that 1[-M.M]

* 1/f(t)-

1 =

f

t+M

1/f(y) dy-

l

t-M

<

<

Propositi on 2.5.22. Iff the Tauberian condition

E

lim M

1/f(y) dy

lR

(!<X> + ~t-M) 1/f(y) dy t+M -<X> eN (1 + It- MI)2N-I.

L 1 (ll~n)

+ L 2 (1Rn), and the Fourier transform satisfies

f

}M:::_2rrii;I~M+l

lf(~)l d~ =

0,

then S'J.tf(x) -+ 0; if in additions -+ u(s; x) satisfies a Dini condition at s = 0, then we have pointwise Fourier inversion: limM SM}'(x) = f(x). Proof. From the above computatio ns, S'i,/(x)

= {

jR"

[11-M.MJ

{

~eN JR"

* 1fr(l~l)]f(~)e 2 ,.u--~ dx 1

A

o +II~ I- MI)2N-l lf<~>l d~.

Iff E L 2 (1R") this can be estimated by Cauchy-Sc hwarz and applying the dominated conver$ence theorem, whereas for f E L 1 (lR") it can be estimated by removing the supremum off(~) and applying the dominated convergenc e theorem. •

Example 2.5.23. Suppose that f E L 1 (1R 2 ) is defined by a smooth function on the interior of a convex region with a smooth boundary with nonzero curvature, and defined to be zero outside. Then the Fourier transform satisfies the asymptoti c estimate lf(~)l < C/1~1 3 1 2 . Hence ·

f

JMsl2rrl;i:::. M+l

li
<

M

~12 M= o ( v ~) M

__.,. o.

M-+ oo.

This example satisfies the Tauberian condition, hence we have Fourier inversion and localizatio n at every point.

152

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

2.5.4

Bochner-Riesz Summability

Closely related to the techniques of spherical Fourier inversion is the notion of BochnerRiesz summability of the Fourier integral. This is a natural substitute for the Fejer means, which have no direct counterpart in IRn if n > 1. The Bochner-Riesz means of order a > 0 are defined by

B'f.tf(x)

(2.5.13)

=

1 (1- l~l:)aJ
lf;I::OM

By Fourier reciprocity, this operator can also be represented as the convolution with K~. the inverse Fourier transform of the function ~ ~ (1 - 1~ 12 I M 2 )a 1 (O.M) (I~ 1). In detail, the Bochner-Riesz kernel is

K~(x)

=

1

(1- 1~12)a d~

e2rrix-t;

= C

{M ( 1 -

n

2

M

lf;I::;M

J.L 2

Jo

M2

= 2:rr lxl-(n- 2 ) 12

1

)ot

n-1 J.L

J(n-2)/2 (2:rr J.LX) (2:rr J.Lx)
d J.L

1

(1 - s 2 )'" J(n-2)/2 (2:rr lxls)sn 12 ds.

This integral is evaluated in the appendix to this chapter, with the result (2.5.14)

Proposition 2.5.24. If a > particular for every f

E

(n - 1) /2. then K~ is an approximate identity. In LP(JRn), 1 =::: p < oo, IIB'f.tf- fliP~ 0 when M ~ oo.

Proof. From the properties of Bessel functions, we have the bound Mn

iK:,(r)i :S Cncx (1

Therefore

{

J.a• iK:,(x)l dx :S Cna

+ Mr)cx+(n+l)/2.

1"" (l + 0

sn-l s)"'+(n+l)/

2

< 00,

which proves that the L 1 norms remain bounded when M --+- oo. In particular the integral of KM can be computed from the Fourier inversion at ~ = 0:

(1- ~1::122)a = 1 '>

.

K:,(x)e2rri<x dx,

JRn

1

= f

J!R."

1~1 :S M

K:,(x) dx.

Finally for any 8 > 0

1

1->l>li

which proves that

K'M

iK:,(x)ldx =

1

Kf(y)dy--+- 0,

Jy/>o/M

is an approximate identity.

M--+- co,

•

FOURIER TRANSFOR MS ON THE LINE AND SPACE

153

One can also prove a.e. summab ility, as follows. Proposi tion 2.5.25. /fa> (n- 1)/2 andf E IJ'(lRn) with 1 < p < oo, then for every Lebesgu e point, limM Uf.,f(x) = f(x). In particul ar this holds at almost every x E lRn. Proof. Without loss of generality we can suppose that x = 0. Furtherm ore we may replace 2 - j(O)e-1x1 to reduce to the casej(O) = 0. Define

f by j(x)

(r) E(r) := - -

lf(y)l dy,

(r) := {

r"

Jly\9

From the hypothes is of Lebesgue point we have E (r) --+ 0 when r --+ 0. Now if 1 for larger we can use Holder's inequality to write (r) ~ Cn,p

E(r) < C

r" 1P',

II fliP

-

n,p

r-n(l-lfp')

'

~

p < oo,

p' =pf(p- 1)

so that E is a bounded function. In case p = oo it is immediat e that E is a bounded function. Now we can write UXJ(O) = jUXJ(O)I

{

JIR"

K'f-t(x)j(x ) dx

~ c,.M" Ln =

+ Mlx~)a+
(1

{00

c,.M"

Jo

1

(1

+ Mr)"'+
= c,.,aM"+I {ooo

Jo

= c,.,aM" {ooo ] 0

-

c

{oo

n,a

(1

(1

dll>(r)

1 It>( ) d + Mr)a+(n+3 )/2 r r

(sjM)

+ s)"'+(n+3)/ 2

s"E(sjM)

Jo (1 + s)ot+(n+3)/2

ds

ds

where we have integrated -by-parts and used (0) = 0 in the fourth line. The final integrand tends to zero when M --+ oo and is dominate d by an integable function since ex > (n - 1) /2 and Eisa bounded function. Hence IJM/(0) --+ 0 when M --+ oo, which was to be proved .

•

2.5.4.1

A general theorem on almost- everywh ere summab ility

It is possible to abstract the features of the Bochner -Riesz kernel to prove a general theorem on almost-e verywhe re summab ility, originall y due to Caldero n and Zygmun d (1952).

Theorem 2.5.26. Suppose that kr(x), T which is majorize d in the form

~

oo is an approxim ate identity on Rn X E

JR"' T > 0

where K : [0, oo) ~ [0, oo) is a decreasi ng function with K(O) < oo, IIK11 := 1 f1Itn K(lxl) dx < oo. Suppose thatf E L 1 (lR") and that 0 is a Lebesgu e point off.

154

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Then lim T--+00

f

}JR.n

kr(x)f(x) dx =f(O).

Proof. The approximate identity can be applied directly to the bounded continuous function x ~ e-lxl 2 ; hence we may replace/ by the integrable functionf(x)- j(O)e-lxl 2 , and reduce attention to the case/(0) = 0. Having done this, we let (r)

lf(x)l dx,

(r) := {

E(r) := - - . rn

Jixi::Y

Clearly (r) :::: 11/11 1 and, since 0 is a Lebesgue point, lim,~o E(r) = 0. Now we write

€

is a bounded function with

1

00

lin kT(x)f(x)dxl :Sin T"K(Tixl)lf(x)id x =

T"K(Tr)d(r).

This will be integrated-by-part s, following the remarks that (i) since IIK11 1 < CX), we must have lim,_. 00 K(r) = 0 and (ii) since <1>(0) = 0 and is bounded, both terms at the limits vanish and we can write

1oo

T"K(Tr)d(r) =

-1

00

T"(r)dK(Tr)

=-

1

= -

1

00

(Tr)"E(r) dK(Tr)

00

s"€ ( ; ) dK(s).

But another partial integration shows that

-1M

s"dK(s)

=

-~ K(M)

+1M

K(s}s"- 1 ds ::=::

CIIKIIt

<

CX)

so that s"dK(s) is a finite measure on [0, CX)). Meanwhile, the function E(s/T) is uniformly bounded and tends to zero when T ~ CX). Therefore by the Lebesgue dominated convergence theorem, the last integral tends to zero when T ~ CX), completing the proof.

•

Exercise 2.5.27. Extend the previous result to f E LP(JRn), 1 < p < oo with a Lebesgue point at x = 0.

2.6

BESSEL FUNCTIONS

Here we give a self-contained development of the necessary facts about Bessel functions. In Chapter 1 we encountered the so-called modified Bessel function lm(t), defined by a power series with positive coefficients. The standard Bessel function Jm(t) is defined by (2.6.1)

oo

Jm(t) =

. (t /2)2j+m

~(-1)' -.,....:..-.+_m_)_! 1

0

m > -1.

FOURIER TRANSFORMS ON THE LINE AND SPACE

155

This power series converges in the entire complex plane. If m is not an integer, we define the factorial as the Gamma function m!

=~a= tme- dt 1

= r(m

+

1).

Proposition 2.6.1. lfm, v > 0, the Bessel functions satisfy the relations d -[tmJm(t)] = tmJm-J(t) dt

(2.6.2)

t > 0

d

-[t-m Jm(t)] = -t-m Jm+l (t) dt

(2.6.3)

t > 0

(2.6.4) J:;,(t)

+

1 tJ:r,(t)

+

( 1-

m2) Jm(t) fZ

= 0

(' = : , )

t>O

(2.6.5) (2.6.6) The first three are obtained by termwise differentiation of the power series. For the fourth, we substitute the definition of J m into the integral and integrate term-by-term:

1

1

o

Jm(Rs)sm+l(l- s2)vds =

1' ( o

1

00

~~)-lY (Rsj2)m+2j) ., . I sm+l(l- s2)vds j=O J.(j + m).

oo

.

(R/2)m+2j

1' .

= - L(-1)' . . rm+1 (1- r)vdr 1 + m).1 o 2 j=O J.(j 2vv! oo . (R/2)m+v+l+2j = Rv+l L j=O( - l ) ' J."I(m + v + J· + 1)1. 2vv! = Rv+l Jm+v+l (R) The final integral formula is obtained by making the substitution s = sin 8 and recognizing the power series coefficients from Chapter 1.

Exercise 2.6.2. Prove (2.6.2) by termwise differentiation of the power series definition (2.6.1). Exercise 2.6.3. Prove (2.6.3) by termwise differentiation of the power series definition (2.6.1 ).

Exercise 2.6.4. Prove (2.6.4) by termwise differentiation of the power series definition (2.6.1 ). Exercise 2.6.5. Complete the details ofthe proofof(2.6.6).

156

INTRODUcriON TO FOURIER ANALYSIS AND WAVELETS

The asymptotic be.havior of J m(t), t --+ oo is most efficiently deduced from a differential equation. Let y(t) = v'fJm(t). From (2.6.4). we have by successive differentiatio n Cy 2 1 (2.6.7) C . ·-m 0 < t < 00, -y" +Y = 2• 4" t

This implies that y and Y 1 remain bounded when t --+ oo. since d -(y2 + 12) = 2 I+ 2 I fl = 2Cyy < _ICI _(y2 +y12). dt y yy y y t2 t2 I

Hence fort > to. y(t) 2 + y (t) 2 < [y(to) 2 + Y 1 (to)] exp 1

(1c1 i' ~:)

< [y(to)

2 + y'(to)]e 1 CI/to,

which proves the required boundedness . From this we have the representatio n (2 .6. 8)

y(t) =

l

~

r

.

0W

.

sui(t-s)-2--ds+A 1 cost+A2sm t. S

Indeed. the first term on the right of (2.6.8) is a solution of the differential equation (2.6. 7). so that it differs from y(t) by a solution of the homogeneou s equation z" + z = 0. whose general solution is A 1 cost+ A 2 sin t. From this follows an asymptotic formula for y(t). since the integral term is bounded in the form

~~~ sin(t- s) c~;s) dsl ~ ~~ IC~;s)l ds =

o( ~),

t--+

00.

We summarize the above work as follows.

Proposition 2.6.6. The Bessel function Jm(t) satisfies the asymptotic relation .JiJm(t) = A1 cost+ A2 sin t

+

o( ~),

t--+

00

for suitable constants A1, A2. Equivalently , we may write Jm(t) =

~ cos(t- &) + o(t31/2)'

t--+

00

for suitable constants A, &.

The constants A, e can be explicitly identified in the case of integer m by the method of stationary phase. described in the next section. In the case of half integer m. it is often possible to identify the constants from elementary formulas. beginning with J1;2(x) = ...;'21irX sinx.

Exercise 2.6.7. Prove that the derivative of the Bessel function satisfies the

asymptoticf ormula (d/dt)(v'tJm (t)) = -A 1 sint +A 2 cost+ 0(1/t), t--+ oo. Hint: Compute Y 1 (t) from (2.6.8).

FOURIER TRANSFORMS ON THE LINE AND SPACE

2.6.1

157

Fourier Transforms of Radial Functions

Iff e L 1 (.!Rn), the Fourier transform j is
Proposition 2.6.8. Jfcp

f(~) =

(2.6.9)

Iff

foM

E

S(O, oo), and we setf(x) = cp(lxl), then

r=

Jo

lp

(r)J(n-2)j2(2.rrrl~l) rn-l d. (rl~ i)
IF(lRn), 1 < p < 2, then this integral is interpreted as a limit in LP' (1Rn) of when M ~ oo. E

This follows from the representation of Bessel functions, specifically (2.6.6) with = .rrrl~l- For details, see Stein and Weiss, 1971, p. 154. We can use the representation (2.6.9) to prove additional properties of Fourier transforms of radial functions for restricted values of p e (1, 2). m

=

(n/2)- 1, t

Proposition 2.6.9. Suppose that 1 < p < 2nj(n+1). ThentheFouriertransfor m (2.6.9) is a continuous function for~ =I= 0, which vanishes at infinity and satisfies

Proof. Recalling the bounds on the Bessel function and applying Holder's inequality, we have

J(n-2)/2(2rrrf~f) ~-1 d f

{M

I

Jo

cp(r)

(rl~ i)
r

~

M

(1

-

1 0

r

<

(1

MI

-

-

lcp(r)f~-l

{M Cn Jo

0

+ rl~ i)
f.n(r)fr
cp(

r

)I

P rn-l

dr

r
+ rf~ l)
dr )l/p(1M (1 0

d

r

~-I d r

+ rj~f)«n-l)p')/2

)1/p' ·

The final integral converges when M -+ oo if and only if (n- 1) - p'(n- 1)/2 < -1, which is equivalent to 1/p' < (n- 1)/2n or ljp > (n + l)j2n. The continuity for~ =1= 0 now follows from the dominated convergence theorem. To prove that] vanishes at infinity,

158

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

we write for any

E

> 0,

r + 1,

r J(n-2)/2(2rrrl~l) r"-1dr

00

/(~) A

r

IJo

cp

-

(

)

Jo

(r)J(n-2)/2(2rrrl~l) rn-1

drl

12

cp( )

(rl~ i)
r

icp(r)lrn-1 dr


Jo

-

=C.,

::5 C.,

=

11 •

cp

(rl.;i)
-

(1' f1 (1'

icp(r)IPrn-l dr

n

1= •

YIP' (1' r"- 1drYIP'

icp(r)IPr"- 1 dr

Yip'

lcp(r)l rn-1 dr (rl.;i)
C., ~~~(n-1)/2

<

lcp(r)lr
C.,E

(r) J(n-2)/2(2rrrl~l) rn-1 drl < C

oo

1'

C., l.;l(n-1)/2

1oo (1oo

lcp(r)lr(n-ll/p r
,

,

icp(r)lprn-1 dr) 1/p

(1oo ,

rq dr) 1/p'

J

where q = n - 1 - p'(n- 1)j2. Taking E = 1 shows that tf(~)IP ::5 C.,P 000 lcp(r)IPr"- 1 dr provided thatp' > 2nj(n- 1). On the other hand, for this same range ofp we have for any E > 0,

Taking E --+ 0 shows that the limit is zero, as required. To see that this range of p is sharp, let cp(r) = a

n Jn;2(21f ra) (ra)n/2 ,

which is the Fourier transform of the indicator function of a ball of radius a. From the asymptotics of Bessel functions, we have cp(r) = [C + 0(1/r)] cos(r- 8)jr1 2 when r--+ oo so that cp(l · I) E LP(JR.") if and only if oo > / 00 r1 2dr, which happens if 1 and only if p > 2nj(n + 1). But] is discontinuous at the sphere of radius a, which provides the required counterexample. •

The above results can be extended, in a suitable form, to nonradial functions by means of the restriction theorems, described below. These results depend in part on the complex interpolation method, to be developed in Chapter 3.

2.6.2

! 2 -Restriction Theorems for the Fourier Transform

The Fourier transform off E L 1 (lRn) is a continuous function that satisfies the pointwise bound tf(~)l < llfllt· However iff E LP(lRn) with p > 1 we cannot expect any such

FOURIER TRANSFORMS ON THE LINE AND SPACE

159

pointwise bound in general. However iff is a radial function, we have shown in the previous subsection that lf(~)l :S: Cnpll/llp for 1 < p < 2nj(n + 1), hence this same estimate applies to the average over a sphere. The restriction theorems generalize this, by bounding the L 2 norm of the Fourier transform on a sphere in terms of suitable LP norms. We have the following proposition, attributed to Tomas (1975).

Proposition 2.6.1 0. Letw(d()) be the uniform surface measure on the unit sphere. Letf

E

S, and 1 < p < 4nj(3n

+

1). Then for some constant A= Ap, we have

(2.6.10) Proof. The Fourier transform of the surface measure is given by w(x)

= {

Js•-•

e-2:trix'<w(d~) =

c

J(n-2)/2(2rrlxi) lxi
and satisfies w(x) = 0(ixi 0 -n>1 2 ), lxi --+ oo. Hence w E Lq(IR."), provided that q > 2n/(n - 1). Now by Fourier reciprocity, Holder's inequality and the Young convolution estimate, we have

.::o llf *fllq· llwllq .::o llfll;llwllq where (2/p) = 1 + 1/q'. But q > 2n/(n- 1) implies that 1/q' > (n 2/p > (3n + l)/2n, which was to be proved.

2.6.2.1

+

l)/2n so that

•

An improved result

We can extend the range of p-values in the previous result by the use of Stein's complex interpolation method, which will be developed in Chapter 3. In this method, we imbed the given problem in a one-parameter family of operators. A family of convolution operators is defined by the kernels Kz(X) = Cz Jcn-2)/2 + z(2.rr I~ I) ~~ l(n-2)/2+z

(2.6.11)

1-n < Re z :S: 1. 2

At the endpoints we have

1-n

zo = -

Zl

- , Kz(x) = cos(2.rrlxl), 2

a bounded function,

= 1 K (x) = Jn;2(2.rrl~l) ' z ~~~n/2

with a bounded F.T.,

Kr

= lro.IJ(Ixl).

Therefore Kzo maps L 1 --+- L 00 , and Kz, maps L 2 --+- L 2, so we can use the complex interpolation method with Po

=

1, qo

=

oo, zo

Pr

=

2, qr

=

2,

Z1

1-n -, 2

= -

=

1.

160

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Thus 1

1-t

t

1

1-t

t

t

p=-I-+2= 1 -2' t

-=--+-=-. q CXl 2 2 1-n

z = (1 - t)-2-

+

t.

In particular p and q are conjugate exponents, 1/p + 1/q = 1. In order to have z = 0 we must take t = (n - 1)/(n + 1), hence 1/p = 1 - (t/2) = (n + 3)/(2n + 2), p = (2n + 2)/(n + 3). We now apply this to the Fourier reciprocity formula and apply HOlder's inequality to write

1"-'

2

lf(8)1 w(d(}) = <

L" if

*f)(x)w(x) dx

11/llp 111 * wllp'-

w

The Fourier transform of w is given by = K 0 • Each member of the one-parameter family maps the Lebesgue space into its dual, in particular

from which we conclude the following theorem. Theorem 2.6.11. Let w(d(}) be the uniform surface measure on the unit sphere. Letf E S, and 1 < p < (2n + 2)/(n + 3). Then for some constant A = Ap, we have

(2.6.12) In Chapter 3 it will be proved that the Fourier transform can be extended as a bounded operator from LP(lRn) to LP' (!Rn) if 1 < p < 2. In this setting the restriction theorem is also valid, by taking limits in the space S. Many of the properties of the Fourier transform on L 1 (!Rn) have counterparts in this wider setting. In particular, the restriction theorem may be used to prove the following result on the average decay of the Fourier transform of a function in LP for a suitable range of exponents. Proposition 2.6.12. Suppose that/ E LP(!Rn) where 1 < p:::;:: (2n Then J;~I=I lf(rg)l 2 w(dg) ---+ 0 when r---+ CXl. Proof. Definef,.(x) = r-"f(xjr) for r > 0, x

],(~) =

r-" {

Jan

e- 2:n:ix·l;f(xjr) dx

E

+ 2)/(n +

JR.". Then

= {

}Rn

e- 2 :n:iry 'f(y) dy

= f(r~)

3).

FOURIER TRANSFORM S ON THE LINE AND SPACE

161

Applying the L 2 restriction theorem, we have

Lf(r~)l 2 w(d~) ~ Cpll/rll 2 =

1

Cpr 2n(l-p)/PIIf1 1;-+ 0

1/il=l

•

sincep > 1.

Corollary 2.6.13. Suppose that f

E LP(JRn)

Then J;I;I=J(r~)w(d~) ---. 0 when r---.

where 1 < p < (2n

+

2)/(n

+ 3).

<X>.

One may note that the results on radial transform s are valid on a wider set of l.J' spaces than the L 2 -restrictio n theorem, since 2n + 2 2n n> 1 ==> n+3
Exercise 2.6.14. Suppose thatf E l.J'(lRn) with 1 < p < 2nj(n J;I;I=J(r~)w(d~) ---. 0 when r ---. ex>.

+

1). Show that

Hint: Interchang e the orders of integration to reduce to the case of a radial function.

2.6.2.2

Limitatio ns on the range of p

The sufficient conditions on the range of p-values for the restriction theorem are also necessary. as revealed by the following example.

Example 2.6.15. Let f be the Fourier transform of the indicator function of the rectangul ar region R defined by the inequaliti es 1- €

where 0 <

E

<XI <

1,

< 1.

In detail f(x) =

= =

lf(x)l =

l e-2:nil;·xd~ {'

11-E

e-2rrix,i;,d~l fi (1.../i e-2:n:ixil;,d~j) j=2

-...ji

e-2:nix,(I- E)) 2rr x 1

(e-2:n:ix, -

I

sin rrx 1E 1r X1

IfJ I

rl (sin2rrxj-V'€) j=Z

sin 2rrxj-V'€

j=Z

1r Xj

1r Xj

I·

To compute the LP norm, we note that by direct computati on

11 IR

sin rrxE 1l"X

IP dx =

€

p-I

11 JR

sin rry IP dy = C 1ry

Ep p

_1 .

162

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Applying this to both factors above yields

= CpEp-l_JE(n-l)(p-l) = CpE(p-l)(n+l)/2,

IIJII~

IIJIIP =

CpE(n+l)f2p'.

On the other hand, the L 2 -norm of the Fourier transform on the unit circle is

Hence

If the restriction theorem (2.6.12) were valid, then we would have E(n-1)/4

This can be true for all equivalently

E

>

n-1

<

AE(n+l)f2p'.

0 if and only if we have (n 1

1

- - - - >- = 1-2(n 1) - p' p

+

+

l)/2p' < (n -

1)/4, or

1 n+3 > --p - 2n + 2'

<==>-

which is precisely the range of admissible exponents in the L 2 -restriction theorem (2.6.12).

2.7

THE METHOD OF STATIONARY PHASE

Often we have occasion to make an asymptotic evaluation of a one-dimensional Fourier transform, when I~ I ---+ oo. This can often be done by a simple integration-by-parts, for example iff E L 1 [a, b] is absolutely continuous withf' E L 1 [a, b], we can write

= lb f(x)d a

-

(e-2JTi~x) -27l'l~

J(b)e-2JTit;b- f(a)e-2JTil;a

- 27l' i~

1

lb

27l' l~

a

+ --.-

J' (x)e-2JTil;x dx.

The new integral is o( 1 I~) when I~ I ---+ oo and we have obtained an asymptotic formula. Iff has higher derivatives in L 1 , then this may be iterated to obtain an asymptotic expansion.

163

FOURIER TRANSFORM S ON THE LINE AND SPACE

In many problems of interest, we encounter Fourier transform s when/' ¢ L example the Bessel function

1

,

for

j'

1 eirx Jo(t) = d.x. n - I ~1- x 2 If we make the change of variable X = cos e, 0 < e < 7r, we obtain the formula

17f eirco,o de,

Jo(t) = -1

0

7r

which looks less forbidding , however it is no longer written as a Fourier integral. If we try to integrate- by-parts, we find that the contributi on to the integral from any interior interval tends to zero like 1/t, but this analysis no longer applies near the endpoints e = o, e = n. To handle this and more general "oscillato ry integrals," we develop the method of stationary phase. Specifical ly, we consider complex-v alued functions of the form (2.7.1)

1b

f(t) =

eitrp(x)g(x) dx,

where VJ is a real-value d function called the phase function. The function g(x} may be either real- or complex-v alued. If VJ'(x) # 0, then we may integrate- by-parts and conclude thatf(t) = 0(1/t), t ~ oo. However, if VJ'(x) = 0 for some x, then this conclusio n is no longer valid. In order to find the correct result, we focus attention upon those points x1 where VJ'(x) = 0, the so-called stationary points.

2.7 .1

Stateme nt of the Result

The complete statement of the result is given as follows:

Theorem 2.7.1. The method of stationar y phase: Suppose that g(x), fP(x) have two continuou s derivative s for a < x < b, that lp(x} is real-value d, and that VJ'(x} # 0 except for a .finite number of stationary points x , where VJ"(x ) =f:. 0. 1 1 Let these be labeled so that fl'"(x1 ) > 0 for 1 < j < K and VJ"(x ) < 0 for 1 K + 1 < j < K + L. Then when t ~ oo, (2.7.2)

1b

earp<x> g(x) dx = 1+ (t)

+r

(t)

+

0 (

~)

where (2.7.3)

(2.7.4)

K J+(t) = ~ ~ ;=I

(

2n tm"(x-) .,. J

)

112

earp<x1 >ei1rf4g(x-) :J

164

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

If either of the endpoints x

= a, x = b are also stationary points, then they contribute to (2.7.3) and (2.7.4) with a factor of!·

A simple tool to remember this complicated formula is to observe that the result is identical to what is obtained by replacing (/>(X) by its two-term Taylor expansion and replacing g(x) by its value at each stationary point, then doing the resultant integrals (one for each stationary point), and then summing the results. We illustrate with a typical example.

Example 2.7.2. Apply the method of stationary phase to find an asymptotic formula for the integral

l

rr/2

(2x

+ 3 )e-itcosx dx.

-Jr/2

In this case we have g(x) = 2x + 3, (/>(X) = - cosx, (j>'(x) = sinx, (j>"(x) = cosx. The only stationary point is x = 0, where (/> 11 (0) = +1, g(O) = 3. Applying (2.7.2), we have

t-+

2.7.2

00

Application to Bessel Functions

As a primary application of the method of stationary phase, we propose to identify the constants in the asymptotic behavior when t-+ oo of the Bessel function Jm(t), which is represented by the integral (2.7.5)

m = 0, I, 2, ....

Proposition 2.7.3. The Besselfunction has the asymptotic behavior

(2.7.6)

Jm(t)

=If cos (t- rr/4- mrr/2) + 0 ( ~).

t-+

00.

Proof. From the integral representation (2. 7 .5) we have (2. 7.1 ), where rp(x) = cos x, g(x) = (lj2rr)e-imxe-im:n:f 2 . Since rp'(x) -sinx, rp"(x) -cosx, there are three stationary points: x 0, x 1r, x -1r, with rp"(O) -1, rp"(rr) 1 rp"(-rr); also g(O) (1/27r)e-im7r/ 2 ,g(7r) (l/2rr)e- 3 im,./ 2 g(-1r). Weapplythemeth odofstationaryph ase,

=

=

=

=

=

=

=

=

=

=

=

FOURIER TRANSFORM S ON THE LINE AND SPACE

165

noting that the endpoints contribute with a factor of ~. Hence e''e-i:n:/ 4

J (t) = m

{2ir

27r

+

(

1 1) Ff-7r . . 14 e-31m:n:/2 ( 1) - +-e-"e':n: + 0 2 2 t 27r t

+ {2ir e-it ei:n:/4 eim:n:/2 +

Vt

H,

(.!.) t

v~

=

0

2rr

f l (ei(t-:n:/4-mrr/2) + e-•
=

2.7.3

27r

it -•:n:/4 e-•m:n:/2

Vte e

=

e-im:n:/2

(.!.) t

cos(t- rr/4- mrr/2) + 0 (

~).

•

Proof of the Method of Stationa ry Phase

We now outline the steps used to prove (2. 7 .2). The idea is to reduce the study to each stationary point, where we can approxima te using with the Taylor expansion s with an error of 0(1/t).

Step 1. If the interval c < x < d does not contain any stationary points, then

ld

eit
=

0 (

~)'

t --+

<X>.

Proof. We multiply and divide by cp'(x) and integrate-b y-parts as follows:

ld

g(x)e''v>(x)d x

=

c

ld .g~x) c

=

d (eitv>(xl) dx

ttcp (x)

.!..id

g(x) eitv>(x) lx=d _ itcp'(x) x=c it

c

eitv>(x) !!._ dx

(

g(x) ) dx.

cp'(x)

Both terms are 0(1/t), t-+- oo, and can therefore be included in the remainder term. Therefore we can restrict attention to contributio ns from intervals containing the stationary points. Assume that x 1 is a stationary point for which cp" (x 1 ) > 0, and let ~ > 0 be chosen so that cp(x)- cp(x 1) > 0 in the 'interval x 1 - ~ < x < x 1 +~.We introduce a new variable of integration v through the equation v = (x- x 1 )

cp(x) - cp(x1) (x- x 1 ) 2

X1 -

~

<

X

<

X-

l

+ ~.

The function x-+- v(x) vanishes atx = x~o with v'(x 1 ) = ~cp"(x 1 )/2 > 0. Therefore there exists an inverse function x = X(v) with X(O) = x~o X'(O) = ../2/cp"(x ). • 1

Step2.

where 82

=

v(xr

g(xr )yf2/q;"(x r ).

+

8), -8r

=

v(xr -

8) and G(v)

=

g(X(v))/v '(X(v)), G(O) _

166

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. We make these substitutions in the integral and change variables to obtain the result .

•

Step 3. t ---+

00.

Proof. We write G(v) = G(O) + vh(v), which defines the differentiable function h(v). The second term contributes to the integral

•

as required.

Step 4. t ---+

00.

Proof. This is the Fresnel integral. Readers familiar with complex-variable methods will 2 find this a one-liner: apply Cauchy's theorem to the functionj(z) = e" on the crescentshaped contour formed by the ray z = reirr/ 4 , 0 _:::: r _:::: R, the arc of the circle lzl = R, and the real axis from (0, 0) to (R, 0), when R -+ oo. We now outline a proof that does not use complex-variable methods. The qualitative fact of convergence of the improper integral is established by the following partial integration:

1N

e'x2 dx =

M

1N M

2

=

_;._d(eu2) 2zx '

2

e rN e rM 2iN - 2iM

+

1 2i

1N

eu

2

x2

M

dx ·

The final integral is less than or equal to 1IN, so that the right side tends to zero when 2 00 M, N-+ oo. This proves that the improper integral J0 eix dx is convergent. Letting N-+ oo shows furthermore that

1

e

e'x2 dx

=

oo

ix2

M

e;M2

1

dx=---+2iM 2i

1oo M

e'x2

-d x. 2 x

Both terms on the right are 0( I I M), so that we have the required speed of convergence:

1M

1oo

e'x2 dx

+ O(IIM),

M-+

00.

FOURIER TRANSFORMS ON THE LINE AND SPACE

167

It remains to compute the numerical value of the improper integral. To do this, we let p > 0 and examine the double integral Jp =

100 100

e-p(x2+v2)el(t2 +,.2)dxdy.

On the one hand. we can take polar coordinates x = r cos 0, y = r sin 0 and compute Jp

=

1= 1"

= 7r 2

-

lao

12

2 2 e-P' e" rdrd0

re -r2(p-•)dr

0

1 2 2(p- i)

7r

On the other hand, the double integral is the square of a single integral: JP =

Letting I =

J00 e'x 0

2

(1oo

e-P
dx)

2

dx, we conclude that

But the complex number I has both positive real and imaginary parts, so that the appropriate .square root is

To apply this to Step 4, write t-+ oo.

The equality of the two limits follows from Abel's lemma (see below), which completes the proof. •

2.7.4

Abel's Lemma

We give two forms of Abel's lemma for integrals, the second of which is applied above. Proposition 2.7 .4. • Suppose thatf(t), t > 0, is a locally integrable function and lim 1 ~ 00 f(t) = L. 00 Then limpJ.O j 0 f(t)pe-P 1 dt = L. • Suppose that g(s), s > 0 is a locally integrable function and that the improper 00 integral ] 0 g(s) ds converges to some L. Then

lim pJ.O

roo g(s)e-psds =

lo

L.

168

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. The proof consists of writing

1""

f(t)pe-P' dt- L

=

1""

(f(t)- L)pe-P' dt.

Given E > 0, we split the region of integration into the two regions 0 < t < T and T < t < oo, where Tis chosen such that 1/(t) - Ll < E fort > T, so that the second integral is less than E. The first integral is less than p J:J" 1/(t) - Ll dt which tends to zero when p ,J.. 0 and the first statement follows. To prove the second statement, we setf(t) = J~ g(s)ds. By hypothesisf(t) - 4 L when t - 4 oo. Interchanging the orders of integration yields

1""

1 = 1"" 1"" 00

f(t)pe-P'dt

=

= Applying the first limpJ.O fo"" g(s)e-P-'ds

statement L.

=

gives

the

pe-P'

(1'

g(s) ds) dt

g(s) ([""pe-P' dt) ds g(s)e-ps ds.

result:

if

lim,_ 00 J~ g(s) ds = L,

then

•

CHAPTE R

3 FOURIER ANALYSI S IN LP SPACES

3.1

MOTIVATION AND HEURISTICS

Much of modem Fourier analysis is concerned with bounded linear operators on the Lebesgue spaces LP('JI') and LP(lRn). This chapter is devoted to the development of systematic methods for proving the boundedness of relevant operators by the method of interpolation. Following M. Riesz, if we can first prove that an operator is bounded on two different pairs of Lebesgue spaces, then we can often deduce boundedness on the intermediate spaces. A more general concept, that of weak boundedness, can often be used in place of strict boundedness, following the work of Marcinkiewicz. These techniques are applied to prove the LP boundedness of the classical Hilbert transform, both on the circle and on the real line. This yields theM. Riesz theorem on LP convergence of Fourier series and integrals in one dimension. This chapter also includes the HardyLittlewood maximal inequality. which proves the weak L 1-boundedness of a fundamental operator that underlies the Lebesgue differentiation theorem and many other almosteverywhere convergence results in Fourier analysis.

3.2

THEM. RIESZ-THORI N INTERPOLAT ION THEOREM

In order to introduce the ideas, we first develop some elementary properties of LP spaces. Lemma3.2.1. Supposethatf

E

LP"(lRn)nL 00 (1Rn). Thenf E LP'(lR")forany

PI> PO· Proof. Letting M

= llfll""' we write {

}R''

lf(x)IP' dx

~

Mr,-ro {

~n

lf(x)IPo dr: < oo.

• 169

170

INTRODUCTION TO f'OURIER ANALYSIS AND WAVELETS

We say that/ lives on a set offinite measure if l{x : f(x) =I= 0} I < Lemma 3.2.2. Suppose that f E If' (ll~.n) lives on a set Then f E LP" (JR.") for any Po < p 1.

CXJ.

B of finite measure IBI.

Proof. From Holder's inequality we can write

{ if(x)i"" dx = ~

{

1.

if(x)i"0 1s(x) dx _:: : : ( {

1.

Lemma 3.2.3. Let 0 < p Thenf E LP(JR..").

1

< p < p

1

if(x)lf' 1 dx)POIP• (IBI)(pl-Pol/Pi <<X>.

•

and suppose that f

E

LP0 (lR")

n

LP' (lR").

Proof. We write/= jl 1111 sll + jl 11 fi>IJ = / 1 + /2. Both/1 and/2 are dominated by f, in particular fi E L"" andf2 E L"'. Butj1 is bounded and/2 lives on a set of finite measure, since l{x :f2(x)

¥=

0}1 = l{x: if(x)i > 1}1 ::0 {

Therefore by the preceding lemmas,j1 space, hencej E L"(!Rn).

I~.n

E

L"(IRn) andh

E

if(x)l"0 dx <<X>. L"(!Rn). But LP(!Rn) is a linear •

Lemma 3.2.3 has a sort of converse, as follows. Lemma 3.2.4. Let 0 < Po < p < p 1 and suppose that f E LP(IR"). Then there existfo E LPo and/1 E LP' such thatf =fo + j,.

•

Proof. It suffices to take fo = f lu1:s 1 and /1 = f llfl > 1.

The last two lemmas can be restated as follows: if Po < p < p

1,

then

(3.2.1) The above lemmas show that for any measurable function/, the set {p : II/ lip < CXJ} is a connected subset of the real line. The theory of M. Riesz-Thori n quantifies this by showing, for example, that the mappingp--- + log II fliP is a convexfunct ion of 1/p. At the same time we deal with linear operators that are simultaneou sly defined on two different LP spaces, developing the interpolation properties of these linear operators. Exercise 3.2.5. Use Holder's inequality to show directly that ifl :::=Po < Pt < andO
CXJ

Conclude that p---+ log 11/11~ is a convex function.

The basic convexity result of M. Riesz will be developed in the following general setting. Suppose we have two me·asure spaces (M, J-L), (N, v) and two pairs of indices

FOURIER ANALYSIS IN U SPACES

171

(p 0 , q 0 ), (p 1, q 1) where 1 ~ po, Pt, qo, q1 ~ oo with Po =I= PI, qo =I= qJ. Further we assume given linear operators Ao : LPo (M) ~ L qo (N) and A 1 : LP 1 (M) ~ L q 1 (N) so that Ao = A 1 on the common domain LPo (M) n LP 1 (M). Let k; = IIA lip, .q, be the respective operator norms, i = 1, 2. An interpolation is defined by a real number t E (0, 1) giving rise to indices (p 1 , q,) defined by the convex combinations -

1

Pr

=

t

1-

1 q,

t

-+--.

PI

t

1- t qo

-=-+--.

Po

q1

Theorem 3.2.6. M. Riesz-Thorin: Under the above hypotheses there exists a linear operator A, : LPr (N) ~ L qr (N) that coincides with A; on LPo (M) n LP 1 (M) and whose operator norm satisfies

IIA I II Pr.qr -< kt-re 0 J• The M. Riesz-Thorin theorem can be applied to prove the Hausdorff-Youn g inequalities for Fourier series in one dimension. This is illustrated in the following two examples. Example 3.2.7. Suppose that M = 'II' with Lebesgue measure and that N = Z with counting measure. Let Af(n) = 1/2n J1f f(B)e-ine dB be the discrete Fourier transform. Letting (p0 , q 0 ) = (1, oo) we see from the contraction property that Ao is bounded with ko = 1. Letting (Pt, q1) = (2, 2), we see from the Parseval identity that A 1 is bounded with k 1 = I. Applying the theorem, we take 1jp1 = t + (1 - t)/2 = (1 + t)/2 and ljq, = (1 - t)/2. These are conjugate exponents, satisfying 1jp1 + 1/q, = 1. Since LP('JI') C L 1 (1I') the operator A must agree with the discrete Fourier transform, so that we conclude for any I ~ p < 2 A: LP('JI') ~ V" (Z) where 1/p + 1/p' = I. Equivalently

(

~ tf(n)IP' )

1

l/p'

< (

2n

i

1/p

lf(B)IP dB)

The roles of 'II' and Z can be reversed to obtain another example. Example 3.2.8. Suppose that M = Z with counting measure and that N = 'II' with Lebesgue measure. For a bilateral sequence {en}, n E Z, define Af(8) = LneZ Cnein&_ From the first properties of absolutely convergent series, this is bounded from 11(Z) to L 00 (1I'); also from the basic properties of L 2 it is also boundedfrom / 2 (Z) to L 2 (1I'). Therefore we can take Po = 1, qo = oo, p 1 = 2, q, = 2 to obtain the same conjugate exponents 1/p, = (1 + t)/2, ljq1 = (1- t)/2 and the conclusion that for 1 ~ p < 2

p' : = __!!___ . p-1

The next exercise treats the convexity of the p norm by M. Riesz-Thorin.

172

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 3.2.9. Let f E LP(E), where E is a measurable subset of R" and 1 < p < oo, with llfiiP := (jE lf1P) 11P.

JE

(i) Prove that llfiiP = supllllp':;:l lf¢1. where p' = pj(p- 1). (ii) Use theM. Riesz-Thorin theorem to prove that iff E Lp0 (E) n Lp 1 (E) with 1
To prove the theorem, we first prove the maximum-modulus theorem. Lemma 3.2.1 0. Suppose that f(z) is an analytic function in an open connected subset S of the complex plane. This means that about every point zo there is a disk lz - zol < r so that f is the sum of a convergent power series f(z) = 0 an(Z - zo)". Then f cannot have a local maximum at any interior point unless f is constant. In particular, lf(z) I attains its maximum on the boundary.

,L::,

Proof. Iff is nonconstant, then there is a smallest value of n 2: 1 so that a,. # 0. By translation, we may suppose that Zo = 0. Near z = 0, j(z) = ao + a,z" + O(z"+ 1 ), 1/(re;11 )1 2 = laol 2 + 2la0 a,.lr" cos (nO-
Now we can prove the three lines theorem. Lemma 3.2.11. Suppose that F is an analytic function defined in the stripS = {z : 0 ::::; Re(z) < 1} and such that I F(iy)l < mo. I F(l + iy)l < m1 for -oo < y < oo where mo > 0, mr > 0. Then I F(x + iy)l :::::: m6-xmi for 0 < x :::::: 1, -00

< y < 00.

Proof. Let F1 (z) = F(z)/m6-'m1, which is an analytic function with IF(iy)l ::;: 1, + iy)l ::: I, so that we can assume mo = m 1 = 1. First we prove the lemma under the added condition that F(x + iy) -+ 0 uniformly when IYI -+ oo, 0 ::;: x ::;: 1. Then we must have IF(x ± iM) I ::;: forM large enough. Therefore by the maximum principle, we must have IF(x + iy)! ::;: 1 for 0::;: x::;: 1, -M ::;: y ::;: M. But M was arbitrary, so the inequality holds in the entire strip 0 ::;: x ::;: I, -oo < y < oo. In the general case, we let F,.(z) = F(z)efn. Then IF(l

k

!F(x

+

iy)!::;: !F(z)!e_Y2fne(x2-l)fn::;: e_Y21n,

which tends to zero when IYI -+ oo. Hence we can apply the previous paragraph to conclude that IF,(x + iy)l ::: 1 in the entire strip. But IF(z)l =lim, IF,(z)l::;: lim sup 1 = 1, which completes the proof. • Proof of the theorem. Having made the necessary preparations, we first note that for any measure space (M, Jl..), A can be defined on the spaces LP'(M), since we can write f =fllfi:~I + fllfi>I· By Holder's inequality, this is the sum of an LPO function and an u1 function, for which A is defined. Now the Lq norm can be computed as

llhllq= sup Jhgdt-·, Jigllq'

~I

N

FOURIER ANALYSIS IN LP SPACES

173

where the supremum is taken over all simple functions, i.e., finite linear combinations of indicator functions of sets of finite measure. At the same time we have

=

IIAIIp.q

sup

1

(Af)gdv.

11/llp=l.llgllq'=l

N

We extend the interpolated exponents to the complex plane by defining 1

z

p(z)

PI

z

1-

z

1

--=-+--.

1-

z

----+-~

q'(z) -

Po

q;>

q;

0::::; Re(z)

'

f - L

~

1.

a1 e'a1 IA,. g = /31 E (0, 2rr]. The functions/. g are extended to the strip in the complex plane by first definingp = p,, q' = q; and setting

It is sufficient to prove the theorem for simple functions:

1:::,;:::N

Ll:::.i:::N bje•f3, 181 , where A 1 , Bj are measurable sets, aj. b1 :=::: 0, and a 1 ,

N

N

r/>(-.z) := La:'IPe;a,IA, = La:'/p(Z)ej. j=l

]=I

N

N

1/F(·,z) := Lb'J'/q'e' 13'1o1 = L b f 1q'cpj• J=l

J=l

where we have set

It is immediate that r/> ( ·, z) Therefore the integral

(3.2.2)

F(z)

=

E

LP1 (M), 1/f ( ·, z)

1

Ar/>(·. z)l/f(-, z) dv

E L

=

q; (N),

t

in particular A 1/f ( ·, z)

dj 1Pbeq'

j,k=l

N

E

U1 (N).

1

(AE>j)<J>k dv

N

is a finite sum of exponential functions, in particular an analytic function in the open strip 0 < Re(z) < I and is bounded and continuous on the closed strip 0 ~ Re(z) < I. On the boundary we have by direct computation

llr/>(·,

iy)llp0

llr/> (-. 1 +

iy) lip,

111/F(·, iy)llq~ 111/F(·, 1 +

iy)llq'

I

= 111/lp/poiiPO = IIJII;1Po = 1, = II IJIPIPI lip, = llfii;1PI = I. = II lglq'!q~llq~ = llgll!jqb = I, = lllglq'/q', llq' = llgllq/q', = 1. q I

1

By the definition of F(z) above and Holder's inequality, we have from (3.2.2)

IF(iy)l ::::; liAr/>(·, IF(l

iy)!lq0

111/f(iy)llqb ::::; leo

+ iy)l:::::: liAr/>(·, 1 + iy)llq,lll/F(·, 1 + iy)llq;

:::::: kl.

On the other hand, fort E (0, 1), r/> (x, t) = f(x), 1/f(y, t) = g(y), so that F(t) = JN Afg dv. Applying Lemma 3.2.11, we conclude IF(t)l::::; k~-~k~. which completes the proof. •

Now we examine some important applications of theM. Riesz-Thorin interpolation theorem.

174

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

3.2.0.1

Generalize d Young's inequality

The interpolation theorem of M. Riesz-Thorin can be used to prove the generalized Young's inequality for convolutions : (3.2.3)

+ .!.r

( 1

=

.!. + .!.) q

p

where 1 < p < ex:>, I < q < ex:> with 11p + I I q :::: I. Here f, g are measurable functions on JR.n and the integrals are taken over all of JR.n. Proof. To prove (3.2.3), we begin with the elementary estimates from Lebesgue integration theory:

f

II/* gll1 ::: ll/ll1 llgll1

E

L 1, g E L 1

/EL=, gEL 1 •

11/ *gil= ::: 11/11"'-llgiii

This shows that, for a fixed g E L 1 , the map f ---+ f * g defines a bounded operator on L 1 and L 00 , with norm less than or equal to one. Therefore, by the M. Riesz-Thorin theorem, this map can be extended to LP, with the same operator norm, to yield (3.2.4)

f

In other words, the map g ---+ f inequality, iff E LP, g E U',

E

LP, g

E

L 1•

* g is bounded from L 1 to LP. On the other hand, by HOlder's

(3.2.5)

p' = pj(p- 1).

Therefore the map g ---+ f * g is bounded from LP' to L oc. Hence we can apply theM. RieszThorin theorem with Po= 1, qo = p, p 1 = p', ql = oo. In detail, I 1 1-t t -=-=--+-

(3.2.6)

q

I I - =r q,

(3.2.7) Solving (3.2.6) for t have

E

[0, I], we have t I

I

p,

p'

I - t

= --+0. p

= pI q', which is possible since p

I

I-t I I -r = -q, = - = p- - p q'

which proves (3.2.3).

I = -

p

I

+-q -

::;: q'. Finally, we

I,

•

Beckner ( 197 5) has shown that the value of the "best constant" in the generalized Young's inequality (3.2.3) has the precise value Mpq = (p 11P lq 11q)nl 2 • The proof is beyond the scope of this book.

3.2.0.2

The Hausdorff- Young inequality

The M. Riesz-Thorin theorem can be immediately applied to the setting of Fourier transforms on lRn. Let fl.! = N = lRn with Lebesgue measure and p 0 = 1, q = ex:>. The 0 Fourier transform! ---+ f is a bounded operator from L 1 (M) to L 00 (N) with norm 1. Also from the Plancherel theorem the Fourier transform is a bounded operator from L 2 (M) to

FOURIER ANALYSIS IN U SPACES

175

L2(N) with norm 1. Therefore we conclude that

Hausdorff- Young: If 1 < p < 2, the Fourier transform is defined and is a bounded operator from LP(lRn) to LP' (lRn) where p' = pj(p- 1) is the conjugate exponent.

Theorem 3.2.12.

Beckner ( 1975) has shown that the best constant in the Hausdorff-Young iw•quality has the value (p 11P jq 11q)nl 2 where q = p'. 2

Exercise 3.2.13. Letf(x) = e-rrlx1 , so that](~) = e-rrl~ 12 . Show that llfllq/IIJIIP

attains the Beckner bound, where 1 < p < 2.

3.2.1

Stein's Complex Interpolation Theorem

The M. Riesz-Thorin theorem deals with a single operator A, initially defined and bounded on yo n LP• and subsequently extended to LP, p 0 < p < PI. E.M. Stein discovered a remarkable extension to a family of operators Az which depend analytically on a complex parameter z. where 0 < Re(z) < 1. For the complete theory, see Stein and Weiss ( 1971 ), Chapter 4. The following is a special case of the general theory. Theorem 3.2.14. Let (M, J.L) and (N, v) be measure spaces with a family oflinear

operators Az defined on the class ofsimple functions S(M) so that z ~ JN (AJ)g dv is analytic and bounded for 0 < Re(z) < 1 whenever f e S(M) and g e S(N). Furthermore suppose thatforsome 1 < po, qo,pi, qi < CXJ we have IIA;y/llqo < MollfiiPo

f

E

S(M), y

E

lR

~Mill/lip,

f

E

S(M), y

E

JR.

IIAI+iy/llq, Then for all t e (0, 1)

IIA,fllq, < MJ-'M~IIfllp,

f

E

S(M),

where 1

1- t

t

Pt 1

Po 1- t

Po t

-=--+-

-=--+-. q, Qo QI

Proof. We can follow the steps of the proof of the M. Riesz-Thorin theorem, defining g E S(N) with 11/IIPo = 1, llgllqb = 1 we set


F(z) =

L

(A,
which is an analytic function in the strip {0 ::;: Re(z) ::;: 1} and for which IF(iy) 1 ::;: M 0 , IF(l + iy)l ::;: M,. Hence by Lemma 3.2.11, IF(t)l ::;: MJ-'M[ for 0 ::;: t ::;: 1, which completes the proof. •

176

INTROD UCTION TO FOURIER ANALYS IS AND WAVELE TS

3.3

THE CONJ UGAT E FUNC TION OR DISCRETE HILBERT TRAN SFOR M

We now pass to a particu lar operat or of centra l import ance in harmo nic analys is, the so-cal led Hilber t transfo rm. In the setting of Fourie r series this is define d on the set P of trigon ometri c polyno mials by the formu la (3.3.1)

+

H (Lcne inO) = -iLcn einO nEZ

i L

n:o:I

Cneine.

n~-1

This gives a conven ient way of expres sing the projec tion operat or J

=

L

Cnein8 ~

L

nEZ

CneinO

= PJ

n?:_l

by writin g f

+ iHJ =co+ 2 LCnei nO

=f(O)

+ 2Pf.

n:O:I

J. .

If we restric t to the subspa ce of functio ns with f = 0, then we can write 2 Pf f + iHf. The operat or H is more conven ient to work with, since it is inverti ble on this subspa ce: H 2 = -1. Exerci se 3.3.1 . Show that the Hilbe n transfo rm is skew-a djoint, in the sense that

for any trigon ometri c polyno mials f, g, we have the identit y /.... Hf(8)g (8) dB = -

i

J(8)H g(8) dB.

Now we will show that the Fourie r partial sum can also be expres sed in terms of H by writin g for any trigon ometri c polyn omial /

L

eiNO H(e-iN OJ) = -i L Cnein8 + i Cnein8' n>N n-N n<-N

L

L

When we subtra ct, we obtain eiNO H(e-iN OJ) - e-tNO H(eiN€ 1/) = 2i

N

L

CneinO - icNeiN€1 - ic-Ne-i NO

n=-N

This allows us to repres ent the partial sums SNf in terms of the normpreser ving operat ors f ~ e±iNOf and the fixed operat or H. (3.3.2) SN/(8 ) =

~ 21

(eiNO H(e-iN fJJ)- e-iNfJH (eiNfJJ ))

The last two terms tend to zero when N any p 2: 1.

~ ex:>

+

.!.](N)e iNfJ 2

+

.!.](-N )e-iNf J 2

and are bound ed by the

If

norm off for

FOURIER ANALYSIS IN V SPACES

177

The representatio n (3.3.2) shows that H cannot be bounded on L 1 (1I").

Proposition 3.3.2.

IIH/III

(3.3.3)

sup II/III

= +oo

where the supremum is taken over all trigonometri c polynomials f =I= 0. Proof. Suppose not; then we would have the estimate IIH/IIt ::: Cllfllt for some constant C and all trigonometric polynomials f. Referring to (3.3.2), this implies that IISN/IIt :S (1 + C)ll/11 1 • which implies that the (1,1) operator norm of SN is bounded by a constant, independent of N. But we showed in Section 1.6.3 of Chapter 1 that IISNIII,l = LN ,.....,

(4 log N) j rr 2 when N -+ oo. Thus we have a contradiction, which proves that the supremum in (3.3.3) is infinite. •

3.3.1

LP Theory of the Conjugate Function

In order to prove uniform boundedness of SN on the space LP(1r) for 1 < p < oo, it suffices to prove that the operator H can be extended to a bounded operator on the space LP(1r). This will be accomplishe d by interpolation , as follows: Lemma 3.3.3. The operator His bounded on L 2 (1r). Proof. From Parseval's identity, for any trigonometric polynomial/

IIH/11~ =

L

L

lcnf :S

0#neZ

lcnl = 11/11~2

nEZ

•

This allows us to extend the definition of H to the space L 2 ('][') as a bounded operator with norm at most 1. But this bound is attained, since IIH(e;8 )112 = lle;8 112 = 1. Since LP ('][') C L 2 ('][') for p > 2, we also obtain the existence of Hf when f e LP (']['), p > 2. We will now show by several steps that H is a bounded operator on any LP space for p > 2. We first prove that H is bounded on LP (1r) if p = 2k is an even integer. The following lemma is attributed to M. Riesz ( 1927). Lemma 3.3.4. For any k = 2, 3, ... there exists a constant C 2 k, so that iff is any trigonometri c polynomial, IIHflb < C2kll/ll2k·

~roof. It suffices to first prove this for real-valued functions. If/ is real-valued (f( -n) = /(n)), then so is Hf and we have Pf = ~ (/ + iHf). Expanding this by the binomial theorem we note that (Pfi has no constant term, so that 0

t ( ~) 2

= { (2P/) 2* =

J.r

j=O

1

{ /j(iH/) 2*-;. JT

Taking the real part and writing j = 2r we have

0

=

t (2k) (

-1 )k-r { J2r (Hj)2k-2r.

r=O

2r

JT

178

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

We isolate the term with r

=

0 and apply Holder's inequality to the remaining terms:

t (2k) {

{ (H/)2k :::::

J2r (H/)2k-2r

J.rr:

r=l

J.rr:

2r

Dividing both sides by llfii~Z. we have the polynomial i~equality x2k :::::

(3.3.4)

t

(2k)x2k-2r 2r

r=l

where X:=

2

If X ::::: 1 there is nothing to prove. If X >

(;~) xzk-z, so that the sum is bounded by x2k :::::

x2k-2

L k

r=l

equivalently X 2

:::::

2 2k

-

I

IIHflb = (f-r;(Hf) k) llflb !-r;J2k

:rr

1, then each term in (3.3.4) is bounded by

(2k) =

x2k-2

<2 2k

_

1>

2r

1. The lemma is proved with C 2k

=

.../2 2k

-

1.

•

Since the space of trigonometric polynomials is dense in lP('li'), we immediately obtain the extension of H as follows. · Corollary 3.3.5. The map f k = 2, 3, ....

-+ Hf is a bounded operator on L 2 k('li') for any

Proof. In order to prove the boundedness of H on the intermediate LP spaces, we can apply the M. Riesz-Thorin theorem to conclude that H can be extended to a bounded operator from LP('][') to LP('][') for any I < p < 2k. But k was arbitrary, so we conclude boundedness for any p > 2. In order to prove boundedness for 1 < p < 2 we use the duality of the norms, namely IIHJIIP =

sup O,CgeLP' ("IT:)

where the supremum is taken over all trigonometric polynomials. We can apply HOlder's inequality and the boundedness result on LP', p' > 2 to see that the numerator is bounded by

and thus conclude that

IIHfllp ::::: which completes the proof.

We list the result as follows.

Cp' II fliP

f

E

U' (']['), I < p < 2

•

FOURIER ANALYSIS IN LP SPACES

179

Propositi on 3.3.6. The mapping f---+ Hf is a bounded operator on LP('Jr) whenever 1 < p <

CXJ.

This is now used to deduce the following result on LP convergen ce of onedimension al Fourier series.

Theorem 3.3.7.

M. Riesz: Suppose that 1 < p < CXJ andf E LP(Jr). Then the Fourier series off converges in the norm of LP(1f): limN llf- SNfllp = 0.

Proof. It suffices to note that we have convergenc e on the (dense) set P of trigonomet ric polynomia ls and that the partial sum operators have uniformly bounded norms. But from (3.3.2), we have supN>I I!SNfllp.p :':: l + CP < oo . Therefore we have norm convergenc e on the entire space U' (1r). •

Exercise 3.3.8. Let Qr be the conjugate Poisson kernel, defined on trigonome tric polynomia ls by (3.3.5)

Qrf(B) = - i

L

sgn (n)f(n)rinl einiJ.

O¥nEZ

Prove that for any k E z+, there exists a constant c2k such that 11Qrfll2k :::; Ckilfll2k for all f E P and 0 < r < 1. Hint: Prf + iQrf = 2 Ln>.J(n)r 1"1em 11 . Now copy the proof of Lemma 3.3.4 and use the U' boundedne ss of the Poisson kernel from Chapter 1.

Exercise 3.3.9. Prove that for any 1 < p < IIQrfliP < Cpllfllpfo r allf

E

CXJ

there exists a constant Cp so that

P and 0 ::S r < 1.

Hint: Use the result of exercise 3.3.8 and theM. Riesz-Thor in theorem for 2 < p < oo. The duality argument takes care of 1 < p < 2.

3.3.2

L 1 Theory of the Conjuga te Function

An alternative method for proving LP boundedn ess of the conjugate function is to first prove the following inequality of Kolmogor ov: (3.3.6)

1{8: lHf(B)l

>

a}l

Cilfllt.

<

a

This is then combined with the Marcinkie wicz interpolat ion theorem, to be proved later in this chapter. For this purpose, we need to define the conjugate function Hf when f E L 1 (1f). In order to develop these ideas, we begin with the Poisson integral Pr and its harmonic conjugate Qr: Prf(B) =

L

rlnlj(n)e"' 6

,

nEZ

Qrf(B) = - i

L 0¥onEZ

sgn (n)r nlj(n)einB. 1

180

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 3.3.1 0. Prove that both PrJ(()) and Qrf(O) are harmonic functions and that PrJ+ iQrf is an analytic Junction oJz = reiB.

Note that, if J is a trigonometri c polynomial, then limr~ 1 Qrf(O) = HJ(O), as defined in (3.3.1). The next theorem extends this definition to any J E L 1 ('II'). The proof, which is adapted from Katznelson (1976), uses the properties of harmonic functions, which are developed in an appendix to this chapter. Theorem 3.3.11. LetJ E L 1 ('II'). Then HJ(O) := limr~ 1 Qrf(O) existsJor almost every() E 'II'. For any a > 0 the weak ( 1,1) inequality (3.3.6) holds. Proof. Since any L 1 function can be written in terms of four nonnegative functions, we first assume thatf :::: 0. In particular ~

1

.rr

{ J/ =

=f(O)

+2L

Prf(O) =/(0) =

2

ll/ll1·

It is readily verified that Prf(B)

+ iQrf(B)

00

rn](n)em 0 ,

n=l

which is a holomorphic function of z = re'e. By the Abel summability of L 1 functions, we see that limr-+ 1 Pr f(B) = /(B) for almost all (}. Now define G(z) = e-Prf -•Qrf, a holomorphic function in D. Since/ :::: 0, then IG(z)l ~ I and we can assert the existence of a radial limit g(B) = limr-+l G(re; 11 ) with lg(B)I = e-f > 0 a.e. since/ is finite a.e. If Qrf(B) were unbounded when r ~ 1, then the set of accumulation points of e-•Orf would fill out an interval, which contradicts the convergence of G(z), hence Qrf(O) remains bounded-in particular has at least one point of accumulation when r ~ 1. If Qrf(B) had two different accumulation points when r ~ 1, then the set of accumulation points of e-iQ,J would fill out an interval, which contradicts the existence of the radial limit of G(z). From this it follows that there exists a radial limit of Qrf(B) for a.e. B. A general f E L 1 (11') can be written/ = / 1 -12 + if3 - if4 where J; E L 1 (11') is nonnegative. We define Hf = Hf1 - H/2 + iH/J - iHf4 , which completes the proof of the a.e. existence of Hf(B). Note that the exceptional set, which comes from the Fatou theorem, may be different from the set where PrJ fails to converge when r ~ 1. To prove the weak (1,1) inequality (3.3.6), we first suppose that/ ;::: 0 and set F(re' 11 ) = Prf(O),

F(re' 11 )

=

Qrf(B).

Then we have a holomorphic function z ~ w = F(z) + iF(z) which maps the disk D to the right half plane Re w ;::: 0. For any A > 0, consider the harmonic function H;.., which equals 0 on the segment of the imaginary axis from -i).. to i).. and equals 1 on the twp complementa ry rays. Equivalently H;..(W)

=

.rr-<1> .7r

.

where is the angle that the point w makes with the points ±iA. Note that H;.. has the constant value of ~ on the half circle w = )..e''~', -.rr /2 < ({) < .rr j2 and outside of this circle is strictly larger than ~. At any point w of the positive real axis we have

FOURIER ANALYSIS IN LP SPACES

181

H>.(w) = (2/rc) arctan(w/ .A) < 2wjrc .A. The compose d mapping z -+_H>.(F : iF) is a harmonic function in the disk D. From the mean-val ue property ofharmo mc functtons , we have

On the other hand,

2:

4

1 8 rc 1{0 : !F(re' )I 2: .A}I,

from which we conclude that

But Hf(O) = lim,~ 1 F(re; 8 ) exists a.e. Therefore we have 1{0: IH/(0)1::: : .A}I::: which proves (3.3.6) in casef 2: 0. A general/ for which Hf = Hf1 - H/2 + iHj3 - iH/4. Then

I{O: IHJ(O)I:::: .All:::

E

811 111 { ,

L 1 ('1I') is written/

t.!{o:

IH.fj(O)I::::

=ft - h

+if:,- if4,

~}I

to which (3.3.6) is applied four times to obtain the result in general.

•

Theorem 3.3.11 provides an extensio n of the Hilbert transform to the entire space L 1 ('li'). If, in addition , f E LP (1I') for some 1 < p < oo, then we can also compute Hf as the LP limit of trigonom etric polynom ials, from Proposit ion 3.3.6. By taking subsequ ences, we see that the two definitio ns of Hf agree forf E LP(1I'), 1 < p < oo. The discrete Hilbert transform is integrab le under a slight addition al conditio n, where we use the notation log+ x = max{O, logx} for x > 0. Theorem 3.3.12. lfflog+

lfl

E

L 1 (1I'), thenHf E L 1 (1I').

Proof. To prove this result, we introduce the distributi on function of a nonnegat ive measurab le function /, defined by (3.3.7)

Af(a)

= 1{0 :f(O) > a}j.

182

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The map a follows:

-+ A.; (a)

is decreasing and can be used to express the LP norm (0 < p < oo) as

(1 (1""

1

11

[

II
=[

1= ([ =1

=

(!1)

paP-' da) de

paP-'Itc>
ltu
da) dO

de) pap-Ida

00

pap- I A.1 (a) da.

We also use the observation that

Having made these preparations, we make the a-dependent decomposition I where

=

Ia

+ lu,

lu =I 11/l>u · Since H is a linear operator we have

IHII

s

IHiu I + IHr i,

A.IHJI(a) S AIH/al(a/2)

The L

1

+

)c!Hf"!(a/2).

norm of H F is expressed in terms of the distribution function as

It remains to estimate the last integral in terms of the distribution functions of IHiu I and IHiu 1. Since 1r has finite measure andlu is bounded, it follows thatlu E L 2 ('lr). From this it follows that Hla E L 2 ('lr) with IIHia i12 S lila 112- Now we transform by the Fubini theorem as follows:

<

00.

FOURIER ANALYSIS IN U' SPACES

183

Similarly, from (3.3.6) and the Fubini theorem, we have

c

1=

Ct

AIHJUI(a/2) :::S -11/ lit a AIHJ"I(a/2) da :::S

1= ~ (i {

=cIT if(O)i = C

if(O)iiiJ<9ll>a dO) da

(J.If
-;:;-

£

1/(0)ilog+ 1/(0)1 dO

< 00.

•

The proof is complete.

Exercise 3.3.13. Iff

L

E

1 (1['),

prove that Hf E LP (F) for 0 < p < 1.

Hint: Begin with the representation

IIHJII~

1

00

=p

aP-I)..Hf(a)da :::: 2rr

foo pap-lAHJ(a)da

and follow the steps of the proof of Theorem 3.3.12.

3.3.2.1

Identificatio n as a singular integral

We close this section by proving the a.e. representatio n. (3.3.8)

1

. Hf(6) = - 1 hm j(O- ¢) sin¢ d¢, 2rr ,-..o l4>l><' 1 - cos¢

Proof. To do this, we take r = 1 -

E

and write

Qrf(O)- _1_ { j(O- ¢) sin¢ d¢ =II +lz, 2rr 1 1l>< 1 -cos¢ 2rr

where I1 :=

h :=

1

1l:s•

f

1<1>1><

[/(0- ¢) - j(O)]

2rsin¢ 1 + r 2 - 2rcos¢

[j(O- ¢)- /(0)] (

d¢

~rsin¢

1+r -2rcos¢

-

sin¢ ) d 1- cos¢ ¢

where we have used the oddness of¢ -+ sin¢. To estimate I 1 , we write 1 + (1- r) 2 + 2r(l- cos¢):::: (1- r) 2 , so that

lid :::: :::S

il~< (1 ~;)21/(0- ¢ ) - /(0)1 d¢ 2

~ E

{ lf(O- ¢ ) - /(0)1 d¢, 11I:S<

which tends to zero for almost every 0

E

'Ir, when

E

-+ 0.

r 2 - 2r cos¢

=

184

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

To estimate /z, we use the inequality 1 -cos c/> ~ ¢ 2 /:rr 2 for let> I ~ :rr to write

1z = 1/zl

~

-(1 -

r) 2

1

l4>l >< (1

(1- r)2 [ 2 (11 1l4>l>< r

~ :rr4(1 _ rf {

SID

~

:n¢1

[f(e-

- cos c/>)

[

f(O - c/>) -

/(O)j

de/>

¢)- f(O)]dc/>

lf(8 - c/>) - /(8)1 de/>

= :rr4(1 - r)21 14>1 ><

lhl

A-.)2

cos'+' 1 I¢ 1-

}14>1>•

where we have set F(¢) =

c/>

+ r- - 2r cos c/>) (1

J;}'

dF(c/>) 3 let> 1

1/(8- u)- /(8)1 du. Integration-by-parts shows that

~ :rr4(1- r)z

( iFjlll4>1=rr

let> I

+ 3 [

11<1>1 ~•

ll=•

IF(~)I

de/>).

ct>

But F(lj))jlj) := '10/v) --+ 0 a.e. when lj)--+ 0 and 1- r = E shows that the first term tends to zero. Similarly, the second term = u17(u) du = o(E- 2 ) when E --+ 0, which shows that h --+ 0. Recalling that Hf(8) = limr__. 1 Qrf(8) a.e. completes the proof of (3.3.8). •

J/j;

Exercise 3.3.14. Prove that 1- cos¢~ ¢ 2 j1r 2 for 1¢1 <Jr. Exercise 3.3.15. Let '7 limx-+oo x- 2

3.4

E ~~(lR)

with 'f'/(X) ~ 0 when x ~ oo. Prove that

J; U'fl(u) du ~ 0 when x

~ oo.

THE HILBERT TRANSFORM ON IR

On the circle we developed the conjuga,te function beginning with its Fourier representation Hf(n) = - i sgn(n)](n) for trigonometric polynomials f. eventually leading to the singular integral representation (3.3.8). When we pass to the corresponding problem on the real line, the relevant operator is the Hilbert transfonn, defined formally as the singular integral (3.4.1)

1 H'.1r+( X ) = -

1'lffi

1f "__.Q,M__.oo

Proposition 3.4.1. Iff

E

[ <I<M

f(x - y) d y. Y

S, the limit ( 3.4.1) exists.

Proof. Denoting the integral in (3.4.1) as H 01 f(x), we have forf :rrH,,Mf(x)

=

1

<
f(x- y)

E

dy

Y

= - fI<M f(x; y) dy

2:rrH,,M f(x) =

1 <
f(x- y)- f(x Y

+

y) dy.

S

FOURIER ANALYSIS IN LP SPACES

If/

E

S, this family of integrals converge when l Hf(x) = 2rr

(3.4.2)

Example 3.4.2. Let f x =I= a, b.

=

1

E

--+ 0. M--+ oo and we have

f(x - y) - f(x

Then Hf(x)

l[a,bJ·

+ y)

d

Y

JR

185

=

•

Y·

(lj:rr) log (ix - alflx - bi) for

Exercise 3.4.3. Prove this. Explicit calculation reveals that in this example Hf(x) ~ (lj:rr)(b- a)jx when x ~ oo, showing that Hf fj. L 1 (lR). The same behavior is generically true whenever frrtf =I= 0.

Exercise 3.4.4. Suppose thatf

E

S(lR). Prove that limx-+oc xHf(x) = 1/:rr J'Rf.

Exercise 3.4.5. Suppose that frrt lf(x)l/(1 + lxl)dx < oo and thatf satisfies a Dini condition at x. Prove that the integral in (3.4.2) is absolutely convergent.

3.4.1

L2 Theory of the Hilbert Transform

In order to define H on L 2 (JR), we let K".M(x) = (lj:rrx)l"
---dx lrX

= -2 l

·1M sin 2:rrx~ "

=

dx

:rrx

-i(Si(2:rrM~)- Si(2:rrE~)).

From the properties of the Si function, we have lim k".M(~) = -in sgn(.;). €-+O,M-+oo We can use this to define H on the space L 2 (1R) as follows.

Proposition 3.4.6. For f Hf

=

Proof. For any 0 <

E

E

L 2 (lR) there exists the L 2 limit

lim H.,,Mf E-+O,M->oo

=

lim { K<'.M(,;)j(,;) e 2 "ii;x d.;. €->O,M-+oo lrrt

< M < oo

and by the dominated convergence theorem for any f E L 2 ,

IIH.. Mf(~) + i sgn(~)f(~)ll~--+

0,

E ........

0, M ........

00.

186

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Therefore the L 2 limit of fi•. Mf exists, especially H,,Mf is a Cauchy sequence. By Plancherel's formula it follows that H •. Mf must also be a Cauchy sequence in L 2 • Hence there exists g E L 2, g = lim.--.o.M--.oo H •. MJ, which was to be proved. • Proposition 3.4.6 can be paraphrased as the representation formula

Hf(x) = - i

l sgn(~)j(~)e "il;x d~ 2

valid for f E L 2 (l~). This shows, in particular, that H is norm-preserving: IIHf 112 = II f 112 and that H (Hf) = -f whenever f E L 2 (R).

3.4.2

LP Theory of the Hilbert Transform, 1 < p <

(X)

We now develop the tools to prove the LP boundedness of the Hilbert transform. For any E > O,let

S£ = {f

E

S :j(~) = 0 forl~l < E}

and S 0 = U£ >OS£. A typical element of So is written

and the Hilbert transform

Proposition 3.4.7. For any p > 2, S 0 is dense in LP(R). Proof. Since S is dense in U', it suffices to prove that So is dense in S in the U' norm. For p = 2 this follows immediately from the Plancherel formula, since

which can be made arbitrarily small with lf(x)- J.(x)l

~

E.

For any p

~

/_: li
2, we can write

~

CE,

which proves that So is dense in the supremum norm, hence also in the U' norm on compact sets. But if p ~ 2 we can write lf(x)- f.(x)IP ~

II!- .t. II~ ~ which completes the proof.

Ep-

2

1/(x)-/.(x)l 2

(CEP- 2 )11/- .t. 11 2 __. 0,

•

FOURIER ANALYSIS IN LP SPACES

f

187

To prove LP boundedness , we can follow the proof in the case of the circle. For E

So, we write f(x) =

{oo f(~) e2rri~x d~ +

Jo

f(0 e2rrt~x d~'

-oo

so that f

lo

21

+ iHf =

00

f(~)e 2 rri~x d~.

Now if p = 2k is any even integer, we can write (j

l

T (f -T

+ iHf)2k

+ iHf)2k dx

1

00

= 22k

=

(!* ... *f)

(~)e2rri;x d~,

r= sin (2nTI;) (f * ... *f) dl;. nl; \:

Jo

The right side is the partial Fourier inversion at~ = 0 of a smooth function that vanishes for small I;, hence lim T __. 00 J~T (f + iHf) 2k dx = 0. On the other hand this integral is absolutely convergent, since f E So and Hf E So. Expanding by the binomial theorem we have

o=

t

j=O

(2~) ]

{

UHfY(f)2*- ;.

}R

Iff is real, then Hf is also real, so that we can restrict attention to the even powers and write { (Hf)2k =

jR

E (2~) jR{

(f)2k-2j (Hf)2j.

j=O

2]

Applying the Cauchy-Sch warz inequality as before, we can estimate JR (Hf) 2* in terms of JRf 2*. This completes the proof that the Hilbert transform is bounded on the space L 2k (R) for any k = 1, 2, .... As before, we can apply theM. Riesz-Thorin interpolation theorem to conclude that/~ Hf is bounded on any intermediate space LP for 2 < p < 2k. But k was arbitrary. So we conclude boundedness on LP for any p > 2. By duality, it also follows, as before, that/ -+ Hf is bounded on LP for any l < p :=:: 2. We have proved the following theorem.

Theorem 3.4.8. Forany 1 < p < oo, theHilberttr ansformcan beextendedf rom the space So to LP (R) as a bounded operator.

3.4.2. 1

Application s to convergenc e of Fourier integrals

In parallel with the case of the circle, the Hilbert transform on R can be used to study the convergence of the partial sums of the Fourier integral in the space LP(R), 1 < p < oo. This developmen t is described in the following exercises.

188

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 3.4.9. Letf

E

S(lR) and let M > 0. Prove the identity

Exercise 3.4.10. Prove that for any f E S(JR.), we have the bound IISM.fllp < Cii/llpfor 1 < p < CXl, where C = Cp. /nparticular theoperator SM has a unique extension as a bounded operator on LP (JR.), with operator norm independent of M. Exercise 3.4.11. Jf2 < p < CXl andf E S(JR.), prove that IISMf- flip M

~

~ 0

when

CXl.

Hint: Use the Hausdorff-Yo ung Theorem 3.2.12 to estimate IISM/ -fliP in terms of its Fourier transform. Exercise 3.4.12. Combine the previous exercises to show that for any f 2 < p < CXl, we have IISM.f- fliP~ 0 when M ~ CXl.

E LP(JR.),

Exercise 3.4.13. Use the duality of LP and LP' to prove that for any f E Lp(lR), 1 < p < 2, we have IISMJ- /lip~ 0 when M ~ CXl.

3.4.3

L 1 Theory of the Hilbert Transform and Extensions

It remains to discuss the Hilbert transform in case p = 1. At the same time we will identify the Hilbert transform with the limit of the conjugate Poisson kernel, equivalently as the imaginary part of the boundary value of an analytic function in the upper half plane. This extension will be carried out in the Banach space (3.4.3)

B1 =

{!: JR{

if(x)i dx < 1

+ lxl

CXl}

which contains all of the Lebesgue spaces LP(JR.), 1 < p < the absolutely convergent Fourier integrals: y > 0,

' To do this, we begin with

CXJ.

X E

JR.,

y > 0,

X E

JR.

For f E L 1 (lR), the Poisson integral and conjugate Poisson integral are written

FOURIER ANALYSIS IN LP SPACES

The defining equations for P yf, Qyf are also meaningful for f P y f . + l.Qy f

-1 -

1R

f(x- . t) d t -y -

·-1

l

lt

1

E

B 1. It is immediate that

f(u) .

+

1R X

189

lY - u

d u,

which is an analytic function of the complex variable z = x + iy for y > 0; hence the name conjugate Poisson kernel. In particular, the defining integrals are absolutely convergent and can be differentiated repeatedly to show that Py and Qy are both harmonic functions for y > 0, x E JR. In casef E LJ'(R), the harmonic property follows from repeated differentiation of the Fourier integral representation. In order to study the conjugate Poisson kernel, we first develop the necessary properties of the operator 1 Pyf(x) = -

(3.4.4)

1 1R

Jr

yf(x- t) dt. t 2 +y 2

This operator is defined on the Banach space (3.4.5)

82 = { f : llflls2 := -1 ;r

Clearly L 1 (R) C B 1

c

i

lf(x)l dx < oo } .

JRl+x 2

8 2 . The Poisson kernel has the following properties.

Proposition 3.4.14. Suppose that f E B2. Then 11Pyfll 82 < 211flls 2 for 0 < Y :S: 1 and for any f E Bz, limy--+0 IIPyf- flls 2 = 0. Furthermore iff E B1, then limy--+OPyf(x) =f(x)foralmostevery x E R Proof. We have

Yl/(t)l

IIPJIIo :5 _1 { ( { 2

rr2

= rr2 -1

la

JR y2

1 (1 R

1/(t)l

1 { = rr JR 1/(t)l (1 :5 -1 rr

+ (x- t)2

1 R

IR y2

1

+

dt)

y + (x-

d.x 1 + x2

t)2 1

+1 x2

dx ) dt

y

+ y) 2 + 12

dt

2 1/(t)l-2 dt

1+ t

= 211fllo2 • where we have used the semigroup property of the Poisson kernel in the form Py

* P1

_

Pl+y· To prove the norm convergence, we first note that if/ = 1ra.bl• then rrPy/(x) _ arctan [(x - b)jy] - arctan [(x - a)/y], which is bounded by 2 and tends pointwise to

Ira.bJ except at the endpoints x = a, b. By the dominated convergence theorem, we have IIP.vf -fllo2 --+ 0 wheny·-+ 0. Similarly for a finite linearcombinationf = ..,..N c·lr b . L...JJ= 1 J aJ. J 1 we have liP)/- j llo, --+ 0. But these functions are dense in the space B~o and we already have proved that the operator norms 11Pyll 82 are uniformly bounded for 0 < y :5 1, hence

190

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

the result. To prove the almost-everywhere convergence, we write

= -l

Pvf(x) - f(x)

rr

iPy/(x) - /(x) I

Jo rr

+

[f(x

+ f(x-

t)

t) -

2f(x)]

0

~ {':xo = -1

ct>, (!) :

100 t

2

1' o

y dt, t 2 +y2

"dct>x(t)

y

+y if(x

+ u) + j(x- u)

- 2/(x)l du.

From Lebesgue's theorem, we have for almost every x, ct>,(t)/t-+ 0 when t-+ 0. On the other hand,/ E B 1 implies that ct>, (t) ~ Ct for all t ~ 0. Now we integrate-by-parts:

=

2ty

1

IPJ(x)- j(x)l ~

(t

0

where the estimate lct>,(t)i in the integration gives

~

+

2

y2

)

2

ct>x(t) dt,

Ct allows one to discard the term at the limits. Setting t = yz

IP,/(x) - /(x)l ~

=

2z

1

" "

ct>,(zy)

O+z-)-

y

dz.

But the integrand is bounded by an L 1 function and tends to zero pointwise when y -+ 0, hence P y f (x) -+ f (x) as required. •

We now introduce a norm on the space B 1 , by defining

11/IIB, =

(3.4.6)

_.!._ {

rr

]TR.

lxf(x)l dx. 1 + x2

Theorem 3.4.15. Suppose that fEB,. Then Qvf(x) converges when y .!- 0 almost everywhere to a limiting function j (x) and we have for almost every x E JR, 1 f(x) = Hf(x) := -lim

(3.4.7)

1

f(x- y)

Proof. Any complex-valued function can be written as f = ~ 0. We begin with the conjugate Poisson kernel operator

J;

(3.4.8)

Qyf(x)

= -1 rr

1 IR

tf(x- !) t2

+y2

dt,

Clearly IQJ(O)I ~ 11/!ls,. Then (3.4.9)

PJ(x)

. + 1Q,./(x) = -i Tr

For any f mapping (3.4.10)

E B~o

y > 0,

1 R

dy.

Y

IYI>€

7l' €-->O

/(t)

.

X+ lY- t

/1 X

-

E

h +

i(/J - / 4 ) where

JR.

dt.

(3.4.9) defines an analytic function in the upper half plane y > 0. The (x, y) --+ exp [ -(P,/(x)

+ iQJ(x))]

is a bounded analytic function in the upper half plane. By the Fatou theorem, it possesses a.e. limits when y ,j.. 0. But Py/(x) converges to a finite limit a.e. whenever f E B 1 c B2. Hence we deduce the existence of the a. e. limit of exp ( -iQ,.f(x)] when y ,j, 0. From this it follows that Qyj(x) can have only one accumlation point ~hen y ,j.. 0. hence the existence of Hf(x) = lim,.'- 0 QJ(x). •

FOURIER ANALYSIS IN lJ' SPACES

191

It remains to identify the Hilbert transform as defined in (3.4. 7), with the boundary values of Qy/. namely to show that for a.e. x E IR, (3.4.11)

i

1

Qyj(x) = -

tf(x- t) dt t 2 +y 2

lR

7r

Lemma 3.4.16. Suppose that/

~

B 1 • Then

E

lim ( { tf(x- t) dt- {

.v~o for almost every x

E

J'R

+ y2

t2

y ~ 0.

Hj(x),

llrl>y

f(x- t) dt) t

= 0

JR.

Proof. We write the above difference as 1 1

+ h where

tf(x- t) dt 12

r (

h =

Jlti?.Y

The function t-+ t/(t

+y

2

2

)

t

2

+

t

y

.!.)t<x- t) dt.

2 -

t

is odd and increasing for ltl < y, so that we can write

1 + 1

=

/1

+ y2

ltl
t

t2

Y

2

[f(x- t) - f(x)] dt

if(x - t) - f(x) I dt -+

ild ::0: __!._ 2y

0

lti
at every Lebesgue point off, especially almost everywhere. To estimate h we note that its kernel is odd, hence for any 8 > 0 andy < 8,

1 /+ 2

-h =

I l>_v

r

1/zl ::::

y2)

[f(x- t) -f(x)]dt

2

Jlti>y

= y2

t(t

1

y 1 if(x- t)- f(x)i dt itidF(t)

lti>y~

where F(t) = J~ if(x- s) -f(x)l ds. Clearly F(t)/t-+ 0 at every Lebesgue point when t-+ 0, whereas F(t) ::;: Ct when for all t. Therefore we can integrate-by-p arts to obtain

1

dF(t) = F(y)

I l>y

itl 3

IYI 3

The term at the limits is clearly o(yF(t)/t = 17(1), v = l/t to obtain

1

ltl>v

F(t) -

t

4

-

dt =

which completes the proof that h

2

)

+3

1

ltl>y

when y

-+

11/v V1'/(ljv)dv =

F(t) dt 4

1

•

0. To analyze the new integral, write

o(y- 2 ),

y-+ 0,

0

-+

0 when y

-+

0 for almost every x e JR.

•

As a special case, we can deduce the properties of the Hilbert transform on LP(R), 1 ::; p < oo.

}92

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Corollary 3.4.17. Suppose that f E £P(JR), 1 ::S p < oo. Then Pyj(x)

--+ f for almost every x E lR and the convergence takes place in LP (lR). Furthermore Qy/(x) converges almost everywhere to a limitingfunction ](x) and the convergence takes place in LP ifp > 1.

The convergence properties of Py/ follow from the properties of the Poisson kernel: It is readily verified that t --+ t/(t 2 + y 2 ) satisfies the properties of an approximate identity. Therefore if B is any homogeneous Banach space, we have Py/--+ fin the Bnorm when y --+ 0. The almost-everywh ere convergence of Qy/ follows from Theorem 3.4.15. • 3.4.3. 1 Kolmogorov's inequality for the Hilbert transform Following the discussion of the conjugate function on the circle, we can establish a corresponding inequality for the distribution function of the Hilbert transform whenever f E L, {lR). This takes the form

l{x: IH/(x)l ::: a}l < c { lf(x)l dx

(3.4.12)

J'R.

a

where Cis an absolute constant. This will be deduced as a limiting case of a corresponding inequality for functions in the space B 1 • Define a weighted measure by

11

(3.4.13)

J.L(A) = -

7r

Theorem 3.4.18. Suppose that f

E

A

dx

1+

.

x2

B I· Iff > 0, then we have the weak inequality

(3.4.14) JL{X:

iHf(x)l::: a}<~ ( 7r

ll/lls2

a - 11/lls,

+

ll/lls2

a

+ 11/IIB,

).

a>

llfll,.

For any complex-valued / E B 1 , (3.4.14) holds with four terms on the right side and with a replaced by aj4. Proof. We consider the harmonic function J,.(w), defined for Re(w) > 0 as the harmonic measure of the two rays {w = iv, v ::: a} and {w = iv, v :5 -a}. This is the harmonic function that takes the value 1 on these rays and takes the value zero on the segment {w = iv, -a :5 v :5 a}. Equivalently, it can be obtained as the imaginary part of (ljrr) log [(wia)j(w + ia)] for a suitable branch of the logarithm. The set {w : la(w) ::: is the exterior of the semicircle described by {w : Re(w) > 0, lwl = a}. On the strip llm(w)l < a, we have

1}

(3.4.15)

la(u +

iv)

=

_!._ [arctan rr

(-u-) + a-v

artctan

(-u-)], a+v

We now consider the harmonic function Uu(X, y)

= la[P,.j(x) + iQJ(x)].

We first recall a basic fact on harmonic functions.

u > 0, lvl
FOURIER ANALYSIS IN LP SPACES

193

Lemma 3.4.19. lfU(x, y) is any bounded harmonic function in y > 0, thenfora ny y 1 , y 2 > 0, we have

(3.4.16) Applying (3.4.I6) with X= 0, Y2 (3.4.17)

=

I,

u = u," we have

. Ja {PI+vf(O ) + tQI+y/(0 ))

=

I

{ J 01 (Pyj(t)

+

iQy/(t))

I+ t2

1T }R

dt.

The right side of (3.4.17) is underesti mated by

_!_ { JT }R

J 01 (Pyj(t)

+ iQy/(t))

I +

t

dt > _I_l.

2

2rr

-

(t

~-

IQ,.f(t)l~<>l 1 + t

Using the inequality Iarctan (x)l :5 Jxl, applied to (3.4.I5), we can overestim ate the left side of (3.4.17) by writing . I ( PI+J(O) PI+y/(0) ) Ja(PI+J( O) + IQI+.vf(O )) :5 1T a - H2I+.vf(O)I + a+ IQI+y/(0) 1 . Therefore we have

(3.4.I8) Recall that Hf(x) = limy ... o Qyj(x) a.e., in particula r we have converge nce in measure. Now from (3.4.6), IQI/(0)1 :5 11/IIB" PI/(0) = II/IIB and the right side of(3.4.18 ) is only 2 increased when we replace Qd(O) by its upper bound 11/IIB • Hence 1

J.dx: IH/(x)l >a} <

-

-

~ rr

(

II/IIB2 + II/IIB2 ) , a - II/IIB 1 a + 11/IIB.

which proves the result in casef 2:: 0. In the general case, we write 4

J.L{X: IH/(x)l 2:: a} :5

L

J.L{x: IHJ(x)l 2:: a/4}

;=I

and apply the result for nonnegat ive functions to each of the terms on the right.

•

The upper bound assumes a more familiar form in case/ is even, as follows. Corolla ry 3.4.20. Suppose thatO 0 we have

E B1

iseven: f(-x) =f(x), Vx

J.L{X: IHJ(x)l >a} <

E

JR. Then

41111182 . JTOt

Proof. In this case we have Qy/(0) = 0 for ally > 0. Thus the right side of (3.4.18) becomes (4/rr)PI+ J(O) ~ 411/11 82 /rr when y ~ 0. The inequality (3.4.I4) contains the classical Kolmogo rov inequality (3.4.I2) as a limiting case, when we introduce a scaling paramete r Y.

194

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

In detail, define (3.4.19)

11

y

= -

t.ty(A)

2 2 ll"At+Y

dt.

Then we have the following scaled replacement for (3.4.14) when a > IQyj(O)I: (3.4.20)

t.Ly{x: IHf(x)i :=:::a} :S

2 ( 7l"

Pyf(O) Qyf(O)

a_

Pyf(O)

+a+ Qyf(O)

) ·

Now multiply (3.4.20) by Y and take Y -+ CXl. For the left side, we note that for any Borel set of finite Lebesgue measure we have from the dominated convergence theorem 0

hm Yt.ty(A)

Y-oo

For the right side, we see that when Y -+ that for any f E L 1 (JR.)

=

1 -IAI. 1f

the dominated convergence theorem shows

CXl,

hm YPyf(O) = -1 0

Y--~>oo

Jr

1

j(x) dx

IR

whereas IQyj(O)i ::: 11/IIL'<JR> x sup tEIR

iti

1

2

+

Hence when we multiply (3.4.20) by Y and take Y -+ form of Kolmogorov's inequality.

3.4.4

y-+

y 2 -+ 0, CXl,

CXl

we obtain (3.4.12), the original •

Application to Singular Integrals with Odd Kernels

The theory of the Hilbert transform can be transplanted to study n-dimensional singular integrals of the form (3.4.21)

K".Mf(x)

J

k(y)f(x- y) dy.

<'
Here k is supposed to be an odd function that is homogeneous of degree -n and satisfies ~xl=t lk(x)l dx < oo. We write k(x) = lxl-nQ(x) where n is odd and homogeneous of degree zero. This can be reduced to the Hilbert transform by the method of rotations, developed by Calderon, as follows: We take spherical polar coordinates y = rw, with dy = rn-l drdw. Then

1

<'
k(y)f(x- y) dy

=1M

r-n ( {

= =

J._,

Q(w)f(x- rw)rn-l dr) dw

Js•-l

"

Q(w)

.!_ { 2 ls•-1

(1M

f(x

~ rw) dr) dw

Q(w) ( / <'
1 . . , ( )(1

f(x- rw)- f(x

+ rw)

dr) dw

r

f (x - rw) dr ) dw = -1 "" w 2 s•-' <'
FOURIER ANALYSIS IN U SPACES

195

make the w-depen dent decompo sition of lR 11 3 x = z + sw where -ex:> < s < ex:> and z is in the hyperpla ne defined by z · w = 0. In detail, s = x ·wand z = x - (x · w)w. Taking E -+ 0, M -+ ex:>, we have Kf(x)

=~Is,_,

Q(w)HF :.w(s)dw .

We estimate the n-dimen sional LP norm by using Minkow ski's integral inequali ty and writing dx = ds dz' as follows:

We summar ize the above computa tions as a theorem . Theorem 3.4.21. There exists an absolute constant Cp such that for eachf E S and 1 < p < ex:>, we have the estimate

Using the oddness of the kernel, we can also write

1[

KE.Mf(x ) = 2

k(y)[f( x- y)- f(x

E
+ y)] dy,

f

ES.

Taking limits, we have the absolute ly converge nt represen tation of the operator : Kf(x) = _!_ { 2

JlR"

k(y)[f( x- y) - f(x

+ y)] dy,

f

E

S.

Example 3.4.22. The Riesz kernels are defined by kj(x) = CnXj/ixin+I for 1 < j < n where C 11 is a constant . Clearly they satisfy all of the above conditio ns of oddness, homogen eity, and integrab ility.

The associat ed singular integral operator is denoted Rj. From Theorem 3.4.21 we have the estimate IIR.JII" < Cpllfll" forf E S. We now show that, to within a constant , the Riesz kernels can be regarded as the formal first partial derivativ es of the operator f -+ JIR,.f(x - y)/lyln- l dy. To see this we use the method of subordin ation. We begin with the heat kernel transform off E S:

196

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Taking the partial derivative with respect to XJ. we have for 1 < j < n, _

i

2

j(x _ y) Yi e -ly1 _/4t dy =

IR"

2t ( 47r t)n/ 2

i

f(~)e-4rr tlld (27ri~j)e2rri~-x d~. 2

IR"

Multiply both sides by t- 112 and integrate over 0 < t < oo. We recognize the elementary integrals

Therefore

which displays the Fourier transform of the kernel. We choose the constant en in the definition so that the Fourier transform is as simple as possible, thus we define the Riesz transform by (3.4.22)

l~j
This leads to a famous application of Calderon and Zygmund, as follows:

Proposition 3.4.23. Suppose thatf (3.4.23)

where L\

E

S(lRn) then we have the a priori bound

l<j,k
=I::;= I a2 ;axJ is then-dimension al Laplace operator.

Proof. We first prove the identity

(3.4.24) The Fourier transform of the left side is -4rr 2 ~j~kf(l;), while the Fourier transform of the right side is

which proves (3.4.24). The proof is completed by applying the LP boundedness of the operators Rj. •

FOURIER ANALYSIS IN U SPACES

3.5

197

HARDY-LITTL EWOOD MAXIMAL FUNCTION

A powerful tool in harmonic analysis is the concept of a maximal function. Strictly speaking, one should speak of a maximal operator, transforming a function into a s4premum over an appropriate indexed family of sets. Iff is a locally integrable function on IR.n, the Hardy-Littlewoo d maximal function is defined by (3.5.1)

}; If/ Mf(x) :=sup _!}__I I B=n

B

where the supremum is taken over all balls containing the point x E IR.n. The balls need not be centered at x. The operator M is subline"ar, in ·the sense that for any two locally integrable functions/, g we have Mif + g)(x) < Mf(x) + Mg(x). It is also clear that Mf(x) = M(/fl)(x), so that we can always assume thatf is positive when studying Mf. To see that Mf is a measurable function, note that both numerator and denominator of (3.5.1) are continuous functions of the radius and center of the ball B. Hence if we restrict the supremum to balls with rational centers and rational radii, the supremum will be unchanged. But for any fixed ball B, the average in (3.5.1) is a two-valued, hence measurable function of x. · It is obvious from the definition (3.5.1) that M is bounded on V'"(lRn): IIMflloo :::;:: llf/1 00 • The following theorem describes the boundedness properties on the other LP spaces. Theorem 3.5.1. Hardy-Littlewo od: (i) There exists a positive constant C, depending only on n such that for each ex > 0, f E L 1 (lRn) l{x: Mf(x) >ex}/ ::: C ex

{

]'Rn

If/.

In particular Mf is finite almost everywhere. (ii) If I < p < OCI, there exists a positive constant Cp.n so that for f

E

LP(JR.n)

(iii) Iff lives on a set B of finite measure and If/log+ lfl E L 1 (B), then fsMf < OCI. Proof. We begin by studying the set Ea = {x : Mj(x) > a}. If x E Ea. then there is a ball Bx ~ x so that f 8 , 1/1 > a/Bxl· Thus we have a covering: Ea C UxeEaBx. Since lR" has a countable dense subset (of points with rational coordinates), we can choose a countable subcollection x* so that Ea C ur:,Bx,· Calling these Bk> we must show that a j(ur:,Bk)i s C/lf/1,. From the countable additivity of Lebesgue measure, I ur:, Btl ~ limN I uf= Btl, 1 so it's enough to show that for any N

(3.5.2)

198

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

If the balls were pairwise disjoint we would be done, since for each k, summing on k we would have

j 8 A If I ?: a

IBk I, and

To complete the proof, it suffices to find a subcollection of balls that are pairwise disjoint and whose union covers at least a fixed fraction of Ea. •

Lemma 3.5.2.

Wiener covering lemma: Given any finite collection of balls

Bk : I _:::: k < N in lRn, there exists a subcollectio n

B; : I

_:::: i < p so that

(ii) {Hi}~= I are pairwise disjoint. Proof. We order the balls in order of decreasing radii, renaming them B, , B 2 , • • • • Let iJ, = B 1 • Assuming that B 1 , ••• , Bm have been chosen, choose Bm+l = Bk where k is the smallest index so that B k n B1 = ljJ for j = I , ... , m. If no such index exists then the process terminates. We now check condition (i): if Bk is a ball that was discarded, then by definition there is a ball lJ, = B 1 such thatj < k and Bk n Bj ¥= f/J. Since the radii are ordered, we have r, = r) ::: rk. We claim that Bk c 3B,. Indeed, there is some z E Bk n B;. Now if X is any point of Bk. we can use the triangle inequality to estimate the distance from x to the center of B; as lxl = j(x- z)

+ zl ::: lx- zl + lzl :::

2r~

+ r, ::: 3r,.

Hence Bk c 3B,, which proves that U~= 1 Bk c uf= 1 3B,. But the measure of a disjoint union is the sum of the measures, and for any B, then-dimensi onal measure of 3B is 3n times the measure of B. The proof is complete. •

Having proved the covering lemma, part (i) of the Hardy-Little wood maximal theorem is complete, since p

II/III

>

L,'=,iJ, lfl

>a

L

IB;I =a

t=l

This proves (3.5.2) and hence part (i). Taking a ---+

CXJ

shows that Mf <

CXJ

Proof. To prove part (ii), we introduce the distribution function

(3.5.3)

AJ(a) := l{x: 1/(x)l > a}l =

{

'R"

Ica.x>(lf(x)l) dx.

The l.J' norm can be written in terms of the distribution function by writing

a.e.

FOURIER ANALYSIS IN LP SPACES

199

and applying Fubini's theorem to obtain {

lrrt"

lf(x)IP dx

(roc

= {

Jrx, Jo

=

1=

=

1

puP-

1

puP-ll(O.If(x) l)(u)) dx

(in

l(u.oo)(f(x))d x) du

00

pup- I AJ(u) du.

We will apply this identity to estimate liM/lip· To do this, we first decomposef into a bounded part and an unbounded part (without loss of generality, we can assume thatf 2: 0). (3.5.4) Since the operator M is sublinear, we have Mf(x) ::::: Mfc,(x)

+ Mf"' (x)

::::: ex

+ Mf"' (x).

Hence if Mf > 2ex, then Mfa > ex. In terms of the distribution function, we have AMJ(2ex) = l{x: Mf(x) > 2ex}l::::: l{x: Mf"'(x) > ex}j.

Butf"' lives on a set of finite measure, since l(x :f"(x) > 0}1 = l{x :f(x) > ex}l ::::: llfiiP exP

while l/0 1 Sex+ lfl so that/a E U(lRn) to which we can apply the Hardy-Littlew ood maximal inequality; we change ex to 2ex and write

IlM/II~ =

1

00

p

(2ex)p-I A.M1 (2ex)d(2cx)

1 (i. 1 p2~'C l, (1f<•J ex~'- dex 1 dx 00

S p2P C

exP-

2

f"' (X)

dx) dex

00

2

= p2PC =

exP- (LJ(x) l(a.oo)if(x))

2

Cp2P = p- }

)f(x)

dx) dex

dx

f(x)P-lf(x)

IR"

Cp2P

= --111/11~. pwhich was to be proved.

•

200

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. To deal with (iii), we write

i

Mf =

i

+

MflMJs2

i

:5 2181

+

1oc

= 2181

+

2.A.Mf(2)

MfiMf>2

AMJ(max(a, 2)) da

+

1':>0 .A.Mf(a) da.

The last term is estimated from the Hardy-Littlewood maximal inequality and the decomposition/ =fo, +

r

2

1=

.A.M1 (2a)da :5 2C / " "

~

(l_.f"'(x)dx

)da

00

=

lt

= 2C

=

.!._ ( { j(x) 1 (a,oo) if(x))

2C {

ex

JR"

dx)

da

l. (ff(x) ~a )t(x) dx

2C {

Jan

j(x) log+ f(x)

dx

= 2CIIf log+ fll•,

which completes the proof.

=

•

L 1(Rn) if and only iff 0. To see this, letf E L (Rn) and choose a (large) ball B centered at 0, so that f 8 Ill > ~11/llt· Then for any x ¢ B, we have

Remark. It is important to note that Mf

E

1

Mj(x) > fn 1/1 > fn 1/1 > const 111/xii~I - IB(O; 21xl)l - IB(O; 2!xl)l where the constant is half the reciprocal of the volume of the ball B(O; 2) in Rn. Hence Mf(x) 2: Cflxln for large x, which contradicts fR· IMfl < ex:>. The upshot of this last remark is that no matter what decay condition we impose onj, it is never possible to achieve Mf E L 1 (Rn).

3.5.1

Application to the Lebesgue Differentiatio n Theorem

The Hardy-Littlewoo d maximal inequality can be used to give an efficient proof of the differentiability of the integral, as follows:

Proposition 3.5.3. Iff is a locally integrable function, then for almost every X ERn

(3.5.5)

lim

JB(x:r)f

r->0

IB(x; r)l

=J(x).

Here B(x; r) is the ball of radius r centered at x

ERn.

FOURIER ANALYSIS IN LP SPACES

201

Proof. To apply the Hardy-Littlewood weak ( 1, 1) estimate, we need to replace f by an integrable function. This is easily accomplished by writing the exceptional set in (3.5.5) as the union of the exceptional sets SM in the ball lxl ::S M, M = 1, 2, .... On each SM we can replace the locally integrable function/ by the integrable functionf(x) lro.M+JJ(Ixl), to which Hardy-Littlewood can be applied. To prove (3 .5 .5 ), we first note that if g is a continuous function, then the indicated limit exists and = g(x) for every x E !Rn. For an arbitrary f, define JB(x;r>f

Qf(x) :=lim supfr(x)- lim inff,.(x).

fr(X) := IB(x; r)l'

r__.,.o

r-+0

Given E > 0, there exists a continuous function g such that II f g + h with llh III < E, we have

- g III

<

E.

Writing

f =

lim supfr(x) = g(x) +lim sup hr(X), r-o r-o lim inff,.(x) = g(x) +lim inf hr(X), r-0

rlf(x)

=

r-0

Qh(x) ::::; 2 sup lhr(x)l ::::; 2Mihl(x).

Therefore for any 8 > 0, l{x: rlf(x) > o}l ::::; l{x: Mlhl(x) But the left side does not depend on

E,

~

8/2}1 ::0

2E

8

hence we conclude that for every 8 > 0,

l{x: Qf(x) > 8}1 = 0 which means that Qf(x) = 0 almost everywhere, which was to be proved.

•

The above reasoning can be strenthened and clarified in terms of the Lebesgue set of the locally integrable function f. This is defined as (3.5.6)

Leb{f) =

{

X E

n

.

JR : hm

fa(x;r)

r---+0

lf(y)- f(x)jm(dy) } = 0 . IB(x; r)l

Proposition 3.5.4. For any locally integrable f, i (Leb(f) Y! = 0. Proof. For each real number c we apply the previous proof to the function I f - c!. Thus we obtain for almost every x E !Rn,

(3.5.7)

.

fBix r)

hm r-o

·

lf(y) - clm(dy) IB(x; r)l

=

lf(x)- cl.

If Cj is an enumeration of the rational numbers, we obtain a countable collection of excep· tional sets Ej· Then E := U1 E1 also has Lebesgue measure zero. But the right side and left side of (3.5.7) are continuous functions of c; the right side is obvious, but so is the left since the triangle inequality gives llf(y)- c,l- if(y) -· c2!1 ::::; lei - c 2 1. Hence if x ¢Ewe let

c1

~

f(x) to conclude that

lim r-o which was to be proved.

JB\x r)

lf(y)-:- /(~/m(d):!._ IB(x; r)/

=

O

x ¢E.

•

202

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The Hardy-Littlewood maximal function can also be used to investigate certain questions of nontangential convergence. Normally this is considered in the framework of convergence in the unit disk or a higher-dimensional space. But the basic ideas are already present in the above framework, in the context of convergence of the type when Xm--+ x. We do not expect that this will be true unrestrictedly, since x --+ f 1;m(x) is a continuous function whereas the limit f is not continuous in general-hence we cannot expect uniform convergence. Nevertheless we have the following proposition.

Proposition 3.5.5. Suppose that Xm f

E L 1 (1Rn),

--+ x so that lxm -xi < lfm. Then for any limm ft;m(Xm) =f(x) atalmosteveryx.

Proof. Repeating the above steps, we write f Define

=

g

+

h where g is continuous and II h 11 1 <

E.

But

The hypothesis

lx -

Xm I

< 1 I m ensures that

x

E {y :

IY -

Xm I

< 1 I m}, hence

Qj(x) ~ 2Mh(x)

from which the proof can be completed as above: 28 l{x: Qj(x) > 8}1 ~ l{x: Mh(x) =:::: 8/2}1 ~ - . €

But the left side does not depend on

E,

hence we conclude that for every 8 > 0,

l{x: Qj(x) > 8}1 = 0.

3.5.2

•

Application to Radial Convolution Operators

The Hardy-Littlewood maximal function can be used to estimate more general convolution operators of the form (3.5.8)

(k, *f)(x) =

{

' R"

k,(y)f(x- y) dy,

where the radial kernel k, E L 1 (lRn) is obtained from a monotone decreasing nonnegative function K : JR+ --+ JR+ by setting k,(y) = t-nK(yft). The example of the HardyLittlewood operator is obtained when we set K(x) = l[O.I](Ixl), the indicator function of the closed unit ball. The following lemma shows that the Hardy-Littlewood maximal function can be used as a universal bound for any radial convolution. The original form is attributed to K.T. Smith ( 1956), and appears in Stein ( 1970). The proof below is attributed to S. Saeki, reproduced by Banuelos and Moore ( 1999).

FOURIER ANALYSIS IN U SPACES

203

Lemma 3.5.6. For any radial kernel, we have the estimate lkJ(x) I < (Mj)(x) Ilk !II 1,

f

E

1

L (lR.n).

Proof. Let J-1.-x be the measure defined by J-1.-,(A) =fA lf<x- y)l dy, for any Borel setA. In particular if A is a ball, we have J-1.-x{A) ::::0 IAI(Mj)(x). Applying this to the ball defined by B>. = {y E Rn : k,(y) > ).}, we have

l(k, *f)(x)l

::::0 .(., k,(y)lf(x-

= {

}fi>.n

=

k,(y)f.lx(dy)

r=

Jo

y)l dy

px(B;,_) d).

::::0 (Mj)(x) {""

Jo

IB"I d).

= <Mf)(x) Ilk, ll1, where we have used the definition of Mf and twice used the representation of the L 1-norm as the integral of the distribution function. • A typical application is to the n-dimensional heat kernel, where K(x) = j(4rr)nf2. Lemma 3.5.6 can then be combined with the Hardy-Littlewood maximal inequality to give a new proof of Proposition 2.2.35, that for any f E L 1 (JR.n), and for almost every x E IR.n, (k 1 *f)(x) --+ f(x) when t --+ 0. e-lxl

2

/

4

Exercise 3.5.7. Complete the details of this argument. Hint: Define Qj(x) =lim supt- 0 (k, *f)(x)- lim inft-o(k, *f)(x) and argue as in the proof of the Lebesgue differentiation theorem to prove that lim,~o(k, * f)(x) exists a.e. Then identify the limit by a density argument.

As a second application of Lemma 3.5.6, consider the case of then-dimensional Poisson kemel-K(x) = Cn/0 + lxl 2 )
Exercise 3.5.8. Complete the details of this argument.

3.5.3

Maximal Inequalities for Spherical Averages

Stein ( 1976) has shown that there exist LP maximal inequalities for the spherical maximal function (3.5.9)

A*f(x) =sup 1>0

Ill.vl=l { f(x- ty)dw(y)l

where dw is the normalized surface measure on the sphere sn-l c IR.n. Results of this type can be used to prove Fatou-type theorems for solutions of the wave equation u 11 = l::!..u for suitable values of n, p. In the treatment below we restrict attention to the simplest

204

INTROpUCTION TO FOURIER ANALYSIS AND WAVELETS

case, where p = 2, n > 4. A complete account of the subject for more n,p can be found in Stein (1993), Chapter XI. We begin with the spherical averaging opeator f

~

jvl=l

(AJ)(x) :=

f(x- ty) dw(y) = (f

genera~

values of

* dwr)(x).

We will also make use of the square function, defined by (3.5.10) which is well-defined whenever f Lemma 3.5.9. For any f

E

E

C 1 has compact support.

C 1 (lRn) with compact support, we have

IArf(x)l < (Mf)(x) +Sf::; ...;2n where Mf is the Hardy-Littlewood maxima/function. Proof. We write

1

=

t-n

d -(snAJ(x)) ds p ds ·

1

=I, +/z where

1 1

I,= t-n

nsn- 1 (AJ)(x)ds

d

]z = sn ds (Asf)(x) ds.

I, is majorized by the Hardy-Littlewood maximal function, since for any Cn I, = -;;t

1

t >

0,

f(y) dy .::; (Mf)(x).

ly-xi9

Meanwhile, [z is estimated in terms of Sf by using Cauchy-Schwarz: 1

liz I = tn

1 o

.:S tn .::;

(

d sn-o; 2 >Js- (Asf)(x) ds ds

Jo{ ' s 2

n-J

ds

) 1/2 (

s

[

d ds (A_,f)(x)

J ds 2

)

l/2

1

r,:o-(Sj)(x),

v2n

which completes the proof.

•

FOURIER ANALYSIS IN LP SPACES

205

Lemma 3.5.1 0. /fn > 4 andf is C 1 with compact support, then

where the L 2 nonn is taken over all of~" and the constant Cn depends only upon the dimension. Proof. The Fourier representation of the spherical average is

where w(~) = c,.J12 (I~ I)/I~ l1 2 is the Fourier transform of the normalized measure w. Now by the chain rule

(3.5.11) where we have set n

J.L(~) := L:~j j=l

aw -. a~,

Now from the asymptotic behavior of the Bessel function, we have

aw a~j On the other hand, when we can write

I~ I

--. 0, we have

= O(I~I(J-n)/ 2 ).

=

aw;a~j

0(1~1).

1~1 __. 00.

Combining these estimates,

(3.5.12) Applying Plancherel's theorem to (3.5.11), we have

Integrating both sides with respect to the measure s ds and applying Fubini, we have (3.5.13)

{ ISJ(x)l 2 dx = { lf(~)l 2 }JRn }JRn

(

{co IJ.L(s~W ds).

Jo

S

But the inner integral can be estimated using (3.5.12). For any M, we have

:S

Cni~I 2 1M sds + Cni~IJ-n Leo s:~2

= Cni~I 2 M 2 + Cn

~~~ -n 3

(n -

2)M"- 3

206

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

The two terms are comparab le to one another by taking M

1= iJ.t(s~)l ~s ~ 2

Cn ( 1 + n

~ 2)

=

I /I~ I, leading to the estimate

= C,.:

=~ ·

Referring to (3.5.13) and applying Planchere l 's theorem once again, the proof is complete .

•

Combin ing Lemmas 3.5.9 and 3.5.10, with the L 2 bound for the Hardy-L ittlewoo d maxima l function , we have proved the followin g result.

Theorem 3.5.11. Suppose that n :?: 4. Then we have the followin g estimate for any f E C 1 (IR") with compact support:

II sup(AJ) (x) liz

(3.5.14)

< Cllfllz

t>O

where the constant depends only on the dimensio n. The estimate (3.5.14) can be extended to any f function s with compact support is dense in L 2 .

E L 2 (IR:")

by noting that the set of C 1

Example 3.5.12. Taking n = 1 and f an unbound ed function shows that sup, AJ(x) = +(X) for every x E JR. 1, hence the estimate (3.5.14) cannot hold for general ! E L 2 (1R.). Example 3.5.13. To obtain an example in higher dimensio ns, take f(x) = lxl 1-n[log( l/lxl)r 1 I[O.IJ(Ixi ), which fails to be in L 2 (1Rn) andfor which it is verified that sup, AJ (x) = +(X) for every x E JR.n.

3.6

THE MARCI NKIEW ICZ INTERP OLATIO N THEOR EM

In this section we develop the notion of weakly bounded operator s and apply it to prove the theorem of Marcink iewicz. To orient the thinking , we first define the notion of weak Lebesgu e space.

Definiti on 3.6.1.

for some C > 0, where 1 < p <

(X).

In case p

=

(X),

we set wkL 00 (JR.n)

=L

00

(.!Rn).

This definitio n can be rewritten in terms of the distribut ion function 'Ar in the form 'Aj(cx) < Ca-P.

Clearly LP(.!Rn) C wkLP(JR.") since iff E LP(JR."), then by Chebysh ev's inequali ty we have for any ex > 0

11!11~

=

{

]'R..

if(x)IP d.x >

1

1/1>0'

if(x)i" d.x:?: cxPi{x : if(x) :?: cx}j.

FOURIER ANALYSIS IN U' SPACES

207

But the converse is not true. For example f (x) = 1 I ( 1 + lx I) is wkL 1 (IR 1 ) but is not integrable . In parallel with the discussion preceding the M. Riesz-Tho rin theorem <~t the beginning of this chapter, we have the following elementar y properties . Lemma 3.6.2. Suppose thatf f E IY' (Rn)for any PI >Po·

E wkLP0 (Il~n)

and that

1/1

< M for some M. Then

Proof. In terms of the distribution functions. we can write {

Jan

lf(x)lf'' dx =p 1 {M aP 1 - 1 AJ(a)da:: : Cp, {M ap,-l-Po)._ f(a)da < oo.

Jo

Jo

•

Lemma 3.6.3. Suppose that f E wkLP' (IRn) lives on a set of finite measure B. Thenf E LPo(JRn)fo r any Po
:S AJ(l)

+ C p 0 fc>o aPO-p,-l da

< oo.

In p 1 = oo, then the result is immediate, since on a space of finite measure, any bounded function is in U 0 for any p 0 • •

Lemma 3.6.4. Suppose thatf Thenf E U(lRn).

E

wkLPo(JRn) andf

E

wkLP' (lRn) where Po < p <

PI·

Proof. We write/ = /1 1111 ~1) + /Ir 1JI>Il = /1 + /2- Both/1 and/2 are dominated by f. In particular / 1 E wk.LPO and/2 E wkU 1 • But/1 is bounded and/2 lives on a set of finite measure, since l{x :j2(x)

=F

0}1

=

l{x: 1/(x)l > 1}1 :S C < oo.

Therefore by the preceding lemrnas,/1 E U(IR.n) and/2 E U(IR.n). But Lp(1Rn) is a linear space, hence f E LP. •

These lemmas will be applied in many ways. As an elementar y instance, we note that the Hardy-Lit tlewood maximal function is in the space wkL 1 (lRn) and also in the space L 00 (1Rn). Therefore we may conclude that Mf E LP(Rn) for any 1 < p < oo. In order to include the Hardy-Lit tlewood maximal function and related operators, we first formulate the general notion of sublinear operator as follows.

Definitio n 3.6.5. An operator Tis said to be sublinear if, whenever Tf1 and Th are defined, and cis any complex number, then

IT(/1 +h) I

:s IT/II+ IThl.

We also need the notion of type (p, q).

IT(cj1) I :s clfJI.

208

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Definition 3.6.6. Let (M, J.L) and (N, v) be measure spaces with a sublinear

operator T: LP(M, J.L) ~ Lq(N, v). We say that Tis of type (p, q) if there exists a positive constant C so that for every f E LP (M, J.L ), (3.6.1) As a consequence , it follows from Chebyshev's inequality that for each a > 0, (3.6.2)

aqJ.L{X:

ITf(x)l

>a}< CPIIfll~-

The converse is not true. This leads us to the notion of weak type (p, q).

Definition 3.6.7. An operator for which (3.6.2) holds is said to be ofweak type (p, q).

For example, the Hardy-Little wood maximal operator is of weak type (1, 1) but does not satisfy (3.6.1) with (p, q) = (1, 1). In case q = oo, condition (3.6.2) implies that Tf E L <Xl so that the definition can be taken in the strong sense.

Theorem 3.6.8. LetT be a sublinearop erator that is defined on the space LP0 +LP' so that it is weakly bounded of type (po, Po) and weakly bounded of type (Pt, Pt). where 1 ~Po 1, where the first term is in LP' and the second term is in LPO. Hence LP c LPO + LP• . We first prove the theorem in case p 1 = +oo. Dividing T by a constant, we may assume that 117JIIoo ~ 11/11 00 • Now we make the a-dependent decompositio n f=fo+/1

where

/1 Now IT/I

~

ITfol

=/l{lflsa/2}·

fo =/luJI>a/21·

+ ITf.l, so that the distribution functions

satisfy

But 1/d ~ aj2 and IITIIoo.oo ~ 1 so that the second term is zero. Now from the weak (po.Po) hypothesis,

and

11/oll::g

=

L

IJU)IP 0 1 (lff>a/21/-l(dt).

FOURIER ANALYSIS IN LP SPACES

209

To compute the norm of Tf, we write liT/liP= p {(X) A.u(a)ar-• da

lo

P

p?P-PO

=

-

p-po

llfiiP. P

This completes the proof that liT/liP .:::0 C(p,p0 )11/IIP in casep, = oo. To discuss the case p 1 < oo, we make the same decomposi tion off =fo + f,:

li T/11~ =

p

1= 1

A.rt(a)aP-I da

00

.:::0 p

A.rt0 (a/2)ap-J da + ATft (a/2)aP-I da

The term involvingf0 can be estimated in precisely the same fashion as above. For the term involving/~. we use the weak (p1,p 1) hypothesis to write

and hence

1""

pal'

1

a-P•

ll/1ll~:da.::::

1

00

pap-l-p,

roo

= { (

JM

(L t,lf(l>l~amlf
da

pap-l-p, da) 1/(t)IP' J,t(dt)

121/U)I

2P-Pl

=

3.7

PI-p

11/11~-

•

CALDER ON-ZYG MUND DECOM POSITIO N

In previous sections we had to decompos e a function into a bounded part and an unbounde d part. This suffices to prove the Marcinkie wicz interpolat ion theorem, for example. In other problems one needs a more sophistica ted decompos ition of an integrable function, due to Alberto Calderon and Antoni Zygmund in their seminal work on singular integral operators.

210

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

Propositi on 3.7. 1. Let f exist disjoint subsets F,

n

L 1 (Il~n) be nonnegati ve and let a > 0. Then there so that E

(i) lf(x)l < aforx E F. (ii) Q = UkQk, where Qk are cubes with disjoint interiors so that

-1

1

a < IQkl (iii) ~n =

n u F, IQI

<

f(x) dx < 2na.

Q,

llfiii/a.

Proof. We begin by decomposi ng IRn into cubes Q whose sides are parallel to the coordinate axes with IQI ::: 11/11 1 jcx. Clearly this is possible since 11/11 1 < oo. In particular, we have

(3.7.1)

1~1 ~ f

:S

i.

1~1

f :Sa.

For each of these cubes, subdivide into 2n congruent cubes Q'. For each of these cubes, it's clear that (3.7.2) for otherwise we would violate (3.7.1). These cubes are of two types: Type(i):

I~' I fo,i

1 T\•pe(ii): - ~:~

IQ'I

{ f JQ'

:Sa, > a.

If Q' is of type (ii), then one does not subdivide further and it is added to the list of cubes Qk. If Q' is of type (i), then we subdivide it into 2n congruent cubes whose sides are parallel to the axes and repeat the decision process. As before, the discarded cubes satisfy (3.7.2). Continuing inductively defines the cubes Qk; we let Q := UkQk. F := IR.n\Q. If x E F, then every cube containing xis of type (i). By the corollary to the Lebesgue differentiat ion theorem we have for almost every x, /(x) =lim Q

-/J

1

IQI


-

X

E

F.

Finally we note that since aiQki :S JQ*f for each cube Qb we can sum these over k to obtain the estimate that a IQ I :S fn f :S II f 11 .. which was to be proved. •

This can be restated as a decompos ition of the function f=g+b

where the good part g(x) =f(x) for x E F and g(x) = JQJ /IQjl for x E int(Qj). Hence the bad part b must satisfy b(x) = 0 for x E F and JQ, b = 0 for each cube Qj.

3.8

A CLASS OF SINGUL AR INTEGRA LS

The Calder6n- Zygmund decompos ition can be used to treat a class of singular conv~­ lution operators that generalize the odd kernels which were discussed in Section 3.4.4.

FOURIER ANALYSIS IN U' SPACES

211

We will only state the results, referring to Stein ( 1970) for a complete treatment of the details. We begin with a real function K (x), x E JR" \ {0}, which satisfies the following growth, smoothne ss, and cancellati on properties : (3.8.1)

0

(3.8.2)

JV'K(x)J <

f

(3.8.3)

JR,
=/=X

E

JR",

c

-~

JxJ"+l

K(x)dx = 0

0 < R1 < Rz < oo.

A truncated convolutio n operator is defined by setting (3.8.4)

T J(x) =

1

K(y)f(x- y) dy.

fyl><

Iff E l/'(JR") for 1 < p < oo, then this integral is well-defin ed by Holder's inequality , in light of (3.8.1). Iff E L 1 (lR"), then T,f is defined almost everywhe re as the convolutio n with a function of class L 2 (lR"). However the naive bounds in these estimates depend on E, when E ~ 0. It is remarkabl e that, under hypothese s (3.8.1) through (3.8.3) the operators T, have uniformly bounded operator norms and are convergen t in LP for I < p < oo.

Theorem 3.8.1. Suppose that the kernel K(x) satisfies (3.8.1) through (3.8.3). Then for each p

E (

1, oo ), there exists Ap < oo such that for each f E LP (lR"),

(3.8.5) In addition, there exists the LP limit Tf = lim,_. 0 T J and the operator T also satisfies the inequality ( 3.8.5).

The conditions (3.8.1) through (3.8.3) are by no means necessary. In fact, (3.8.2) can be replaced by the weaker Hormande r condition: (3.8.6)

1

JK(x- y) - K(x)i dx ::5 C,

lxf~21yl

Exercise 3.8.2. Prove that ( 3.8.2) implies ( 3.8.6). Hint: Apply the mean-value theorem to the integrand in (3.8.6}, noting that the segment from x to x- y is outside of a sphere of radius lxl/2.

We now discuss some of the broad outlines of the proof of Theorem 3.8.1. The basic strategy is to apply the Marcinkie wicz interpolati on theorem, first proving the boundedn ess on L 2 (1R") and the weak boundedn ess on L 1 (1R"). The first of these is proved by showing that the truncated Fourier transform s

k, (~)

:= {

Jfxl~€

K(x)e-Zrri~ x dx

212

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

remain uniformly bounded: IK, (0 1 ::: M, VE > 0, ~ E iR". This is proved in Lemma 3.3 of Stein (1970), pp. 36-37. The weak L 1 boundedn ess is proved by applying the Calder6n- Zygmund decompos ition at level a: f = 8a + ba and doing each term separately . We have l{x: ITf(x)l ::: a}l < l{x: ITgcr(x)l ::: a/2}1

+

l{x: lTba(x)l > a/2}1 ·

We outline the treatment of the first term. From the properties of the Calder6n- Zygmund decompos ition, we can write ll8a IIi =

2

{

!gcr (x) 1 dx

}R,n

i

l8a(x)l dx

+

i


+

2 "a 1QI

=

2

< allfllr

= a(l

2

lga(x)l dx

2

2

+ 2 2 "allfllr

+ 2 2")11fllr·

This is combined with Chebyshe v's inequality and the L 2 -boundedn ess to write l{x: ITga(x)l:: : a/2}1::: (a; ) 2 11Tg"ll 2 2

Mz

.,

< (a/ 2 ) 2 llga

112

2

<

M

(a/2) 2

(1

+2

2

")1lfll 1

'

which proves the weak L bound for 8a· A more lengthy argument using the Marcinkie wicz integrals shows that the correspon ding estimate holds for Tb, and allows one to complete the proof of boundedn ess for 1 < p < 2. Then a duality argument is applied to prove the boundedn ess in case 2 < p < oo. For details see Stein (1970), pp. 30-33. 1

3.9

PROPERTIES OF HARMO NIC FUNCTI ONS

For completen ess we state and prove the basic properties of harmonic functions m the disk.

3.9.1

General Propertie s

A twice differentia ble function u(x, y) is called harmonic in an open set D if it satisfies Laplace's equation u.u. + Uvy = 0 for every (x, y) E D. The prototype examples of harmonic functions are the real and imaginary parts of f(x, y) = (x + iy)". It is clear thatf satisfies Laplace's equation, hence each of the real and imaginary parts do also. These are abbreviate d in the polar form u, = r" cos n() and v, = r 11 sin n&. Taking the

213

FOURIER ANALYSIS IN U SPACES

derivativ es off and fonning the real and imagina ry parts shows that iDxun I < nrn-l, IDx 2 unl < n(n- 1)rn-Z and similarly for Vn. Hence we can form more general harmoni c function sbythes eriesu(x ,y) = Ln>oan un(x,y). lflani < MandR < 1,thenth isseries converge s uniform ly in the closed disk x 2 + y 2 < R 2 together with the different iated series, thus defining a harmoni c function in the open disk x 2 + y 2 < 1. In particula r, a useful class of harmoni c function s are provided by the Poisson integrals of integrab le function s (or more generall y of finite measure s):

( 3 ·9 · 1)

u(x, y)

=

1 [ 2JT Jlf' R2

2

+

r

2

R r 2 - 2rR cos (8- ¢/(¢) d¢

Proposi tion 3.9.1. Suppose thatf

"""" (

=~

r) lnl f(n)ern A

•

R

0

·

L 1 (1f') and u(x, y) is defined by ( 3.9.1 ). Then u is a harmoni c function in the disk x 2 + y 2 < R 2 and limr....,.R u(rei0 ) = f(8) for almost every 8 E T. Iff E C('JI'), the converge nce is uniform in 'JI'. Iff E LP(1f'), 1 < p < ex::>, then the converge nce is in the norm of LP('JI'). E

Proof. We simply note that lf(k)l::; 11/11 1 so that we can infer that the sum of the series in (3.9.1) is a harmonic function. The converge nce to the boundary values was already proved in Chapter 1 as part of the Abel summabi lity of Fourier series in the spaces L 1 ('lr), C('lr), and LP('lr). •

Exercise 3.9.2. Suppose that f E ux'(1I') and that u is defined by ( 3.9.1). Prove that iu(x, y)l :::::: llfll 00 in the disk x 2 + y 2 < R 2 . Exercise 3.9.3. Suppose that m(d8) is a finite Borel measure on 'JI'. Defining 1

u(x, y) := 2JT

1 1f'

R2

+

R2- r2 m(d
show that u is a harmoni c .function in the disk x 2 + y 2 < R 2 . Show by example that u is not necessar ily a bounded function . Hint: First show that u(x, y) = Lnez(r/R ) 1n1mnem(i for a bounded sequence mn. If m is a point measure, then u is unbounde d near that point.

We now turn the picture around, assumin g only that u is a given harmoni c function in the disk. The first problem is to prove the uniquen ess of harmoni c function s with given boundar y values.

Proposi tion 3.9.4. Suppose that D is a bounded and connecte d region with a piecewis e smooth boundar y. Suppose that v, ware twice different iable in D and the first derivativ es have continuo us extensio ns to b. If v, ware harmoni c in D and satisfy v = w on the boundar y. Then v = w in D. Proof. Let u = v - w, so that u is harmonic in D with u = 0 on the boundary . We will apply Green's theorem to the vector identity

div(u gradu) = u(uxx

+ Uyy) +

(u;

+ u;).

214

INTRODUCTION TO FOURIER ANALYSIS ANLJ WAVELET'>

The first tenn on the right is zero since u is harmonic. Applying Green's theorem transfonns the double integral of the divergence into a line integral on the boundary. Thus (

lao

u gradu dS

= ( (u; + u~) dx dy.

Jn

But the left side is also zero, since u is zero on the boundary. Hence we conclude that the continuous function u7 + u: = 0 in D, meaning that u is constant on D. But the boundary value is zero, hence u 0 in D. •

=

Corollary 3.9.5. Any harmonic function in the disk x 2 + y 2 < 1 can be represented on the closed disk x 2 form

+

y 2 :S R 2 <

1 by a Poisson integral, in the

(3.9.2) u(x, y) = - 1 ~ 2n 1f R 2

2

+

..,

r~

2

R - r u(Re"") d¢, - 2rR cos (8 - ¢)

Proof. From the above discussion, the right side of (3.9.2) is a harmonic function in the disk x 2 + y 2 < R 2 and has the same boundary values as u. Hence by Proposition 3.9.4 the equality (3.9.2) follows. •

Corollary 3.9.6. Any harmonic function in the disk x 2 + y 2

1 has the mean

<

value property:

(3.9.3)

R
Proof. It suffices to take (x, y)

3.9.2

=

•

(0, 0) in (3.9.2).

Representa tion Theorems in the Disk

The Poisson integral can be characterize d in each of the spaces L 00 (1f), LP(1f), (1 < p < oo) and the space of nonnegative measures. We enumerate these results separately, beginning with the classical Fatou theorem for bounded harmonic functions.

Theorem 3.9.7.

Fatou: Suppose that u is a harmonic function that is bounded in theunitdisk x 2 +y 2 < 1: iu(x,y)l < MforsomeM . Thenthereex istsu 1 E V)()('[') so that

( 3 ·9 ·4 )

1 [ u(x, y) = 2n J1r 1 + r 2

-

l - r2 2r cos (8 - ¢) UJ (¢) d¢,

In particular we have limr-. 1 u(re; 0 ) = u 1 (8) for almost every 8

..,

.,

X~+ y~

E

< 1.

'IT'.

Proof. We will make a compactness argument based on duality of L 1 (1!') and L""('ll'). Consider the linear functionals on L 1 (1!') defined by

(3.9.5)

LRf

2

= ~ ~ f(O)u(Re'") de.

0

:s R

< 1.

Clearly iLRfi :5 M llfllr, so that this family of linear functionals is a bounded subset of the ball 1/LI/ :5 Min the dual space of L 1 (If), namely L'"('Jf). But this ball is compact in the

FOURIER ANALYSIS lN LP SPACES

215

weak* topology, which can be seen directly by taking a countable dense subset of L 1('JI') and applying Cantor's diagonal argument. Thus we obtain a weak* convergent subsequence LR1 with R-J --+ 1 with a weak* limit, call it u1 with the property that for every f E L 1('li'), limj LR,f = 2~ 1"J f(())u 1 (()) d(). We apply this to the Poisson integral representation of u: (3.9.6) u(x, y) = -

1

2rr

1 "J

R2r2 1

R}

+

r2 - 2rRj cos(() - cp)

u(R1 ei
For any fixed (x, y) the Poisson kernels PR, converge uniformly on 1I' when Rj Poisson kernel P 1 ; this follows from the series representation

::::: 2(1 - R)

L

--+

1 to the

r" ;,.

n~l

::::: 2(1 - R)

L> (~)" n~l 1+ r

provided that R > (1 + r)/2. For any fixed r < 1 the last sum is finite, which proves the uniform convergence of the Poisson kernel. Hence we have established (3.9.4). The convergence of u(re' 0 ) follows from the almost-everywher e Abel summability of the Fourier series. •

The next result concerns representation of nonnegative harmonic functions.

Theorem 3.9.8. Suppose that u is a nonnegative harmonic function in the disk x 2 + y 2 < 1. Then there exists a nonnegative Borel measure m on '['so that

1

u(x, y) = - 1 2n 'lf 1 + r 2 For any f E C('JI'), we have (3.9.7)

(3.9.8)

lim r--.1

f

}']f

2

-

1- r m(df/J), 2r cos (B- f/J)

u(rei0 )f(B) d() =

f j(B)m(dB).

}']f

Proof. Again we consider the linear functional (3.9.5), now defined on the space C('li') whose dual space is the set of finite Borel measures on 'JI'. We have ILRfl ::::0 llflloc

2~ 1u(rei

0

)

d().

But from the Poisson integral representation of u in the disk x 2 + y 2 :::; R 2 , we have u(O) = 1/21r 111: u(reiiJ) dO, hence ILJI :::; u(O)IIflloo· Applying the weak compactness argument once again yields a measure m on 1I' and a weak* convergent subsequence LR· so that LRJ --+ J..Jf'f(O)m(d()). As in the previous proof we have (3.9.6) on which w~ can take the limit R, --+ 1 to conclude (3.9.7). To prove the convergence, we multiply (3.9.7) by a continuousf(O) and integrate on the circle of radius r. The left side is Lrf = 1-r u(re;0 )f(8) dO. The right side can be written as 1"J PJ(8)m(d8). But Prf converges uniformly tof, so that the right side converges to 11ff(8)m(d8), from which it follows that limr-.1 Lrf exists and is given by (3.9.8). •

216

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

To complete the picture, we state and prove the theorem on LP boundedn ess. Theorem 3.9.9. Suppose that u is a harmonic function in the disk x 2 with the property that

where 1 < p < ex:>. Then there exists u 1

(3.9.9)

1

u(x, y) = -2 n

i1r

E

+ y2

< 1

l!'("JJ') so that

2

1

+

r

2

l-r -

2

cos

r

(8

-

¢)

UJ

(¢) d¢,

In particular

lim [ iu(rei0 r-+ I

J1r

andlimr-+ l u(re' 8 ) = Ut(8)fora .e. 8

) - UJ

E

(8)1P d8

=

0

"JJ'.

Proof. Again we consider the linear functional (3.9.5), now defined on the space U' (1r) which is the dual space of £P (1r) where p' = pI ( p - 1). We have for any f E u' (1r),

ILRfl :5 Mil/lip· so that the linear functionals LR have bounded norms. Applying the weak compactne ss argument once again yields an U function/ on 1r and a weak* convergent subsequenc e LR so that LRif--+ f..rf(O)u 1 ((})dO. As in the previous proof we have (3.9.6) to which we can1 apply take the limit Rj --+ 1 to conclude (3.9.9). The convergenc e follows from the £P and a. e. Abel summabilit y of the Fourier series proved in Chapter I. •

3.9.3

Represe ntation Theorem s in the Upper Half Plane

The results in the previous section can be transform ed to obtain representa tion theorems for harmonic functions in the upper half plane JR 2 + = { (x, y) : -ex:> < x < ex:>, y > 0}. To see this, we write z = x + iy and introduce the fractional linear transform ation (3.9.10)

z-i z+i

x+i(y-1 ) x+i(y+ l)

w= ---- = --------- --

which maps ito 0 and maps the real axis -ex:> < x < oo to the unit circle lwl = 1, deleted by the point w = 1, which correspon ds to the point z = ex:>. The upper half plane JR 2 + is mapped 1: 1 conformal ly onto the unit disk D = {w: lw) < 1}. If U(z) is a harmonic function defined for z = x + iy E JR 2 +, we obtain a harmonic function on D by setting u(w) = U(z). This can be seen directly by computing the partial derivative s by the chain rule or by observing that u is the real part of the holomorp hic function obtained by compositi on of a holomorp hic function with the fractional linear transform ation (3.9.10). If U is a bounded harmonic function on JR 2 +, then u is a bounded harmonic function in D. If U is a nonnegati ve harmonic function in JR 2 +, then u is a nonnegati ve harmonic function in D. We state and prove the correspon ding Fatou theorems.

217

FOURIER ANALYSIS IN LP SPACES

Theorem 3.9.1 0. Suppose that U is a bounded harmonic function in IR 2 +. Then there exists U 1 E L 00 (IR) so that U(x, y) =

(3.9.11)

y 7r

f

JJR

Ur (t) (t - x)2

Proof. From the Fatou theorem in the disk, we set w (3.9.12) 1 u(w) = 2rr

1 1f

+ y2

=

re' 11 and

2

1

+ r2

-

dt.

1- r 1 u 1 (¢J)d¢J = 2r cos (0 - ¢J) 2rr

1 1f

2

1. - lwl UJ(¢J)d¢J. je•

It remains to transform this to the z = x + iy coordinates. Direct computation shows that the numerator of the Poisson kernel is computed from

l

w 12 -

4 1 -lwl 2 -X y -::-2-+-(y.:....__+--:-1 )--::-2

x_2=--+_(-'y_-_1)-=-2 x2+(y+1)2 '

-

= ( t - i)j(t + i), w =

The denominator is computed by writing ei

le'

(z- i)j(z

+

i) to obtain

z-i

-z+i

-

2y + 2i(t- x) (tx- y - 1) + i(ty + x

=

4y2 4(t - x) 2 ---=--:-::--- -'-----'-----: ::

+

t)'

+

(tx - y - 1) 2 + (ty + x + t) 2

y2

4 = 1 + t2

x2

+

( t - x) 2 + ( Y + 1)2 '

resulting in the identity

1 - lwl 2

+

y(l t2) y2 (t -x)2.

+

lei
It remains to compute the Jacobian of the mapping t ~ f/J. This is computed directly by writing iei
The representatio n theorem for nonnegative harmonic functions contains a new term, which was not present in the disk. Theorem 3.9.11. Suppose that U is a nonnegative harmonic function in IR 2 +. Then there exists a nonnegative finite Borel measure M on Ill and a nonnegative

constant c so that (3.9.13)

U(x, y) = cy

Proof. Again setting u(w) (3.9.14)

1 2rr

u(w) = -

1 1f

= 1

yl

+-

n

IR

1 (t - x) 2

+ y2

M(dt).

U(z), we apply Theorem 3.9.8 to obtain

+ r2

1-

r2

2r cos (0-

f/J)

1 2rr

m(df/J) = -

1 1f

1 - lwl2 je•
m(df/J)

·

218

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

If the measure m attributes no mass to the point cp = 0, then we can transform this integral exactly as we did in the proof of the previous theorem, by setting M (A) = m(A) if the Borel

set A is the image of the Borel set A under the mapping (3.9.10). If m({O}) > 0, we must compute the contribution as the Poisson kernel corresponding to t = oo:

4y

+ (y +

x2

-X --2_+_(y_+_l_)-=-2

1)2

4

which gives the additional term in (3.9.13). where c

=

= y,

•

m({O}).

The Fatou theorem for the I f norm is also different from the case of the disk. To see this, we first note that the image of a horizontal line y = const. under the map z -+ (z - i) I (z + i) is a circle whose center is on the line Re w = 0 and which passes through the point w = 1. When y -+ 0 this circle tends to the circle lwl = 1. In detail, we write X + iy - i y 1 X - i ( y + 1) = ---+ ---- ---~--~ X + iy + i y + 1 y + 1 X + i ( y + 1) so that the center is at yj(y + 1) and the radius= lf(y + 1). We parametrize the circle by writing -~

1

y

w= --+---e' y+l y+l from which we compute d\11 = 2(y transform according to (3.9.15)

l

rr

+

lu(w)IP d\11 = 2(y

+

l)/(x 2

+

(y

+

1) 2 ) dx. Therefore the LP norms

1) { IU(x, y)IP

JIR

-rr

1

X +(y+l) 2 2

d.x.

Theorem 3.9.12. Suppose that U is a harmonic function in IR2 + such that for each Y > 0,

sup

Y>y>O

1lR

1U(x, y)!P d + x 2 t.::: M < oo. 1

Then there exists U 1 E If(lR; dt/(1

(3.9.16)

U(x,y) =

~ 7r

+ {

JlR

t 2 ))

(

so that

\ t - x)

+ y2

U 1 (t)dt. .

Proof. We have t 2 + (y + 1) 2 > 1 + t 2 , thus

1

1

---------- < -2 - t2+(y+l)2 t +1

so that 2(y + 1)

L. t2 ~~~; ~~:)2 dt :s 1

2(y + 1)

L. ~?.::IP dt :s 4M(Y 1

Transforming the integrals as in (3.9.15), sup Y>y>O

1"' I ( u

-n:

-yy

+}

+

1

· ) ----e'"'

y

+

1

lp d'JJ <

oo.

+ 1).

FOURIER ANALYSIS IN U' SPACES

219

Now we can apply the compactness argument from the previous section, defining a sequence of linear functionals on LP' ('II') by LR.f

=

1"

j('ll)u

-rr

(+

-yY 1

+

1 e •'~') d 'll. -y + 1

We see that their norms are uniformly bounded, hence we can choose a weak* convergent subsequence Lv1 where YJ --+ 0 and Ly1 --+ u 1 E LP('JI'). Writing the Poisson integral representation of u with respect to a fixed but arbitrary point in the interior of the circle Cy1 , we note that the Poisson kernels converge uniformly in 'l1 E ( -rr, rr) so that we can take the limit y1 --+ 0 and conclude that u(w) is represented by the Poisson integral of u, as in (3.9.9). Finally we transform this into the (x, y) variables, exactly as in the proof of Theorem 3.9.9. •

3.9.4

Herglotz/Bochner Theorems and Positive Definite Functions

We can use the representation theorem for positive harmonic functions to characterize the Fourier coefficients of a nonnegative measure on the circle. A bilateral sequence of complex numbers {un}nez is called positive definite if for every finite set of complex numbers {cn}~=-N• we have N

L

(3.9.17)

CmCnUm-n 2: 0.

m.n=-N Exercise 3.9.13. Prove that a positive definite sequence {un} satisfies u 0 lunl < uofor n E Z.

:=: 0 and

Hint: Apply the definition with co = 1, Cn = re' 0 and otherwise ck = 0. By suitable choice of 0, first prove that Un is hermitian symmetric: Un = u-n· Then minimize a quadratic polynomial to obtain the inequality.

Theorem 3. 9.14. Herglotz: A sequence ofcomplex numbers {Un }nez is positive definite if and only if there exists a nonnegative Borel measure M on '][' such that

(3.9.18)

Un =

i

e-inf:l M(d()),

n E

Z.

Proof. If (3.9.18) holds, then we have for any finite set of complex numbers fcn}:=-N•

:::::0. Conversely, suppose that (3.9.17) holds. Let Cn where 0 :5 r < I and set F(r, 0) :=

L

=

rnem 8 for n ::=:: 0 and en

=

0 for n < 0,

Um-nrme•miJrne-mfi.

m,n::::O

Thedoublesumismajorizedbyu 0 L::m.n.o::o~+m

= u 0 j(l-r) 2 < oo.ltisalsothe(N--+ oo) limit of the finite sum for 0 :5 m, n :5 N, hence F(r, 0) ::=:: 0 by (3.9.17). We can rewrite

220

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

F(r, (}) as a sum along 45 degree lines in the first quadrant:

0 :5 F(r, 8)

=

L

L

JEZ

=

L

Um-nrmeim!Jrne-in!J

m.n~O,m-n=J

L

Ujeij!J

rm+n.

m.n~O.m-n=J

JEZ

If m - n = 0 the inner sum is I:::=o r 2 m = 1/ (1 - r2 ). Otherwise we have m - n = j ::;6 0 and we can remove the common factor rill from the inner sum to obtain 1 F(r 8) = - --

""'u·rl! 1 19 > 0 l-r2L.., 1 ei '

'

JEZ

which proves that the harmonic function u(r, 8) := L,ez ujrl!1e'19 is nonnegative in the unit disk. Hence by Theorem 3.9.8 there exists a nonnegative Borel measure M on 'II' so that

L

(3.9.19)

u1 r li' e'1·e =

JE Z

1 T

1+ r2

-

1- r2 M(d¢). 2r cos (8 - ¢)

Now we multiply both sides of (3.9.19) by e-iNe and integrate over 'II' to conclude that e-iNe M(d8) for any N E Z, which was to be proved. •

UN =

JT

One can also consider positive definite functionsf(~) on the real line. These are defined by the statement that for every finite set of real numbers~~ •... , ~N and every finite set of complex numbers {cn}~=I we have N

L

(3.9.20)

Cmcn.f(~m - ~n) > 0.

m.n=1

Again it follows thatf(O) 2: 0 and that If(~) I -s_j(O).

Exercise 3.9.15. Prove this. Theorem 3.9.16. and only

Bochner: A continuous function f(~) is positive definite if

if there exists a nonnegative Borel measure M on lR such that

(3.9.21)

~ E

JR.

Proof. Suppose that/(~) is defined by (3.9.21). Then for any finite sets (~.,) and (en) we have

~

0.

Conversely, suppose that (3.9.20) holds. Now for any x e JR, y > 0, define F(x, y) =

1oo1oo /(~- ry)e-
This double integral is majorized by the convergent double integral 00 00 /(0) fo fo e-be-ryy d~ dry =f(O)jy2 • It is also the limit of finite Riemann sums of the

FOURIER ANALYSIS IN LP SPACES

form Lm,nf(l;m -

l;n)e-~mYe•<mxe-
221

which are nonnegative, by hypothesis. Hence

F(x, y) 2:: 0. On the other hand we can write 0

~

F(x, y)

= { f(u)eiux ~

= _!_ { 2y

JR

(1l;.ry~O.I;-ry=u e-y(~+ry) d71) du

f(u)e'=e->"

1

"

1

du,

which proves that 2yF(x, y) is a nonnegative harmonic function in the upper half plane. Hence by Theorem 3.9.11 there exists a nonnegative finite Borel measure M on R and a nonnegative number c such that (3.9.22)

cy

+2 {

}R Y2 +

y (t -

X) 2

M(dt)

= {

JR

eiuxf(u)e-ylul du.

However both integrals are bounded functions of y when y -+ oo. Hence the constant term must be c = 0. To complete the identification, we recall the representation of the Poisson kernel: y2

+

2y

=

(t - x)2

{ e•<x e-•l;r e-yll;l dl;.

JR

Integrate both sides with respect to M(dt) over JR. to obtain i_J(u)eiuxe-ylul du

=

i

M(l;)e'"xe-yll;l dl;

from which we conclude, by the uniqueness of Fourier transforms, that for all/; we havef(/;)e-YI~I = M(/;)e-YI~I. hencef(/;) = M(/;), as required.

E

JR., y > 0, •

CHAPTER

4 POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

4.1

MOTIVATION AND HEURISTICS

Up until now we have treated Fourier series and Fourier integrals independently of one another. Given the strong parallels between the one-dimensional theories, we may ask if there is a systematic link for passing back and forth. This is supplied by the notion of periodization and the closely related Poisson summation formula. For the sake of clarity, we will first pursue these ideas in one dimension where the formulas are simpler. As applications, we will obtain the Shannon sampling formula for band-limited signals and the transformation formulas for Gaussian sums from number theory using these ideas. The higher-dimensional Poisson summation formula allows us to study multiple Fourier series. As a by-product we can derive the famous Landau asymptotic formula for the number of lattice points in a large sphere in Euclidean space. In order to simplify the notations, we will consider Fourier series for functions of period 1; these are defined by their restriction to the interval (0, 1). In this setting a Fourier series is written

f(x) ,...._

L nEZ

222

1

1

f(n)e27rinx,

f(n) :=

j(x)e- 21rinx dx,

223

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

and the Dirichlet kernel of Fourier series is written DMFS (x) =

sin((2M + l)rrx) sinrrx

M

"""' e2:rik.x = - - - - - - ~

k=-M

THE POISSON SUMMATION FORMULA IN 1R 1

4.2 4.2.1

Periodization of a Function

Given a function on the line, one can construct a periodic function by summing over the integer translates, defining (4.2.1)

](x) = L f ( x

+ n)

nEZ

where the series is supposed to converge in the sense of symmetric partial sums. This is called the periodization off. Example 4.2.1. Iff E L 1 (0, 1) and is defined to be zero elsewhere, then simply the periodic extension off to the entire real line.

j is

Example 4.2.2. Iff is the heat kernel, defined by f(x) = (4rrt)- 112 e-x 2 / 41 for t > 0, x E lR, thenj(x) = (4rrt)- 112 LnEZ e-<x+n) 2 / 41 is the periodic heat kernel. Example 4.2.3. Let f be the Dirichlet kernel relative to the Fourier transform, studied in Chapter 2:

f(x) = D[{ (x) :=

sin(2Mrrx) Jl"X

=

1M

.

e 2 :r•~x d~.

-M

In this case the series defining j is not absolutely convergent, but we can compute the periodization] directly as follows: (4.2.2)

t

sin 2rrM(x rr(x

n=-N

+

n)

+ n)

=1M

e2:ri~x (sin(2~ + l)rr~) d~. smrr~

-M

We apply the pointwise inversion of Fourier series on each interval (k- 1/2, k each time obtaining a contribution from the center. If M ¢ Z, we obtain (4.2.3)

lim N-+oo

t

n=-N

f

+

sin 2rrM(x + n) = ez:r,k.x = sin((2[M] + l)rrx) rr(x + n) k=-!Ml sinrrx

1/2),

224

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Furthermore the partial sums (4.2.2) converge boundedly for each M < oo, since the right side consists of a finite number of Fourier partial sums for the function ~ ---+ e 2 :n:ixl; at~= 0, ±1, ... , ±[M]. We have shown that the periodization of the Dirichlet kernel D';J is the periodic Dirichlet kernel D[~ 1 • Exercise 4.2.4. /fM E Z, showthatwemu staddanadditio naltermofcos2M rrx to the right side of(4.2.3).

We can further exploit the bounded convergence to represent the Fourier partial sum of an arbitrary f E L 1 (0, 1):

i

N_

f(x + t)Df.J (t) dt =

1'o

-N

~ sin2rrM(t+n) dt---+ f(x + t) ~ n=-N rr(t + n)

1' 0

f(x + t)Df~ 1 (t) dz

WhenN ---+ oo, the left side converges to the Cauchy principal value, or symmetric improper integral, which can be summarized by writing M¢Z.

As a final example of periodization, we consider the kernel of the Hilbert transform, which was studied in Chapter 3. Example 4.2.5. Letf(x) = 1/xfor x

=I= 0.

The periodization is determined from the identity 1 " " - - - = rr cot rrx, ~x-n nEZ

(4.2.4)

X¢

Z.

This can be proved by residue calculus, which the reader is invited to supply. We offer a Fourier-analytic proof as follows: Proof. Let j(x) = Ln€Z I I (x - n) - rc cot rc x for x ¢ Z. Then x ~ j(x) is an odd function of period I. It is also a continuous function, since Lo;o
=

(4.2.5)

1/2

lim N

/

-1/2

sin(2rcMx)

(

I ) L -n N

-N

X-

dx =lim N

!

N+I/ 2

-N-l/2

sin 2nMx X

dx = rc.

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

On the other hand 1/2

(4.2.6) /

sin(2nMx)ncotn xdx=

!

1 j2

1 2

1

1/2

sin(2nMx) dx ncosnx . sin 7T X

But elementary trigonometric identities reveal that L:~=l cos (2k- l)nx 2 sin JT x, so that by orthogonality

f

(4.2.7)

1/2

sin(2n Mx)n cot

JT x

dx =

l/2

f

225

1/2

2rr cos 2

JT x

dx =

sin(2Mnx)/

JT.

-1/2

Therefore J~;~ 2 j(x) sin(2nMx)dx = 0 forM= I, 2, ... , hencej(x) = 0 almost everywhere. But we already noted thatf is continuous, and a continuous function that is zero a.e. must be zero everywhere, which completes the proof of the required identity. •

Additional examples of periodization are obtained from the Fejer kernel and the Poisson kernel.

Exercise 4.2.6. Consider the Fejer kernel on the line, Kr(x) = (1 -cos 2:rrTx)/

2:rr 2Tx 2 for X =I= 0. Show that ifT E z+, the periodization of Kr is the Fejer kernel of Fourier series: KT-l (x) = sin 2 (T:rrx)jTsin 2 (:rrx). Hint: Begin with the representation of Kr as a Fourier integral: Kr(x) = f~r ez,.,xl; (1

-

1~ 1/T) d~.

Exercise 4.2.7. Consider the Poisson kernel on the line, Pv(x) = yj[:rr(x 2 + y 2 )] where y > 0, x E R Show that the periodization of P,_. is the Poisson kernel of Fourier series. Hint: Begin with the representation

4.2.2

of~·

as a Fourier integral: P,(x)

= JJR e 2rrixl; e-21TiHr d~.

Statement and Proof

The Poisson summation formula allows us to compute the Fourier series of] in terms of the Fourier transform off at the integers.

!'fleorem 4.~8. Suppose that f E L 1 (R). Then f(x) is finite a.e., satisfies f(x + 1) = f(x) a.e. and is an integrable function on any period, e.g., [0, 1]. The Fourier coefficient is obtained as t](x)e-2rrimx dx =](m) =

lo

r= f(x)e-2rrimx

}_=

dx,

mE

Z.

/fin addition L~-= i](n)l < oo, then the Fourier series of] converges and we have the a.e. equality

(4.2.8)

L nEZ

f(x

+ n)

=f(x) =

L nlf:-_'?..

j(m)e2rrim~.

226

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

In particular j is a. e. equal to a continuous function on 1r = ~/Z. Redefining f necessary, then the equality holds everywhere and we have the Poisson identity (4.2.9)

Lf(n) =](0) =

L:J<m).

nEZ

mEZ

if

Proof. We have

t (n~oo lf(x + n)l oo

Jo

)

dx =

n~oo oo

fn+

I

n

1/(x)l dx =

~oo

-oo

lf(x)l dx <

CXl,

which shows that] is finite almost everywhere and integrable on [0, 1]. The same calculation applied to](x)e- 2"'mx allows us to integrate term-by-term:

t

00

= n~oo lo

f(x

+

n)e-2rcimx dx

fn+l f(y)e-2rrtm(y-n) dy

oo

= n~oo n

= J:f(y)e-2rrtmy dy

=f(m), which proves the first statement. Now if the series Lmez lf(m) I converges, then the Fourier series off converges uniformly and in L 1 , in particular the Cesaro means converge to the same limit. But the Cesaro means converge in L 1 to f. Therefore/ is almost everywhere equal to the sum of its Fourier series. In particular, for a dense setS. Finally iff is continuous, we let y-+ x through the setS to conclude thatf(x) = LneZ f(n)e 2rcinx. Finally, set x = 0 to obtain the Poisson identity (4.2.9). •

Exercise 4.2.9. Suppose thatf E L 1 (~). thatf has finite total variation on~ and is normalized so that 2f(x) = f(x + 0) + f(x- 0). Prove the Poisson identity (4.2.9).

J

Hint: Show that satisfies the hypotheses of Dirichlet's convergence theorem from Chapter 1. Check first the uniform convergence of the series (4.2.1).

Example 4.2.10. Jff(x) = (4rrt)- 112 e-x 2 / 41 , then all ofthe conditions are satisfied and we have ](n) = e_ 4 ,... 21n 2 and thus the identity ~

~

nEZ

e

-(x-n) 2 /41

./4iit

=

~

~e

2rrinx -4JT 2 tn 2

e

.

nEZ

This example shows that the periodic heat kernel has a natural Fourier series representation in terms of separated solutions of the heat equation.

POISSON SUMMATIO N FORMULA AND MULTIPLE FOURIER SERIES

227

Remark . It is clear that some continui ty restrictio n is necessar y to obtain the Poisso~ summati on formula. Indeed, if j(x) = 0 for x = 0, ±1, ±2, ... andj(x) = e-rrx otherwis e, then the left side of Poisson' s identity (4.2.9) is zero, but the right side is obtained as LnEZ e-rrn 2 , from the above example . Closely related to the Poisson summati on formula is a bilinear identity similar to the Fourier reciproc ity studied previous ly. Proposi tion 4.2.11. Suppose that either

(a) f (b) f

E E

L 00 ('li') and K E L 1 (IR 1) or L 1 ('"II') and LnEZ K(x + n) converge s bounded ly on 'Ir.

Then

1

l

1

(4.2.10)

j(x)K(x ) d.x =

j(x)K(x ) d.x.

Proof. In case (a) we appeal to Theorem (4.2.8) to obtain the a.e. converge nt series K(x)

=

LK(x

f(x)K(x) = L

+ n)

f(x)K(x

X

+

E (0, 1),

n)

X E

(0, 1).

nEZ

The right side is dominate d by the integrabl e function can integrate term-by-t erm to obtain

t

lo

f(x)K(x) dx =

L t nEZ

=

L nEZ

=

lo

l

11/lloo

f(x)K(x

+

LnEZ

IK(x

+

n)l so that we

n) dx

n+l

f(x- n)K(x) dx

n

i_J(x)K (x)dx.

The proof in case (b) is another applicatio n of the dominate d converge nce theorem.

•

The previous proposit ion can be paraphra sed in terms of adjoint operator s. Let P be the operator that transfor ms/ E L 1 (IR) into its periodiz ation] = Pj E L 1 ('Ir). In each case the dual space is L 00 and ( 4.2.1 0) becomes the identity

l

f (PK) d.x =

i

(P*j) K d.x,

which shows that the adjoint of the periodiz ation operator P is the operator that forms the periodic extensio n off E L 00 ('Ir) to L 00 (IR). These ideas can be applied to "lift" computa tions from the circle to the real line where the formulas may be simpler. For example , iff E L 1 ('Ir) is extended periodically to IR and KM is the Fejer kernel with KM(t) = (sin 2 Mnt)/(n Mt 2 ), then from Exercise 4.2.6, its periodiz ation is the Fejer kernel for Fourier series, defined by kM-1 (t) = (sin 2 Mnt)/(M sin 2 nt) so that we can transform the Fejer mean of the

228

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Fourier series as follows:

1-

2 sin 2 (M:rrt) sin (M:rrt) . 2 dt= f(x+t) M dt. o M sm :rr t IR rr 2 t 2 In the same fashion, the Poisson integral is transformed according to the identity 1

1

1

1

0

f

4.2.3

(x

+

t)

1

+

f(x+t)

1- r2 dt = 2 r - 2rcos2:rrt

1-

f(x

IR

+

t)

y

dt rr(t 2 + y 2)

Shannon Sampling

As a first application of the Poisson summation formula, we consider the problem of reconstructing a band-limited signal from its values on the integers. By definition, these are functions of the form (4.2.11) where F E L 1 (-A, A) and where we set F(~) = 0 for I~ I > A. The number A is the bandwidth. The Shannon sampling formula is the following identity: (4.2.12)

~ sin rr(t- n) f(t) = ~f(n)-rr-(-t_n_)_

t E IR

nEtu

where the series is taken as the limit of the symmetric partial sums and where the fraction is set equal to 1 when n = t. The following example shows that one cannot expect (4.2.12) to hold for an arbitrary bandwidth. Example 4.2.12. f(t) = (sinrrt)2 = Tft

j' (1- l~l)e27Tir~ d~ -1

is a band-limited signal with A = 1. Clearly f is zero on the integers, so that we cannot retrieve! from {f(n)}, n E Z.

The following general theorem gives sufficient conditions for the validity of (4.2.12). Theorem 4.2.13. Whittaker, Shannon, Boas. Suppose that FE L 1 (-A, A) defines a band-limited signal with A < Then the Shannon sampling formula (4.2.12) holds. More generally, if A > ~we have the estimate

i·

(4.2.13)

~

f(t)-L-f(n) nEZ

sin rr (t - n) ( ) Tf t - n

<2

1

1~1>!

IF(~)Id~.

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

229

Proof. Let F(~) = EnEZ F(~ + n) be the periodization ofF E L 1 (IR). The Fourier transform ofF is x -+ f( -x) so that by the Poisson summation formula, we have the Fourier series F(~) ~

(4.2.14)

L

L

j( -n)e2rrml; =

neZ

f(n)e-21rml;.

nEZ

From the results of Chapter 1, any L 1 Fourier series may be integrated term-by-term after by a function g of bounded variation. Applying this with g(~) = e 2rrul;, we have

multiplicat~on

1/2

F(~)e2rrul;

=

1 -1/2

L

.f(n)

11/2

nEZ

=

e2rrrl;(t-n) d~

-1/2

"'"'

sin rr(t - n)

nEZ

Jr(t-n)

L., .f(n)

.

If A.::; ~.then F is the periodic extension ofF to JR, so that F(~) = F(~) for I~ I < ~.and the left side reduces to ~~~~ 2 F(~)e 2 ""~; =.f(t), which proves (4.2.12). Otherwise, we can rewrite the left side as

1

1/2

F(~)e2rru/; d~

=

-1/2

L

11/2

nEZ

= =

L

F(~

nEZ

l

L

e-2rrmt

F(u)e21ru(u-n) du

n-1/2

.f(t) -11/2

F(~)e2rru/; dl;l

-1/2

L n;o"O

=

but

F(u)e2rriru du;

thus

n-1/2

,,..0

::: 2

F(u)e2rruu du;

n+l/2

L

nEZ

l

l

n+l/2

n-1/2

l = j.L(l=

n)e2rrul; d~

n+l/2

nEZ

.f(t)

+

-1/2

i

e-2rrmr) in+l/2 F(u)e2rritu du) n-1/2

n+l/2

IF(u)l du

n-1/2

2 {

J/u/~l/2

IF(u)l du,

•

which completes the proof.

Band-limited signals have the further property that the total signal strength

JJR f(t) dt can be computed by sampling at the integers.

4

Proposition 4.2.14. Suppose that A < and that F e L 1 (-A. A) satisfies a Dini condition at~ = 0 with valueS. Then the series LneZ f(n) converges and we have the identity

(4.2.15)

lim

T-+oo

JT -

T

f(t) dt = S

= ""'f(n). L._,;

neZ

230

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

Proof. From ( 4.2. 11 ), the left side of (4.2.15) is computed as

I

T

-T

f(t) dt

f).

= -). F(l;)

sin 2rri;T 7r

I;

dl;,

which converges to S by one-dimen sional Fourier inversion. The convergenc e of the series on the right side of ( 4.2.15) can be seen directly by computing from (4.2.11 ): ~

~f(n) -N

=

f). F(l;) sin(2N. + t:l)rrl; dt:s· -).

Sin Jrs

Since J... _:::: ~, this is an integral on a single copy of the basic period interval (- ~, ~) and converges to S by applying one-dimen sional Fourier inversion. •

Exercise 4.2.15. Suppose that f(t) = J~). e 2 rrit~ J-L(dl;), where 11 is a finite Borel measure and A. < Prove the identity

1·

1 lim - T--+oo 2T

(4.2.16)

!T

f(t) dt =lim

-T

N

1

2N

+

1

t

f(n).

n=-N

Hint: Suitably apply the dominated convergenc e theorem and identify both sides with J.L({O}).

!

Remark. The formula (4.2.12) for a band-limi ted signal with A. < is not canonical. If instead we integrate (4.2.14) on the interval [-A., A.], we obtain the alternative representa tion (4.2.17)

f

(t) =

"'"'

L.; f

sin 7r J...(t - n) (n) _7r_(_t--'- -n)-

t E JR.

nEu...

Equivalen t formulas can be obtained for any v

E

[A.,

1J.

Exercise 4.2.16. Suppose that f(t) is any band-limi ted signal. Show that we can reconstru ct! from its values at the points nj2A., n E Z by means of the formula

f

"'"'f ( - n

(t) = 2 A. L..., "" nEu...

2A.

) sin ;r(2At- n) . ;r(2A.t- n)

Hint: Apply the Shannon sampling formula to F"(l;) := F(2J...I;).

4.3

MULTIPL E FOURIE R SERIES

Multiple Fourier series are naturally associated with functions on the torus ']f'd = (0, 1)d = JRd ;zd. There is a natural 1:1 correspon dence between functions on ']f'd and functions on JRd, which are periodic in each coordinate : f (x 1 , ••• , x; + 1, ... , xd) = f(x,, ... ,x;, ... ,xd) for I< i < d, (x 1 , ... ,xd) E !Rd. We begin with an integrable function on the d-dimensi onal torus ']f'd = (0, l)d. The L 1 norm is denoted 11!11 1 = f1fd lf(x)l dx and the Fourier coefficien ts of

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

f(n) := {

lu"

j(x)e~2n:m x

231

dx,

and the Fourier series is written f(x) ~

(4.3.1)

L

f(n)e2n:m x.

nEZd

Basic L1 Theory

4.3.1

The elementary properties of multiple Fourier series may be obtained from the periodic heat kernel. We first develop the lemma of Fourier reciprocity in the following form.

Proposition 4.3.1. Suppose that K(x) := LnEZd K(n)e 2:n:mx is an absolutely convergent trigonometric series: LnEZd

(4.3.2)

1

IK(n)l

f(y)K(x- y) dy =

L

<

CXJ.

Iff E L 1 ('lrd), then

K(n)f(n)e2:n:in·x.

nE!Zd

"j["d

Proof. Multiply the defining equation for K(x- y) by f(y) and integrate term-by-term. •

We apply this to the periodic heat kernel

(4.3.3)

The Fourier representation of K 1 (x) is obtained by repeated application of the one-dimensional Poisson summation formula to obtain (4.3.4)

Kr(X) =

L

e2:n:m·xe~4:n:2rlnl2'

nEIZd

since both (4.3.3) and (4.3.4) are absolutely convergent sums, and can be evaluated by multiplication of the corresponding one-dimensional sums. Applying Fourier reciprocity, we obtain (4.3.5)

This can also be written in terms of], the periodic extension off to JRd as (4.3.6)

t

> 0, X E JRd.

232

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 4.3.2. Prove that the series LnEZd

J1fd

2141

if(y)ie-lx-y-nl dy converges 1 whenever f E L (1rd) and use this to show that the left side of (4.3.5) is equal to the left side of(4.3.6).

From this we conclude the following.

Proposition 4.3.3. (i) Jff(n)

(ii)

= 0, thenf =

0 a.e.

If

LnEZ" lf(n)l <(X), then the Fourier series (4.3.1) converges to a continuous function and we have almost everywhere f(x) = Lf(n)e2rrin·x. nEZd

Proof. Both statements are direct consequences of the pointwise almost everywhere summability associated with the Gauss kernel, proved in Chapter 2. •

The periodic heat kernel is a nonnegative approximate identity, meaning that

(4.3.7)

K 1 (x) 2: 0,

{

J.!fd

K 1 (x)dx = 1,

lim t-+0

1

K 1 (x) dx = 0,

Vo > 0.

lxl>li

Exercise 4.3.4. Prove the three properties (4.3. 7) of K 1 • A multiple Fourier series is said to be Gauss-summable if (4.3.5) converges tof when t -+ 0. Since the periodic heat kernel is an approximate identity, we immediately obtain the following properties of Gauss summability.

Proposition 4.3.5. Suppose that f E L 1 (1rd). Then the Fourier series is Gausssummable in L 1 (1rd) to f. If, in addition, f is continuous at x E 'JI"G', then the Fourier series is Gauss-summable to f(x).

The Gauss-summability of Fourier series has the further consequence that the set of trigonometric polynomials is dense in L 1(1fd). A direct proof can be obtained by applying properties of one-dimensional Fourier series, as follows. Lemma 4.3.6. The trigonometric polynomials Llnl:;;N ane 2rrin·x are dense in L' (1rd). Proof. From Chapter I, we know that the Fourier series of an indicator function converges boundedly and in the L 1 norm. Now if we have a product of indicator functions l(a,.b,) each approximated by a Fourier partial sumS~, then we can write

n s~- n d

t=l

d

t=l

d

t(a,.b,)

=

L t=l

[s~-

t(a,.b,)J

c,

POISSON SUMMATJ 0:-.1 FORML1LA AND MULTIPL E FOURIER SERIES

233

where C, contains i - I factors of S~ and d - i factors of I
But finite sums of indicato rs of rectangl es are dense in L 1 (1I'd), which complet es the proof.

•

4.3.1.1 Pointw ise conver gence for smooth functio ns The conditi on of absolut e converg ence: LmEZd lf(m)l < CXJ can be verified in casef is sufficie ntly smooth . To see this, we comput e the Fourier coeffici ents of any mixed partial derivati ve corresp onding to a multiin dex a = (a,, ... , ad). obtaini ng

Applyi ng this twice to each of the coordin ate derivat ives and summin g, we obtain

where ~ =

L:j= I a2 I ax} is the Laplace operato r. Applyi ng this k times we obtain

Iff E C 2 k (1f'd), the right side is the Fourier coeffic ient of a continu ous functio n, hence the estimat e lf(m)l <

c

In particul ar, if 2k > d, then the series LmEZd lf(m) I converg es and we have absolut e and uniform converg ence of the Fourier series. This is summa rized as follows : Propos ition 4.3.7. Suppos e that f E C 2k(1f'd) with 2k > d. Then the Fourier series converg es absolut ely and uniform ly to f. In particul ar, the Fourier series of an infinite ly differen tiable functio n on ']['d is uniform ly converg ent. Iff has fewer than d /2 derivati ves, exampl es show that one may have Fourier series that are diverge nt at a point. This will be discuss ed in Section 4.5.3 in the context of radial functio ns. 4.3.1.2 Repres entatio n of spheri cal partial sums The partial sum of a one-dim ensiona l Fourier series can be written in terms of a correspond ing partial Fourier integral . Restric ting attentio n to () = 0 and f E L 1 (']['),

234

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

we can write SMJ(O) :=

j n sin(2M. + 1)nl1 j(e) dl1 L j(k) = ~~~M ~1/2 smnl1 M

1

A

=

1

1 2 /

sin(2M

+

n11

-I 12

1)nl1 (nl1j(l1)) dB. sinnl1

The last integral is a Fourier partial integral of the associated function 11 ~ [nBf(B)j sin nl1] 1 (~ 112 , 112 ) (11). This allows convergence questions for Fourier series to be reduced to corresponding convergence questions for Fourier integrals. In higher dimensions the spherical partial sum of a multiple Fourier series bears no simple relation to the spherical partial sum of the corresponding Fourier integral, as it does in the case of one dimension. To obtain a suitable substitute for the latter, we consider the quasispherical partial sum of the Fourier integral, defined forf E L 1(JRd) as (4.3.8) where BM is the set of cubes s~ of side I, centered at the integer points k with lk I < M. In one dimension BM is the interval [ - M - ~, M + ~] if M = 1, 2, .... The corresponding quasispherical Dirichlet kernel is -

Clearly we have the representation formula that for any f E L 1 (JRd) (4.3.9)

SMf(x) =

{

JfRd

DM(x- y)j(y) dy.

This is to be compared with the spherical Dirichlet kernel of Fourier series, defined as DM(x) :=

L

e 2 n,L...

II. I :oM

To compute DM in general, we first compute the integral k

Summing these for lk I

and the formula

::s

M, we have

E

'J:/.

POISSON '>UMMATION FORMULA AND MULTIPLE FOURIER SERIES

235

The spherical Dirichlet kernel of Fourier series differs from the quasi spherical Dirichlet kernel by a factor which is smooth and bounded above and below over the basic cube

[-i, it-

This can be immediately applied to write the spherical partial sum of the Fourier series. SMf(x) =

L

f(k)e2ruk

<

lkl:sM

=

L

{ f(y) ( JT"

e2ruA·p-,)) dy

iki:<:M

At x = 0 this is the quasispheric al partial sum of the Fourier integral for the associated function defined by fo(z) = nj= 1 (.rrz1 / sin(.rrz1 ))f(z) ly". From (4.3.8) and (4.3.9) it follows that

L 1(1fd) satisfies the condition that for the associated function fi.h limM~= .ftM-v'd::<:I~I:<:M+v'JJ lfo(.;)l dl; = 0. Then the spherical partial sum of the Fourier series is equiconverg ent with the spherical partial sum of the correspondi ng Fourier integral.

Proposition 4.3.8. Suppose that f

E

Proof. From the above computations , we have

This can be applied to certain two-dimensi onal Fourier series where the Fourier coefficients satisfy LM::<:Ini::<:M+I lf(n)l ~ 0 when M ~ (X) but not the stronger condition that L 11 Ez2 lf(n)l < oo.

Exercise 4.3.9. Letd = 2andf(x) = lto.aJ(Ixl). Showthat.L: nEZ2if(n)l but LM::<:Ini:<:M+I lf(n)l ~ 0 when M ~ (X).

4.3.2

=+(X)

Basic £ 2 Theory

As in the case of the circle, multiple Fourier series have a very satisfactory theory in the space L 2 (1fd). For any finite set of complex numbers c 1 , . . . , CN, and multiindices

236

m 1,

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

••• ,

mN. we have

J _

2

N

L

Cje2rrim1 ·x

J=l

N

= 2

L

N

lc;- Ju>i 2

+ 11tll

2

-

J=l

L

if(j)i

2

•

J=l

Thus we see, as in the one-dimensional case, that the Fourier coefficients minimize the mean-square distance between f and the finite dimensional set spanned by e 2 rrimrx. In particular we have Bessel's inequality LJeZd lf(j)l 2 ::=: 11!11 2 . Proposition 4.3.1 0. Iff E L 2 (1I'd), then the Fourier series converges to f in L 2 (1I'd) and we have the Parseval equality

L

if(n)i 2

= 11!11~-

"ezd Proof. From Bessel's inequality, the series LnEzd lf(n)l 2 converges. Let F LnEzdf(n)e 2 :n:in·x, an L 2 convergent series. The function/- F has all Fourier coefficients equal to zero, hence/- F = 0 a.e. Since LnEzdf(n)e 2:n:mx converges to/ in L 2 ('Jrd), the L 2 norms also converge, which yields Parseval's equality. •

4.3.3

Restriction Theorems for Fourier Coefficients

Zygmund (1974) discovered a universal bound for the L 2 norm of the Fourier coefficients of two-dimensional Fourier series in terms of the l.F norm of the original function, for some p < 2. This is closely related to the restriction theorems for Fourier transforms, which were treated in Chapter 2. To formulate the result, we begin withf E L 1 (1I'd) and its Fourier coefficients

f(~) = { f(x) }yz

(4.3.10)

e-2rril;·x

dx.

For any given r > 0, the set{~ E Z 2 : I~ I = r} is a (possibly empty) finite collection of lattice points on the circle of radius r. Then we have the following theorem. Theorem 4.3.11. For any f E L 413 (1I' 2 ) and any r > 0, we have the bound

(4.3.11)

Proof. If the left side of (4.3.11) is zero, then there is nothing to prove. Otherwise, let c(n =f(t;)!)L: ~ =r 1](1;)12. Then L: ~l=r lc(t;)f = 1 and we have

11

1

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

237

I](~) 12 = J<~)f
L

=}(Oc(~)

1}(~)12

lsl=r

I~ 11<~)12 = (~/(~)c(~))

L

1](~)12 = L:J<~)c(~)

lsl=r

lsl=r

We now apply HOlder's inequality with p (4.3.12)

Lli<~W l
=

~, p'

=

4, to obtain

L 1}(~)12 ~ 11/114/311 L c(~)e2rris·ll I< l=r

I< l=r

4

Therefore we need to show that if
=

L

y(p)e2rrwx

pEZ2

L

=

where y(p) :

c(J-L)c(v),

JJ.-V=p

and where the final sum is over those pairs (J-L, v) with Il-L I = I vi = rand 1-L- v = p. From the complex orthonorma lity of {e 2"ip·x }. we have

The nonzero terms in this sum are of three types: (i} p = 0, (ii) IpI (iii) 0 < IPI < 2r. The contributio n of the terms of type (i) is given by y(O)

L

=

lc(J-L)I

2

=

= 2r

and

1.

l~tl=r

For the terms of type (ii}, the pairs (J-L, v), which enter into the defining sum, are antipodal points (J-L = - v) of the circle I~ I = r, one for each admissible value of p. Therefore the sum of these pairs contributes

L IPI=2r

ly(p)l

2

=

L

2

2

lc(J-L)I 1c(-J-L)I •

l~tl=r

To study the terms of type (iii), note that for a given value of p E z;z with 0 < IPI < 2r, we can have at most two pairs (J-L, v) and (J-L', v'), correspond ing to two nondiametr ical

238

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

chords of the circle with diametrically opposite endpoints; in detail, either J1 11 + v' = 0. Thus, ifO < IPI < 2r, we have

+

y(p) = c(J-t)c(v) ly(p)l

2

+ 11' =

0 or

c(J-t')c(v')

2

::S 21c(J-t)l 1c
+ 21c(J-t')l 2 1c(v')l 2 .

The sum of these pairs contributes at most

L

ly(p)l

2

L

::S 2

0<tpl<2r

2

lc(J-t)l 1c(v)l

2

+

L

= 4

L

2

J.l..,l.> <>
2

lc(J-t')l lc(v')l

2

JJ'.LI':O
2

2

lc(J-t)l 1c(v)l .

JJ.,V 0
On the other hand, we have from the normalization

Therefore, the sum of terms of type (ii), (iii) can be bounded by 4

L

2

lc(J-t)l 1c(v)l

2

::S 4,

JL.v:O
leading to the final estimate

•

which was to be proved.

4.4

POISSON SUMMATION FORMULA IN JRd

The Poisson summation formula in JRd is entirely similar to the one-dimensional case. The periodization off E L 1 (JRd) is defined by j(x) = L f ( x

+ n).

nE"Zl

This is a periodic function, whose Fourier coefficients are computed as

=

L nEZd

1

=](k),

1fd

f(x

+ n)e-2rrik·x dx

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

239

leading to the formal identity

L

](x) :=

(4.4.1)

f(x

+ n)

L

,.....

nEZd

j(k)e2rrik-x.

kEZd

Without any additional conditions onf, we can only interpret this as a formal computation. In order to obtain a pointwise identity, we assume thatf and its Fourier transform satisfy the decay estimates (4.4.2) for positive constants C 1. Cz, E. Then both sides of ( 4.4.1) converge absolutely and uniformly on 'lrd. This is seen from writing the sum as a Steiltjes integral with respect to the lattice point counting function N(R) = Liki~R 1, which satisfies N(R) < cdRd. Hence

L

l](k)l <

1](0)1 + cl

iki~R

lR

lxl-d-EdN(x)

I

= 1](0)1 +R-d-EN(R) + (d+E)

lR

lxl-d-E-IN(x)dx.

The term at the limits tends to zero and the final integral is absolutely convergent, proving that the right side of (4.4.1) is absolutely convergent.

Exercise 4.4.1. Place a cube of unit side with center at the lattice point k E zd to prove that the lattice point counting function satisfies the two-sided estimate cd (R-

tJdr :5 N(R)

< cd (R

+ tJdr

where cd is the volume of the unit ball in~We can summarize the above discussion as follows:

Theorem 4.4.2. Suppose that f E L 1(Rd). Then the Fourier series of the periodization] is given by (4.4.3)

f(x) ,.....

L

](n)e2rrin·x.

nEZd

/fin addition we have the decay estimates (4.4.2), then both sides of(4.4.3) converge absolutely and uniformly on 'Jrd and we have equality in ( 4.4.3) almost everywhere.

4.4.1

*Simultaneous Nonlocalization

When we discussed the uncertainty principle in Chapter 2, we remarked that one can prove that a function and its Fourier transform cannot both be nonzero on a bounded

240

INTRODUCTION TO FOIJRIER ANALYSIS AND WAVELETS

set. We now use the Poisson summation formula to prove a stronger proposition on simultaneous nonlocalization due to Benedicks ( 1985). The statement and proof are carried out in the d-dimensional setting, as follows: Proposition 4.4.3. Suppose that f E L 1(JRd) is supported by a set of finite measure:

(4.4.4)

A := {x : f(x)

=f.

0},

IAI < oo

0},

IBI <

and the same for the Fourier transform: (4.4.5)

B :=

{~ : f(S)

=f.

00.

Thenf = 0 almost everywhere. Proof. By a scaling transformation x ->- ax, we may assume that IAI < 1. Consider the periodization of the indicator function of B:

L

1 8 (~

k) 2: 0.

-

A.EZd

Since 18 E L 1 (IR.d), it follows that except for a set U of measure zero, this sum is finite for all~ E IR.d. But the terms are natural numbers, therefore implies

card{k

But the Fourier transform](~) is nonzero iff~ (4.4.6)

implies

zd : ln(~

E

card{k

E B,

< 00.

therefore

Zd : 1<~

E

# 0}

- k)

-

k)

# 0}

< 00.

Now letfdx) = e 2"'<'J(x) and let]~; be its periodization. The Fourier transform off~; is 1<·- ~) so that the Poisson summation formula gives (4.4.7)

]~; (x)

=

L

e2rTI(x-v) l;j(x- v) ~

vEZd

E yd

=

11; (k)e2rrila

kEZd

Now since l/1; (x) I = lf(x) I for x {x

L

:_h(x)

E

L

1
kEZd

IR.d, lA (x) I ::5 Lvad lf(x- v) I so that

# 0}

u

C

{x

E yd:

lf(x

+ v)l #

0}

vEZd

l{x E 'JI'd :]~;(x)

# 0}1

::5

L

l{x E

r:

1/(x

+ v)l #

0}1

vEfld

=

L

I
+

r : lf(y)l

=P O}l

vE£d

= IAI


so that (4.4.8)

l{x E 'JI'd :.,h(x)

# 0}1

< 1.

Property (4.4.6) implies that for ~ rt U, ]~; is a trigonometric polynomial. But property (4.4.82 implies that this polynomial is zero on a set of positive measure. Hence we must have A = 0 almost everywhere for~ rt U; furthermore (4.4.7) shows that the Fourier transform](k- ~) = 0 for~ rt U, k E ~Z. But the set of translates {k- U; k E Zd} has measure zero. Hence we conclude that f = 0 almost everywhere, hence f = 0 almost everywhere, which completes the proof. •

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

4.5

241

APPLICATION TO LATTICE POINTS

The number of integer lattice points in a ball centered at x N(x; R) =

L

E

JRd is defined by

1[o.RJ(Ix- nl).

nEZd

This is the periodization of the function x ---+ 1 [O.RJ (I xi); hence the Fourier series is found from the Poisson summation formula as (4.5.1)

N(x; R)""'

L

FR(k)e2:ruk·x

kEZd

where FR(k) is the Fourier transform of x---+ 1 1o.RJ(Ixl): FR(k) =

1

e-2:rrik·x dx

lxi::R

_ ~ Jd/2 (2n lkiR)

-

(lkiR)d/ 2

•

For each R > 0, x ---+ N(x; R) is a bounded function, in particular in L 2 (1I'd), so that the Fourier series (4.5.1) is convergent in L 2 (1I'd)_ But the series is divergent at x = 0 if d > 2, as we shall show below. We first do the L 2 theory of lattice points, due to Kendall (1948).

4.5.1

Kendall's Mean Square Error

The Fourier transform of 1 [O.RJ at k = 0 is the volume of the ball, whereas the Fourier transform has slower growth for k =I= 0~ in detail

from the asymptotic behavior of Bessel functions. This sequence is square-summabl e and we can apply Parseval's theorem to N(x; R)- (nR 2 )df 2 /(d/2)! to obtain

d-l

00

=Rd

1 R

s

(1

= O(Rd-l)

We can summarize the result as follows.

+ s)d+l R---+

ds CXJ.

242

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVEJ ETS

Propositi on 4.5.1. The L 2 nann of the error tennis bounded in thefonn N(·· R)- (rrR2)dl2' '

(4.5.2)

,

'I

(d/2)!

< C R
d

This L 2 estimate can be transform ed into an almost-ev erywhere result by using the Chebyshe v inequality : for any 8 > 0

1

N(x; R)- (rrR2)df21

2

d.x > 8 2 x -

(d/2)!

1fd

I{

x E

']['d :

I

N(x; R)- (rrR2)df21 > 8

(d/2)!

-

}I

.

'

If we let R ---+ oo along integer values we can take 8 = R
Therefore the set of points m infinitely many of these sets has measure zero. We summariz e this as follows:

Propositi on 4.5.2. For each E > 0 and for almost every x E ']['d, the number of integer lattice points in a ball centered at x satisfies the estimate

z+ Exercise 4.5.3. Show that if R

3 R -.. oo.

oo along the sequence of squares R = j we have the improved estimate: VE > 0 and almost every x E ']['d, ---+

N(x· R)- (rrR2)df21 < R
,

(d/2)!

-

,

Generaliz e to any power law R = f' with

ot

R = 1·2 ---+

2

,

then

00.

> 0.

Kendall also obtained a formula for the limiting average variance, defined as

(4.5.3)

a

2

=

lim -1 T->oo T

~T dR 1

1 Td

I

2

R 1 -d N(x; R)- (rrR2)d/21 d.x.

(d/2)!

In cased = 2, this can be computed from the asymptoti c expansion of the Bessel function

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

Thus -1 T

~T

(i

iN(x; R)-

R

Td

I

- L Oi'nEJ:2 7r

1 2

lnl

2 2

rrR 1 dx ) dR 3

3

243

_!_ {T [cos 2 (2rrlniR- : ) dR T },

+ o(~)].

The average value of the trigonometric term is :1-, leading to the evaluation

which can be expressed in terms of the Riemann zeta function and Dirichlet's L function (see Kendall, 1948). Exercise 4.5.4. Obtain a formula for the limiting average variance a general case d ::=:: 3.

4.5.2

2

in the

Landau's Asymptotic Formula

A more specific result is obtained if we fix attention on a single point, which we take to be the origin. This leads to the famous Landau estimate, as follows. Proposition 4.5.5. The lattice-point counting function satisfies the asymptotic estimate

(4.5.4)

R-+ oo.

Proof. Ford = I the estimate is exact, so we assume that d > I. The lattice point counting function can be represented as N(O; R) = LfR(v) l!tZ"

wherefR(x) = I 1o.R 1 (1xl) with Fourier transform FR(k) = _RJJd;2(27l'Riki)/( Rikl)'112 • However we cannot apply the pointwise form of the Poisson summation formula directly. Instead we will apply the pointwise form of the Poisson summation formula to the regularized functionfR * p, where p, (x) =E-d p(xjE) and pis a nonnegative ceo function supported in the ball lx I ~ l and of total integral I. Both p, and its Fourier transform are rapidly decreasing, so that we can apply the pointwise form of Poisson's formula to obtain (4.5.5)

N,(R) := LUR

* p,)(v)

= L

vezd

FR(k)p,(k) = L

kezd

FR(k)p(Ek).

kezd

On the other hand, the smoothing density can be chosen so that for any desired integer + 1)/2,

N > (d

lf>(k)l ~

1

cd.N

( 1

+ lkl

)N

244

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

for a constant Cd N. The term of the series (4.5.5) with k = 0 is simply the volume ndf2Rd /(d/2)!. Subtracting this, we estimate the remainder as IN,(R)- (rrR2)d/2/(d/2)!1 =

ILFR(k)p(Ek)l k#O

=

(4.5.6)

On the other hand, since

p,

_

(R)
cd.N

-

E

is supported in the ball of radius

E,

we have

(4.5.7)

This follows from the fact that the middle term is either zero or one; in the first case IvI > R and the ball {lz - vi :::0 E} does not intersect the ball {lzl :S R- E}. In the second case Ivi :::0 Rand the ball {lz- vi :::0 E} is contained in the ball {lzl :::0 R + E}. Summing (4.5.7) over v E zd, we obtain N,(R- E) :S N(O; R) :S N,(R +E).

Applying the estimate (4.5.6) gives the upper and lower bounds

+ El +

(4.5.8)

N(O; R) :::0 cd(R

(4.5.9)

N(O; R):::: cd(R- E)

d

-

Cd.N -

cd.N

The two error terms are balanced when we choose substitutions produces the stated result.

4.5.3

./?. + )(d-1)/2 ( ~

(R ) -

E

,

(d-1)/2

-E-

E

=

Rf.

Making the necessary •

Application to Multiple Fourier Series

We can use the Landau lattice-point formula to estimate the partial sums of multiple Fourier series of radial functions at the center of the torus. We begin with a function F on the real line, which is supported in the interval [0, a], obtaining a radial function on JRd through the formulaf(x) = F(ixi). The Fourier transform is again a radial function:](~) = A (I~ 1), where A is the Hankel transform

A(!~ I) =f(~)

=

1

lxlsa

F(lxl)e-2rri~·x dx

= Cd

{a J(d-2)/2(2rrl~lr) F(r)rd-1

Jo

(nr)1 2

dr.

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

245

We now consider the Fourier series of the periodized function

L

j(x) =

(4.5.10)

L

F(lx- ni) ,...._

nEZd

A(lmi)e2rrim· x.

mEZd

The partial sum of the Fourier series at x = 0 is written as a Steiltjes integral:

=

SM](O)

L

A(n)

= A(O) +

J.M A(fl.) dN(fl.).

lni:::M

4.5.3. 1

I

Three-dime nsional case

We can obtain a simple necessary and sufficient condition for the convergence of the spherical partial sums of a periodized radial function in three dimensions, as follows. The case a < twas treated in Pinsky, Stanton, and Trapa (1993).

Proposition 4.5.6. Suppose that d = 3 and that F is a C 2 function on the interval [0, d]for some a > 0. Then the spherical partial sums ofthe Fourier series ( 4.5.1 0) converge at x = 0 if and only if F(a) = 0. This will be proved by developing an asymptotic expansion for the Fourier transform.

Lemma 4.5.7. We have the asymptotic estimates when fl. A({L) = ,

A (fl.) =

a cos 2rr a fl. Trfl. 2

Proof. We haveA(JL)

{La d + sin 2rr -d (rF(r))(a2 3

F(a)

2 2a sin 2rra{L fl- 2 F(a)

2 JT fl.

+0

=](I;> where

= {a{"

~

(

1 )

fl- 3

r

oo 0)

+

0

(

1 ) 4fl.

,

.

JL = 1~1 and

{2rr F(r)e-zrr,wco slir2sin()drd() d<j)

lo Jo Jo

= =

21 o

-~ JL

a

r

2p

r

Jo

(r)

sin 2rr rjL rJL

rF(r)

dr

~ [cos(2rrJLr)J dr dr

2rr JL

_ -~ (aF(a)cos(2r raJL) (2 ) JL 7TJL

-

1a o

cos(2rrJLr) d ) d-(rF(r))dr . 2 7TJL

r

The final integral can be integrated by parts once again to obtain the indicated form, with the indicated remainder term for A(JL). Finally the last formula can be differentiated to obtain the asymptotic formula for A'(JL). •

246

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Now we use the Landau formula (4.5.4). We must compare a sum of the form

1

A(JL) dN(JL)

M

1

with

M

A(r)47rr 2 dr.

()

{)

To do this we integrate both by parts:

1 1

MA(JL)dN(JL) =

A(M)N(M)-

{M A'(JL)N(JL) dj.J_, ~

=

47l"M' A(M)-- 3

()

M

()

A(JL)47r JL 2 dJL

1M 0

47l"JL3

A'(JL)-- dj.J_. 3

Now we subtract these two expressions and analyze the two terms separately. If F(a) = 0, then we have the asymptotic forms sinaJL A(M) = C2~

(

+0

I )

JL 4

,

•

A (JL)

= C2

acosaJL

JL 3

+0

I )

(

JL 4

•

Therefore the productA(M)(N(M)- 47rM 3/3) = O(M 3 12 x M- 3) = O(M- 3 12 ). Similarly the integral is estimated by 1= r 3 12 r- 3 dr, an absolutely convergent integral. Therefore the difference

J

1 {)

MA(JL)dN(JL)- {M 47l" M2 A(JL) djJ. ~

has a limit when M -+ oo. But the integral is easily seen to be convergent from the form of A(r). To treat the case F(a) =/:. 0, we ex!_lmine the sum LIL<:OinlsM<~ 1 A(lnl) for large k, where JLk = (2k + l)/4a.lfthe series Ln j(n) converges, this sum must tend to zero when k -+ oo. But the above analysis allows one to compare this with the corresponding integral, with a smaller error. Thus A(lnl)-

1-'k 1I"<-

47rr 2A(r) dr I

= 11-'k+ IA(JL)d [N(JL)1-'k

3] .

47f jL --

3

When we integrate-by-parts. we find that the terms at the limits yield A(J.Ld [N(JLd-

4

7l":~] = O(k- 2

x k 312 )

=

O(k- 112 )

X

k-2)

while the new integral is

1:·+

I

[

N(JL)- 47l";3] A'(JL)dj.J_

=

0(k3/2

=

O(k-l/2).

Thus

k-

00.

On the other hand. the explicit form of A(JL) with F(a) =/:. 0 shows that the integral has the explicit asymptotic expression

= 2F(a)(-l/ +

o(~).

which fails to tend to zero when k -+ oo, completing the proof.

•

POISSON SUMMATIO N FORMULA AND MULTIPLE FOURIER SERIES

4.5.3.2

247

Higher- dimensi onal case

In higher dimensio ns, we can use the above method to prove that the Fourier series of the periodiz ed indicato r function of a ball diverges at x = 0.

Proposi tion 4.5.8. Suppose that d > 3. Then the spherica l partial sums SMf(O) =

'"""'

L- a

k Jd/2

d

lkajJ

lkl:s:M

are unbound ed when M--+

CXJ,

(2.rr lkla)

wheref( x) = l[O.aJ(Ixi ).

Proof. We can repeat the asymptot ic analysis done above in case d function A(JL) =ad (Jd 12 (27r JLa))j(aJL )dl 2 has the asymptot ic behavior A(JL) =

A'(JL)

=

JL<::~>/ 2

[cos(27r aJL-

~~:~>~d

3. The Bessel

(d-/)Jr) +0(~)]. -/)Jr) + o(~) J.

[sin( 2JraJL- (d

Letting JLk be the consecuti ve zeros of the above cosine function, we estimate as before:

When we integrate- by-parts, we find that the terms at the limits yield A(JLk) [N<JLk) _ CdJLn = O(k-(d+l) /2 X kd-2+2/(d +l)) = O(k(d-3)/ 2-(d-I)j(d +l)) while the new integral is

Thus we have when k --+ oo,

On the other hand, the explicit form of A (JL) shows that the integral has the explicit asymptot ic expressio n

1:*+'

JLk-IA(JL )

$

=

1:•+'

=

const k1 2(-l)k [1

JL
7f

d

~ 3) + o(~)

J$

+ o(~)],

which is unbounde d, and of larger order than the error term, completin g the proof.

4.6

•

SCHRO DINGE R EQUAT ION AND GAUSS SUMS

In this section we will use the Poisson summati on formula to evaluate some finite sums that occur in number theory. In order to explain the setting, we first formulat e the notion of Fourier series of Schwart z distribut ions on the circle. This will be applied to the

248

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

fundamental solution of the Schrodinger equation, which is a finite linear combination of 8-distributio ns at equally spaced points whenever 2rr tis a rational number. The treatment follows Taylor (1999).

4.6.1

Distributio ns on the Circle

c= ('Ii) is the space of infinitely differentiabl e functions on R that are periodic with period 1. Convergenc e is defined by requiring that all derivatives converge uniformly on any period interval. This can be defined by the metric

L 2-n n=O

d

00

d(¢, 1/f) =

n

1 + dn

dn := sup l¢(n)(x)- 1/f(n)(x)l.

'

O:;:x:;:l

A periodic distribution is a continuous linear functional L on the space Fourier coefficients of a periodic distribution are defined by

c= ('Ii). The

L(n) = L(e- 2"'")

(4.6.1)

and the Fourier series is written L "'"' LnEZ L(n)e 2 "'"x.

Example 4.6.1. The linear functional L(¢) = ¢(0) is the Dirac mass at zero, written L = 80 . Its Fourier coefficients are given by L(n) Fourier series

= 1 so that we have the

Example 4.6.2. Any f E L 1 ('Ii) becomes a periodic distribution by setting LJ(¢) = /1r f(x)¢(x) dx. The Fourier coefficients are given by the usual integrals 4-(n) =

l

f(x)e-2rrinx dx =f(n).

The Fourier representatio n of a periodic distribution is obtained from the Fourier series of¢ E c= ('Ii), which has the convergent Fourier series ¢(x) =

L

/IJ(n)e2rrinx

nEZ

so that we can apply L to both sides to obtain the Fourier representatio n (4.6.2)

L(¢) = LtP(n)L(- n).

This allows us to identify a distribution in terms of its Fourier series, as follows. Proposition 4.6.3. If L1, Lz are periodic distributions with the same Fourier series, then L1 = Lz. Proof. Applying (4.6.2) to L identically zero.

=

Lt - L2 shows that L(t/>)

=

0, ' e C 00 (1l'), hence Lis •

The Fourier representatio n can also be used to define new distributions .

249

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

Exercise 4.6.4. Suppose that Ln is a bilateral sequence of complex numbers of

polynomial growth: ILnl < C(l + lni)N for some C > 0, N > 0. Prove that there exists a periodic distribution L such that L (n) = L, for every n E Z. Hint: For every cf> for every n E Z.

E

C""('JI'), and every k > 0, there exists Ckn > 0 so that lcP(n)l .:S CknO

+

lnl)-k

Exercise 4.6.5. Define the convolution of a periodic distribution L with ¢

C 00 (1I') by L * ¢(x) = L(¢(x - ·)). Show that L have the convergent Fourier series L

* ¢(x)

=

L

*¢

E

E

C 00 ('Jf) and that we

L(n)rP(n)e 2rrmx.

nEZ

An important class of distributions are those that are sums of a finite number of delta measures at equally spaced points; in detail we write L = L:;: 1 cJfJJ/N where 0

The Fourier coefficients are N-1 L(k) =

L

cie-2rrtkJ/N.

j=O

This sequence is periodic with period N, since L(k

+ N) =

N-1

L

Cje-2rri(k+N)j/N

J=O

=

N-1

L

Cje-2rrikJ/N

=

L(k).

j=O

Conversely, suppose that we are given a bilateral sequence Lk with the property that for some N E z+, Lk+N = Lk for all k E Z. The smallest such value N is called the period. Then we can uniquely solve the system of linear equations c·e-2rrijk/N J

O=:::_k
o~~N-1

in the form O=:::_j
to obtain a periodic distribution L =

L:f;;Q 1 c1 81 ;N and conclude the following.

Proposition 4.6.6. Every periodic sequence is the set of Fourier coefficients of a uni7ue distribution that is obtained as a finite sum of delta measures: L = CJOJ/N where N is the period of the sequence.

L:S:,o

Example 4.6.7. Let L = -};; L:S:o 1 OJfN· The Fourier coefficients are given by · h zs · one zif n = O, ±N , ± 2N, ... , and zero L n = IiiI "'N-1 L...J=O e -2rrinj/N , w h zc A

(

)

250

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

otherwise. Therefore we have the Fourier series

4.6.2

The Schrodinger Equation on the Circle

The initial-value problem for the one-dimensional Schrodinger equation on the circle is to find u(x, t) defined for x E 'TI', t > 0 so that

au at

a2 u ax

- i - = - -2

u(x,O) = ¢(x)

E

C""('TI').

Separation of variables produces the factored solutions u = series

u(x, t) =

L

2 2 e 2 JTin.xe- 4 JT irn

and the Fourier

/P(n)e2JT•nxe-4JT2irn2.

nEZ

It is immediately verified that for ¢ E C 00 ('TI') this series converges in c= ('TI') and that limHo u(x, t) = ¢(x). The fundamental solution is the distribution with Fourier series Lr ~

L

e2JTinxe-4JT2irn2'

nEZ

From the definition of convolution, we have u = L,

* ¢.

Proposition 4.6.8. Assume that 2rr t = M / N for some M, N E z+. Then L, is periodic with period N and L, is a finite sum of delta measures at the equally spaced points 0, 1/N, ... , (N- 1)/N where the measure ofj/N is N-1

CMN(X)

:=

L

2

e2JTirxe-2JTir M/N

r=O

for j = 0, ... , (N- l)jN. Proof. It is immediate that if 2rr t

=

MIN, then

which proves the periodicity of the sequence. In order to obtain an explicit evaluation, we write the Fourier series at 2rrt = M/N: L, = limn~oo 'L:=-nL,(k) in the sense that for each¢ E C 00 (1r), we obtain L,(¢) by integrating and taking the limit of the partial sums. This limit can be computed by writing k = Nj + r and considering an arithmetic sequence

POI<;SON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

251

of sums as follows:

L

N(n+l)-l

L L n

e2Jrlkxe-2Jflk 2 <M/N)

=

k=-Nn

N-l

2 e27fi(NJ+rhe -2Jrl(NJ+r) (M/N)

;=-n r=O

L L n

=

N-1

2 e2Jrl(NJ+r)xe-27flr (M/N)

J=-n r=O

=

L

L

n

N-1

e2;nNp.

J= -n

2 e2rurxe-2Jrlr (MJN)

r=O n

=

CMN(X)

L

e2rr1NJ'.

;=-n

When n proof.

--+ CXJ,

the last sum converges to the distribution r_:;=~ 1

811 N,

which completes the •

We now look for another formula for L 1 , by means of the Poisson summation formula. The solution of the Schrodinger equation on the real line can be obtained from the Fourier transform as u(x, t) =

where/ E S. Clearly u(·, t) in x if we identify

E

lj(~)e2rrd;xe-4rr2at;2 d~

S for each

2

e-4rr ir!; 2

=

> 0. This can be represented as an integral

t

1

e2rr1/;x e

-< 2 /411

dx

../4irTt

IR

where the integral is taken as a Cauchy principal value and the square root is taken with positive real part. This gives the explicit representation u(x, t) =

411 { e-:;;:;;; /(y) dy.

j!R

4.rr lt

Applying the Poisson summation formula leads to the representation

L

](n)e2rrinxe-4rr2irn2 =

nE?l.

L

u(x- v, t)

vE?l.

=

{ j(y) JIR

(L

vE?l.

e-(x:,:;;;/411) dy. 4.rr lt

This allows us to represent the distribution L 1 as e-<x-k) 2/411 Lr=L-=--kE?l.

../4irTt

Taking 2.rr t = M / N, we have 1

N.rr 4it- 2Mi'

1

N

4nit

2Mi

252

INTROD UCTION TO FOURIE R ANALY SIS AND WAVEL ETS

The sum is evalu ated by letting k =2M }+ r, leadin g to e -(x-k) 2 j4it

=

e-(x-2 Mj-r) 2 (Nrr j2Mi)

=e =e

-(x2 +r2 -2rx-4M jx)(Nrr j2Mi)

-x2Nrr j2Mt

e

-4Mjx(N rr j2Mt)e iNrrrx/ Metr 2Nrr /2M

There fore l

- ---==.

2M(n+ l)-l

v'47fTt

'""' ~

k=-2M n

e

-(x-k)2 j4it

=

[& 2Mi

n

2

--eN rrix j2M D

MN

(x) """"'

L-

.

.

e21fiNX j

j=-n

where we have set D MN ( X ) -_

2M-I

""""'

L-

e

iNrrrx! Me-r 2 Nrrj2M i

.

r=O

The last sum conve rges to a sum of unit delta measu res at the points 0, 1/N, ... , (N - 1) 1N. Equat ing the two forms of the distrib ution, we obtain the identi ty C MN ( X ) =

v{N 2Mi

e-Nrrix 2 /2M D

MN

(x)

1

x=O , N, ... ,

N -1

N

In detail , (4.6.3 )

for each N = 1, 2, ... and each k

E

{0, 1, ... , N - 1}. In partic ular fork = 0 we have

Speci alizin g this to the case M = 1 yields the most classi cal Gauss sum:

4.7

RECU RREN CE OF RAN DOM WAL K The forma lism of multip le Fouri er series can be comb ined with Lapla ce's asymp totic metho d to study the recurr ent behav ior of a simpl e rando m walk on the intege r lattice . The intege r lattice 'Z/ is the set of all d-dim ensio nal vector s whose coord inates are intege rs: zd = {(k1, ... , kd) : ki = 0, ±1, ±2, ... }. The simpl e rando m walk begin s at the origin and move s with equal proba bility 1 /2d to one of its neigh bors, with the succe ssive steps being indep enden t of one anoth er. In detail , denot ing the coord inate

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

253

vectors by ej, 1 :::;j
Prob[Sn+l - Sn = ±ej] = lj2d

Fourier series are introduced through the characteristic function

L

fn(t) = E[ei(r.S,)] =

e'(r.k)Prob[Sn = k]

kEZd

where (, ) denotes the standard inner product of ~d. The hypothesis of independence allows us to write fn(t) = E [

fJ

e'
j=l

fJ

E[e'(r.(SJ-sJ_,))] =f(tY.

j=l

But f(t)

=

e' 1 '

+ e-ir, + · .. + eird + e-

11

d

cos t1

=

2d

+ · · · + cos td d

.

The recurrent behavior is determined by Prob[Sn = 0], as follows. Proposition 4.7 .1. • lfn is odd, then Prob[Sn = 0] = 0. • If n = 2m is even, then 1immd12prob[Szm

=

0]

=

2I-ddd12rr-d12.

m

Proof. A retum to zero occurs at time n if and only if in each coordinate the number of positive and negative steps are equal, hence the total number of steps in each coordinate must be even, in particular the total number of steps (in all coordinates) must be even. We have

j(t)n

=

L

e'(r,k)Prob[Sn

=

k].

kezd

From the formulas for multiple Fourier series, Prob[Sn

= k] = -

1

-d

(2rr)

1

j(t)ne-t(t,k) dt.

(-rr,rr)d

In particular Prob[Szm

=

0]

= ~ (2:rr)

1

f(t)2m dt.

(-n.rr)d

This can be further simplified by noting thatf is 2:rr-periodic in each variable and satisfies the oddness property /(t, + :rr, ... , td + :rr) = -f(t 1 , ••• , td). The periodicity allows us to write

1

(-rr.rr)d

f(t) 2 m dt =

1

j(t) 2 m dt.

(-rr/2.3:n:/2)d

The oddness property further allows us to write Jill<~ rl j(t) 2 m dt = Jii-Jli<~ r f(t) 2 m dt . Now we can apply the asymptotic method of Laplace. On the cube ( -rr /2, 3rr j2)d, lf(t) 1 ::::; I with

254

INTRODUC TION TO FOURIER ANALYSIS AND WAVI::LI:.TS

=

=

global maxima assumed at t1 0, and t1 rr where 1/(0)1 ltl > 8, it- rrl > 8 we have 1/(t)l ::::; 1- 111 < 1 so that {

=

1, 1/(rr)l

1 . On the sets

j(t)2mdt ::S: (2rr)"(l-r yi)2m_

Jltl>o,lt-rrl >o

Thus md/21

f(t)2m dt = 2md/2 {

(-rr/2.1rrj2) d

j(t)2m dt

+ O(md/2(1

- T/1 )2m}

Jltl<8

=

+ o(l).

j(sj.J,i1) 2 m ds

2 { }!..,1<8mlf2

In order to apply the dominate d converge nce theorem, we first note that the integrand is bounded by the integrabl e function e-lsl 213". To find the limit, we use the Taylor series expansio n and the inequality IA 2m - B 2 ml ::::; 2miA- Bi to write t--+ 0

f

(-m-~1-2) 2m

- e

-1112 /d

=f ( ::::; 2m

Jm) k(J,n) _ 2m -

(e

-1;12/(2md)) 2m

2

e-lsl

<2m (1-

-

=0(~)

J:f_ 2md +

/(2md)l

o(-

1 ~ + o()) m2

1 ) - 1+ m2 2md

m--+

m--+

oo

oo

from which we conclude that lim m" 12

m~oo

f

j(t) 2 m dt

(-rrj2,3rrj2) d

=2

{

}'JI.d

2 e-lsl /d

ds

= 2(rrd)"l 2 ,

•

which gives the required result.

This can be used to estimate the mean occupati on time of the origin, defined by 00

Mean occupati on time

=L

Prob[Sn

= 0].

n=O

= I, resp. d = 2, then the 2mth term of this series is asympto tic to the general term of a divergen t series (1/ -./iii resp. 1/m), thus we have an infinite mean occupati on time. Howeve r if d > 3, then the general term of the series is asympto tic to the general term of a converge nt series (1/md/ 2 ), thus a finite mean occupati on time. In this sense we say that symmetr ic random walk is recurren t in one and two dimensio ns and transient in dimensio ns three and greater. If the dimensio n d

Exercise 4.7 .2. Suppose that a one-dim ensional random walk is defined by indepen dent steps with

Prob[Sn -

Sn-l -

1] = p,

Prob[Sn -

Sn-1 -

1] = 1- p

POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES

255

where p =I= 1/2. Show that Prob[Szm = 0] = CZ,;:')pm ( 1 - p )m. Find an asymptotic formula for Prob[Szm = 0] and conclude the transient behavior. Exercise 4.7 .3. Suppose that a one-dimensional random walk is defined by independent steps with Prob[Sn- Sn-1 = k] = Pk

k=0,±1,±2, ...

where LkezPk = l.Lkez kpk = 0, LkeZ k 2 Pk = o- 2 < 00. Find an asymptotic formula for Prob[Sn = 0] and conclude the recurrent behavior.

CHAPTER

5 APPLICATIONS TO PROBABILITY THEORY

5.1

MOTIVATION AND HEURISTICS

The previous chapters have dealt with the Fourier analysis of functions in one and several dimensions. While this is sufficient for many applications, it does not fully cover problems from probability theory, where we must deal with measures that do not have a density with respect to Lebesgue measure. Unlike the L 2 theory of the Fourier transform from Chapter 2, this theory is inherently nonsymmetric: The Fourier transform of a finite measure is a continuous function rather than another measure. For this reason we change slightly the definition of the Fourier transform for notational convenience.

5.2

BASIC DEFINITIONS

Let m be a finite Borel measure on JRd. This is a nonnegative, countably additive set function defined on the Borel sets of JRd with m(JRd) < oo. The Euclidean inner product is denoted I; · x = L~= 1 l;,x,. The Fourier transform of the measure m is the function (5.2.1)

m(l;) =

{

jfRd

eit;·~ m(dx).

Note the change in the sign convention and the omission of the factor 2.rr in the definition. These adjustments are made in order to conform with the conventions of the theory of probability. If m is a probability measure (m(JRd) = 1), we refer tom as the characteristic function of the measure m. The properties are listed below. Proposition 5.2.1. The Fourier transform has the following properties: 1.

256

mis a continuous function with m(O) =

m(JRd).

APPLICATIONS TO PROBABILITY THEORY

2.

257

mis a positive definite function, meaning that for every set of complex numbers (cJ)I=sJ=sn and vectors (~J)I=sJ:::on n

L

c/ckm(~1 - ~d >

o.

;.k=l

3. If m 1 , m 2 are two measures, the Fourier transform of the convolution is the product of their Fourier transforms, where the convolution is defined by (m1

* mz)(B)

1

=

m1 (dx)mz(dy).

{ (x ,y):x+yEB)

Equivalently , for any bounded continuous function g

f

~d

g(z)(m 1 * m 2 )(dz) =

{

~u

g(x

+ y)m1 (dx)mz(dy).

Proof. The continuity of m follows from the dominated convergence theorem: If l;n ~ l;, then the complex-valu ed functions e''" x are bounded by 1, and converge to e'< x when n ~ oo. The positive definite property is a direct computation:

::::0.

To prove the convolution property, multiply the two transforms to obtain ml (l;)m2(l;)

= {

}fftu

ei< (x+y)ml (dx)m2(dy)

= {

}fftd

e'l; (x+y)(m,

* m2)(dz),

•

which was to be proved.

Example 5.2.2. The centered Gaussian distribution with variance parameter 0 is the measure with density e-lxl 2 12a 2 • Its Fourier transform can be computed

a >

in terms of a product of one-dimensi onal transforms as

m(~) =

=

f

J"R,d

eiS"·xe-lx1 2 /2a 2 dx

n(

f

;=I

j'R

eiS"}x}e-x}/2a 2 dxj)

n( d

=

v'2JTa2e-.;}a 2)

;=I

= (2JTa2)dl2e- a 2 fl;i 2 /2.

258

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Example 5.2.3. The Fourier transform of the uniform measure on the rectangle

nf=l (aj, bj)

is computed as

If ~J = 0 for some j, then the correspondi ng factor is replaced by bJ - aJ. We now prove that the mapping m

----+

mis 1 : 1.

Proposition 5.2.4. The measure m can be retrieved from its Fourier transform by the inversionfor mula

provided that them-measu re of the boundary ofthis rectangle is zero. In particular. m is uniquely determined by m. Proof. Multiply the defining equation (5.2.1) by e-cr 2i
If we integrate this with respect

1

to~

and use the Gaussian Example 5.2.2, we obtain

e_"2HI2e-i~·Ym(~) d~ = (2rr)d

1

-~~21~2

e

m(dx).

iRd ( ../2rru2)d/2

IRd

Now we integrate with respect toy over nj(aj, b1 ) to obtain

(5.2.2)

The integrand on the right side is bounded by 1; it tends to 1 if X E Of= I (aj, b ) and tends 1 to zero if x ¢ n;'= 1 [a1 , bj]. The boundary of this rectangle is supposed to have m-measure zero. The conclusion now follows from the dominated convergence theorem. •

If the Fourier transform of a finite measure is integrable, then the measure has a density that can be recovered by Fourier inversion, as follows.

Corollary 5.2.5. lfm

E

L 1 (JRd) then the measure m has a density, given by -dm =

dy

1 (2Jr )d

1 ..,

e-•s·Ym(~)d~.

JRd

APPLICATI ONS TO PROBABILI TY THEORY

259

Proof. With this extra hypothesi s, we can take the limit under the integral in Propositi on

5.2.4 to obtain

which displays the measure of the rectangle as the integral of the required density function. • These ideas allow us to generali ze Maxwel l's characte rization of the Gaussian density in Chapter 2, Proposit ion 2.2.51, from density function s to the larger class of finite measure s on JR.d. The precise statemen t is the followin g.

Proposi tion 5.2.6. Suppose that m is a finite measure on JR.d, d > 2 with the

followin g two properti es:

(5.2.3) where m 1 , ••• , md are finite measure s on JR. For any orthogo nal transfor mation T ofJR.d,

(5.2.4)

{ f(Tx)m( dx) =

)Rd

{

)Rd

f(x)m(d x),

Then either m =A 80 or m(dx) = Ae-Bixl 2 dx where A > 0, B > 0. Proof. The Fourier transform m(~)

zation

= fRd e•<·xm(dx )

satisfies the correspon ding factori-

(5.2.5)

Takingf( x) = e-•< ", (5.2.4) shows that m(T~) = m(~), hence m(~) = G(l~1 2 ) for some continuou s function G. Now we can follow the steps of the proof of Propositi on 2.2.51 in Chapter 2 to conclude that, since G is bounded, G(x) = Ae-Bx, where B =:: 0. If B = 0, then m is a multiple of 80 ; otherwise B > 0, which complete s the proof. •

Remark . We can render more transpar ent the computa tions in Proposit ion 5.2.4 and Corollar y 5.2.5 followin g by using the notation s of Chapter 2, beginnin g with the Fourier represen tation of the heat kernel H 1 , a bounded function with H E L 1 (JR. d): 1

260

INTRODUCTION TO FOURIER ANALYSIS ANO WAVELETS

Multiply both sides by lA ( y)m(dx) and integrate over JR.d x JR.d: { (Hr

}Rd

*

IA)(x)m(dx) =

{

Jocd

m( -0 iA

m(A) =lim { t---+0

= =

}Rd

(~)Hr(~) d~

m( -~)

L (~)Hr(~) d~

if m(aA) = 0

f m(-~)LC~)d~

if ,n

}Rd

1(ld e rrtl;·xm(~) d~) 2

E

L'CJRd)

by Fubini.

dx

The above methods can also be used to prove the following continuity theorem for Fourier transforms of measures. Proposition 5.2.7. Suppose that mn, n = 0, 1, 2, ... is a sequence of finite measures whose Fourier transforms converge:

lim mn(~) = n

mo(~).

Then the measures converge on every rectangle whose boundary has mo measure zero. Proof. For any ¢

E

S, let 1fr (/;)

= ¢. (-~), so that ¢ = Vr.

Then by Fourier reciprocity we

have

Letting n -----+ oo, the dominated convergence theorem implies that

li~ If R

i.

¢(x) mn(dx)

= li~

i . 1/f(~)mn(~) d~ = i . lfr(~)mo(~) d~ = i .

= n7=1 (a;, b;) is any rectangle, let¢± lim supmn(R) .:::lim sup { n

n

JR."

E

s

cp(x)mo(dx).

so that¢- .::: IR.::: ¢+.Thus

¢+(x)mn(dx) =

{

}TR.n

cp+(x)mo(dx)

liminfmn(R):;:: liminf {

{ ¢-(x)mo(dx). n hn ¢-(x)mn(dx) = hn

n

Now let¢+ ..!- lR, ¢-

t

lR to conclude that

m 0 (R").::: liminfmn(R).::: limsupmn(R).::: mo(R). n n

If m 0 (aR) = 0, then the extreme values are equal, so that the limit exists as required.

5.2.1

•

The Central Limit Theorem

Fourier analysis of measures is particularly well suited to study the convolutions of a single probability measure. In the case of the measure p8 1 + (1- p)8 0 this was effectively studied in Chapter 1 in connection with the DeMoivre-Lapla ce local limit theorem. The central limit theorem extends this to an arbitrary probability measure with a finite second moment, which we now describe.

261

APPLICATI ONS TO PROBABILI TY THEORY

Theorem 5.2.8. Suppose that m is a probabil ity measure on the rea/line with

L

i

x m(dx) = 0

2

x m(dx) = u

Then for any interval A,

* m * · · · * m)(A-vCn) =

. hm(m "

1

~2nu 2

2

<

CX).

1 A

e -x'fa' dx .

Proof. From Propositi on 5 .2. 7 it suffices to compute the Fourier transform of the indicated convoluti on. This is [m(~ 1~n)r. The Taylor expansio n at~ = 0 is

m(~)

=

I - ~ 2 a 2 /2

+

o(e).

The character istic function of the Gau:-.sian measure with me;1n zero and variance a the same Taylor expansion . Now we can write m

I

A

- ~ )n - e -t.'!l<>21 < n m

(

Jn

I

-

A

(

-~

Jn

- e -t.'!2nrr

)

2

has

2 1

where we have used the fact that if a, b are complex numbers with Ia I < 1. lbl < 1, then ian - bn I ::;: nja - bi. But from the Taylor expansion s, m A

(

n ~112

which proves that Iimnm(~/n'1 2 ) 11 continuit y theorem proved above.

Exercise 5.2.9.

2

)

=

e -t.'!2na -_ o (1/ n ) ,

-

e-t.'l 2 a'. The conclusio n now follows from the •

Centra/ limit theorem for Abel sums. Suppose that m is a

probabil ity measure on the real line with

l

xm(dx)

=

1, 2, ... , let m~(A)

For 0 < r < 1 and n any interval A

= 0,

l

2

x m(dx)

=

I

1.

m(A/(r 11 ~1- r 2 )). Prove that for

lim(m~ * m; *···)(A ) = -1r-->

=

.J2ir

1 '

e __,·; 2 dx.

A

Hint: It suffices to show that the Fourier transform satisfies the limiting relation lim,~ 1 n~ 0 m(trny' I- r 2 ) = e-r'1 2 . Use the fact that m(~) = e-t.'1 2> when~~ 0.

5.2. 1. 1

Restate ment in terms of indepen dent random varia/;Jies

The central limit theorem is presente d as a result on the convolut ion powers of a single probabil ity measure . This can also be recast as a result about the measure induced by a sum of independ ent random variable s, as follows:

Definiti on 5.2.1 0. A set of real-valu ed function s X 1 (t), ... , X 11 (t) on a proba-

bility measure space (Q, ~) is mutually independ ent if for every choice of real numbers x 1 , ••. , X 11 , we have

262

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Let m; be the distribution of X;, namely the measure induced on IR by the equation m;(A) = J.L{t : X;(t) E A}. Then the distribution of the sum X, (t) + · · · + Xn(t) is the convolution m 1 * · · · * mn, so that (m * · · · * m)(A.y'n) is the distribution of the sum (X 1 (t) + ... + Xn(t))/Jn. In probability theory, the term "random variaple" is synonymous with "real-valued measurable function." The central limit theorem can now be recast as follows: Theorem 5.2.11. Suppose that {X,.(t)},.= 1 , 2 .... is a sequence ofmutually indepen-

dent random variables with distribution m, where fiT?. x m(dx) = 0, JR x 2m(dx) = u 2 < oo. Then the distribution of the normalized sum [X 1 (t) + · · · + Xn(t)]/u Jn converges to a standard normal distribution when n --* oo.

Independent random variables may be constructed on the unit interval Q = [0, 1] as follows. Let cj> : N --* N 2 be a biject~ve mapping. For example, this may be constructed by listing all of the integers in a doubly infinite array as follows: 1

3

6

10

15

2

5

9

14

20 ...

4

8

13

7

12

11

21 ...

19 ...

18 ...

17 ...

16 In this example we have, for example c/>(8) = (2, 3), c/>(18) = (3, 4) and so forth. Now we expand t = L:~ 1 wk /2k and define

and so forth. Then for every n, {X1 (t),X2(t), ... ,Xn(t)} are independent random variables, each of which is distributed according to Lebesgue measure on [0, 1]. To achieve more general distributions, it suffices to form Borel functions in the form Y;(t) = c/>;(X;(t)) fori= 1, 2, ....

5.3

EXTENSION TO GAP SERIES

The asymptotic normal distribution is not restricted to sums of independent random variables. In this section we consider a class of trigonometric series that are asymptotically normal. More general results are found in the book of Zygmund (1959, Volume 2,

APPLICATION S TO PROBABILITY THEORY

263

Chapter XVI). Here we consider sums of the form k

(5.3.1)

Sk(t)

=

L

a1 cos n1 t

}=1

where (a1 ) are real numbers and n 1 < n2 < · · · are integers which satisfy (5.3.2)

(k

=

1, 2, ... )

for some q > 1. The growth of the sum is measured by the L 2 norm, which is

(5.3.3) where we assume that

(5.3.4)

(k -----* 00).

As preparatio n for the theorem, we first prove a simple lemma.

Lemma 5.3.1. Under the conditions ( 5.3.4 ), we have (k-+ oo). Proof. Given E > 0, let K, be such that (Ak) is increasing, we have for k > K,

-

1

la,I!Ak

max la1 1 ::;

Ak K, ~J~k

I -E Ak

< E fork> K,. On the one hand, since

max IAjl =

K, ~j~k

E,

while (k-+ CXi).

Hence lim

supk~= (l flAk

I) max,~ 1 ~k la1 I ::;

E.

But E was arbitrary, so the proof is complete .

•

This lemma allows one to conclude, for example, that for any p > 2

Theorem 5.3.2. Suppose that the integers (nk) and the real numbers (a ) satisfy 1 (5.3.2) and (5.3.4) with q > 3. Then we have fork-+ oo (5.3.5)

-I

2rr

I{

Sk(t) t E 1I' : YI ::::: - :::=: Y2

Ak

}I

--+ -1-

../iii

1Y2 e-u y1

2

12

du.

264

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. We compute the Fourier transform, defining

It suffices to prove that k(~) --+ e~< 1 2 when k --+ oo. From the power series of the exponential function, we have for small lzl, 2

(I

+ z)e~o

=

(I -

+ z)

= (I

z

+ ~ + O(ld))

z2

3 2 + 0(1zl )

I -

so that (z--+ 0)

and we can write

where the o(l) term is uniform in t E 'lf'. Now we write

noting that I

-2

1

rr

2

Tk(t) dt

1f

I

=8

L k

J=l

a4

~-+ 0

(k-+ 00),

Ak

in particular Tdt) --+ 0 in measure. Meanwhile

n(t

I}=I

2

+

i~ Aka} cos njr) 1

:s

Ii }=I

2

(1

+

~Ak~J)

so that we can write

k(~) = 2 ~ 1Il (1 + i~ ~j 1f J=l

k

2

cosnjt) exp [ - x (1 2

+ ~k(t) + o(l))] dt

APPLICATION S TO PROBABILITY THEORY

265

It remains to analyze the final integral. For this purpose we expand the cosine products using repeatedly the identity 2 cos a cos b = cos (a +b) +cos (a -b) to obtain a finite sum (5.3.6) where the sum is over those indices of the form v = n,, ± n, ± · · · with n,, > n, > · · ·. 2 2

Lemma 5.3.3. Suppose that nk+I > qnk with q > 3. Suppose that an integer vis represente d in two (possibly) different ways

Proof. If all of the subscripts are equal, there is nothing to prove. Otherwise there is a first subscript that differs in the two representat ions. By relabeling the subscripts, we may assume without loss of generality that i 1 > j 1 • By a further relabeling and moving all of the terms to one side, we may write

where the coefficients a,

E

{0, ±1, ±2} and i 1 > k 1 > k 2 > ···.Hence

+···) which is a contradicti on. To complete the proof of the theorem, we note that in the product (5.3.6) the only contributio n to the term ao occurs when all of the frequencies are zero, hence a = 1. 0 Applying the orthogonal ity of cos vt, we conclude that

Il (1

1 { 2 rr }.,. = 1 1

+

which completes the proof of the theorem.

i~ Ak a 1

cos n1

t)

dt = l,

•

Exercise 5.3.4. Suppose that the coefficient s (an) satisfy (5.3.4). Prove that 1

n- log lanl--+ 0 when n--+ oo.

Hint: Write a~/A~ =En ~ 0 and solve for a~ = a~(En/EN )n:Z,+ (1/(l -Ed) for n > N, where 1 0 and Ek < E for n > N.

EN >

Exercise 5.3.5. Suppose that the coefficien ts (an) satisfy ( 5.3.4) and the integers (nk) satisfy (5.3.2). Let Sk(t) = L~=I a cos(n t - (} ) where (} E JR.. Prove 1 1 1 1 that ( 5.3.5) holds for this wider class of series.

266

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

5.3.1

Extension to Abel Sums

The central limit theorem for gap series can naturally be extended to obtain the limiting distribution of the harmonic function oc

(5.3.7)

u(r, t)

=

2:a1rn1 cosn1 t j=l

under the same conditions as in Theorem 5.3.2. We define (5.3.8)

Then Exercise 5.3.5 implies that A(r) < oo for 0 < r < I. The following result was first proved -by Kac ( 1939) and later extended by Salem and Zygmund (1948). Theorem 5.3.6. Suppose that the integers (nk) and the real numbers (a ) satisfy 1 (5.3.2) and (5.3.4) with q > 3. Thenfor any interval C C "JJ', we have for r ~ 1

I{t

(5.3.9)

E '][' :

u(r t) '

A(r)

E

C

}

I~

1

2

e-Y 12

c

-./2]i dy. 2n

We first develop the Abelian counterpart of Lemma 5.3 .1. Lemma 5.3.7. Let bk ::: 0 for k ::: 1 and set Bo = 0, Bn = bt + · · · + bn for n ::: 1. Suppose that Bn ~ oo and bn/Bn ~ 0 when n ~ 00. Furthermore let B(r) = Ln::: 1 bnrn for 0 < r < 1. Then 1

- - supbkr B(r) k:;o:l when r

~

k

~

0

I.

Proof. We set e,. = b,.J B,. for n ~ 2. Without loss of generality we may assume that b = I 1 and e,. < 1 for all n. Then B,_ 1 JB, = 1 - e,., so that we can write for n ~ 2

APPLICATIONS TO PROBABILITY THEORY

Since Bn

---+

oo, we have B(r)

---+

oo when r

---+

267

1. On the other hand we can write for any

NEz+,

B(r)

"' = Lb,r" n=l N

=::: L

(B,- Bn-1 )r"

n=l N

= r'\ B"

+ (1 -

r)

L

r"Bn

n=l

= r

Given then

E

N( 1 - I

> 0, let K, be such that e, <

E

e2

) . . . ( 1 - 1 eN ) ·

fork > K,. If sup, bkrk is attained at some k > K_,

On the other hand, if the supremum is attained at some k

But

E

~

K,, then

•

was arbitrary, which completes the proof.

We can apply this lemma by taking b 11 , =a~ and b 11 = 0 if n ~ {nk,, nk , 2 conclude that r"k akf A (r) ~ 0 when r ~ 1, unifonnly in k E z+.

•• • },

to

Proof of the Theorem. We compute the Fourier transform

We proceed as in the proof of Theorem 5.3.2, beginning with the estimate e 0 = (1 + z)exp[z 2 /2 + O(lzi 3 )] applied to z1 = (ii;/A(r))ajr"' cosn t, noting that z ---+ 0 1 1 when r ---+ 1, uniformly in}. Then

Write

268

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The L 2 norm of \llr is estimated as before r-+1

where we have used Lemma 5.3.7 to replace one factor by the supremum, the remaining sum being equal to 1. Meanwhile, we have the uniform bound

n1 ""

,~

2 I

1

t;uk + i- cosnkt 1

A(r)

n

::: = ( 1

t; u~ ) + --)-~

•=I

1

A(r -

so that we can apply the dominated convergence theorem to conclude that

But this infinite product can be expanded as a sum:

n ""

(

1+

1'/;

= a,r nk cosnkt ) =L:a.,(r)cosvt .

A(r)

k=l

v=O

From Lemma 5.3.3 and the condition (5.3.4) we have cx 0 (r)

=

o(l), so that we conclude

•

which completes the proof.

5.4

WEAK CONVERGEN CE OF MEASURES

To probe the deeper aspects of the convergence question, we develop the following notion of weak convergence of measures.

Definition 5.4.1. A sequence of finite Borel measures (m,) is said to converge weakly to a limit measure m iffor every bounded continuous function g

lim { n

}Rd

g(x)m,(dx) =

{ g(x)m(dx). }'R,d

If A is any set in JR.d, the interior A o is the set of points x E A such that A contains an open ball about x. The closure A is the complement of the interior of the complement; in symbols (A)c = (N') 0 and we have A o c A c A. The boundary is defined as aA = A \A o. The portmanteau theorem gives equivalent conditions for weak convergence.

Theorem 5.4.2. The following conditions are equivalent 1. m, converges weakly to m.

2. For every closed set A, lim sup, m,(A)

::::=:

m(A).

APPLICATI ONS TO PROBABILI TY THEORY

269

3. For every open set A, lim infn mn(A) > m(A). 4. For every Borel set A with m(3A) = 0, limn mn(A) = m(A). The proof of this theorem , which has nothing to do with Fourier analysis, can be found in Billingsl ey ( 1999). It is also helpful to develop the appropri ate notions of compact ness in the context of weak converg ence of measure s. In this setting we refer to a tight family of measure s, formaliz ed as follows:

Definiti on 5.4.3. A sequenc e offinite measure s (mn) is tight if lim supnmn( {x : lxl >A}) = 0.

A--><XJ

Theorem 5.4.4. Suppose that mn is a tight sequenc e of finite measure s with supnmn( Rd) < oo. Then there exists a weakly converge nt subseque nce. Again we refer to Billingsl ey (1999) for the details.

Exercise 5 .4.5. Prove that any weakly converge nt sequenc e of finite Borel mea-

sures is tight.

= 1, let mn = 8n, a point mass at the point n. Then mn is not a tight sequence , since for any n, mn {x : lxl > n/2} = l. Exampl e 5.4.6. With d

5.4.1

An Improv ed Contin uity Theore m

The theory of tightness can be used to formulat e an improve d version of the continui ty theorem for sequenc es of characte ristic function s of probabil ity measure s. In the previous version, Proposit ion 5.2.7, we required that the Fourier transform s mn(~) converge to a limit M(~). which is assumed to be a characte ristic function . We now have the followin g improve d version.

Theorem 5.4.7. Suppose that mn is a sequenc e of probabil ity measure s on Rd with Fourier transfor ms mn(~). with the property that there exists (5.4.1)

M(~)

=lim mn(~) n

and that M is continuo us at ~ = 0. Then there exists a probabil ity measure m so that mn converge s weakly tom and m(~) = M(~). Proof. From the hypothes es, we can write

1-

mn(~)

=

{

JTR.d

{1 - e'" ') mn(d.x).

270

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

This equation is integrated over the cube Cu

1

[I - m,(~)] d~

=

Ca

Id --

(2a)

1 c,

= n;= 1 { 1~1 I s

1( Rd

[1-m,(~)] d~ =

a} to obtain

d 2sinax·) 1 (2a)d - n m,(dx)

1(

J=l

XJ

1 sin ax I - nd - ) m,(dx)

JRd

J= 1

ax1

since the integrand on the right side is nonnegative. If (x 1 •••• , xd) lies outside the cube C 2 a-' , then the indicated product contains at least one factor that is less than ~, the remaining factors are less than I . Hence - -I d

(2a)

1

~

m,(~)] d~

[I -

C,

:::: -I 2

1

m,(dx).

(C , - 1 )' 2

From the dominated convergence theorem we conclude that lim sup {

" J' la

1

m,(dx) < ~ { - (2a)'

Jc

[1 -

M(~)] d~.

u

From the continuity of M, the right side can be made arbitrarily small by taking a sufficiently small. This proves that the measures m, are tight, hence we can extract a weakly convergent subsequence. If we had two different subsequential limits m 0 and no, then both of these measures must have the same characteristic function, hence they must be the same measure by the uniqueness of Fourier transforms, Proposition 5.2.4. Hence every subsequence has a subsubsequen ce that converges weakly to the same measure m 0 • From this it follows that the original sequence converges weakly to mo. •

Exercise 5.4.8. Show that the hypothesis of continuity at~ = 0 in Theorem 5.4. 7 can be weakened to the hypothesis that ~ = 0 is a Lebesgue point for M, in the sense that lima ....... o a-d Jc, II - M(~)l d~ = 0. 5.4. 1. 1 Another proof of Bochner's theorem In Chapter 3 we introduced the concept of positive-defi nite function and used a Fatou theorem for harmonic functions in the upper half plane to prove Bochner's theorem, which affirms that any continuous positive-defi nite function on lR is the Fourier transform of a nonnegative measure. In this section we will give an independent proof of Bochner's theorem using the theory of weak convergence applied to a Gaussian convolution. Proof. We begin with/(~), a complex-valu ed positive-defin ite function that is assumed to be continuous. Define

(5.4.2)

m, (x)

= - 1-

By hypothesis, we have for every 1fr

2n

E

r e-uue-'"

]IR

L 1 (lR),

2

/(u) du.

APPLICATIONS TO PROBABILITY THEORY

271

2 Taking the choice 1/F(~) = e'-"<e-•< 12 we infer that

0

~I12 f(~- T/)e•x<J;-ry)e-•<212e-•ry2/2 d~ dT/

=I

12 f(u)e'-"ue-•
=

j!!;

1

f(u)e'xue-•u2/4 du

= const. x m, 14 (x), which proves that m, (x) ::=: 0. We claim that m, E L 1 (IR) and that the Fourier transform of the nonnegative function m, is precisely e-•u f(~). To see this, we use Fubini to establish the following identity where 8 > 0 2

(5.4.3) Taking~

= 0 and using Fatou's lemma and the continuity at u = 0 shows that

1

m,(x)dx

~

liminf8-o

1

8 2 2

m,(x)e- x 1 dx =f(O) <

oo.

Hence m, e L 1 (IR) and llm,IIL' 0 fixed:

where we have used the continuity off in the last line. Hence the productf(~)e-•< is the Fourier transform of the function m,. Taking E -+ 0 and using Theorem 5.4.7, we see that f is the Fourier transform of a nonnegative finite measure, which was to be proved. • 2

Exercise 5.4.9. Show that the hypothesis of continuity in Bochner's theorem can be weakened to the hypothesis that ~ = 0 is a Lebesgue point for f, in the sense that lima-+0 a- 1 lf(O)- f(~)l d~ = 0.

F'-a

Exercise 5.4.1 0. Show that the above argument can be generalized to 1Rd, to obtain a characterization of the Fourier transform of a .finite nonnegative measure as a positive definite function on JRd.

272

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

5.5

CONVOLUTION SEMIGROUPS

Fourier analysis is particularly well suited to deal with one-parameter families of probability distributions that are closed under convolution. Example 5.5.1. Let P 1 be the probability distribution on lR 1 with density 2 e-x 121 I ../2iit. Then by direct computation, P 1 * Ps has the density e-x 2 /Z(t+s) I .../2rr(t + s), so that P 1 * P, = P,+,. Definition 5.5.2. A convolution semigroup of probability measures is a family (P 1 )r>
when

t-+ 0,

in the sense ofweak convergence. The Fourier transform of the convolution semigroup is defined by ~ .fr(~) = P,(~) =

(5.5.3)

{

}R e

il;x

P,(dx).

This has the following properties. Proposition 5.5.3. • ~ -+ .fr ( ~) is continuous and bounded by 1. • .fr (~) -+ 1 uniformly on compact~ -sets when t -+ 0. • t -+ .fr ( ~) is continuous for t > 0. • t-+ .fr(~) is differentiable fort> 0 and l/f(~) := limt---> 0 • .fr(~) satisfies the identity .fr(~)

= 1 + l/f(~) J; f,(~) ds,

.fr'(~)

exists.

and.fr(~)

= etl/1"(1;).

Proof. The first property follows immediately from the definition (5.5.3). To prove the second, we have for any 8 > 0

l.fr(~)-

11

=I ~{

[e•<x- l]P,(dx)l

where the second term tends to zero when first term is less than M 8. Therefore

::::01

~1<8

t ~

l~xiP,(dx) +2P,({x:

0. On any compact interval 1~ 1 .: : : M, the

lim sup sup l.fr(~)- 11 r~o

lxl > 8})

.:::=::

M8.

I
But 8 > 0 is arbitrary, hence the limsup is zero, proving the uniform convergence. From (5.5.1) and (5.5.2) we have (5.5.4)

lim j, (~) r~o

=

1.

Letting s ~ 0, we see thatj, is right continuous at each t > 0. But for smalls > 0 we can write

APPLIC ATTONS TO PROBABILITY THEORY

273

so that we can take s ---+ 0 and conclude that t ---+ j; is left continuous. In particular t ---+ j; is bounded and measurable, so that we may integrate (5.5.4) on an interval [0, 8] to obtain

f

(5.5.5)

r+8

r

{8 /;(/;)

ds =ft(l;) Jo fs(l;) ds

where 8 > 0 is chosen so that lf,(l;) - II < 1/2 for 0 < s < 8. The formula (5.5.5) displays j; as a differentiable function oft with a continuous derivative. Taking the derivative for t > 0 we obtain

Hence there exists

=

1/f(l;)

limj,'(l;)

=

J:

.fo(l;)-

r-O '

1

,

fs(l;) ds

which displays 1/f as a continuous function with 1/f(O) = 0. Computing the derivative from (5.5.4), we obtain the differential equation!,'(/;) = 1/f(l;)j;(l;). The unique solution satisfyingf0 (l;) = I is

• The next goal is to prepare the proof of the following theorem of Levy and Khintchine. Theorem 5.5.4. Suppose that (P 1 )r>O is a convolution semigroup of probability measures on JR. Then there exists a unique Borel measure M and real numbers J-L, a, so that the Fourier transform has the representation

where a2t;2

1/r(/;) = iJ-L/;- - - +

2

1. IR

[e';x- 1- il; sinx]M(dx)

and {

jlR To prepare the proof, we define Fn

x2

_1_+_x_2M(dx) <

=P

1

;n and Gn

00.

= nx

2

j(l

+ x 2 )Fn.

Lemma 5.5.5. The total mass ofGn is uniformly bounded, specifically

sup n

1 IR

nx2

1 +x2

Fn(dx) < oo.

274

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. We begin with the basic relation 1 - j,(/;) = fn.,[I - e'-"<]P,(dx) that is integrated on the interval [- 1, I] to yield

1>1-

~

(5.5.6)

i(

=

j,(/;)) dl;

1- si:x) P,(dx).

When we divide by t, the integrand on the left side tends boundedly to 1/f(/;). On the right side, we use the inequality sinx 1--x Dividing both sides by t

= I Jn

lim sup ,

we obtain

71

2

1

nx

Fn (dx) ::5:. JRI+x 2 2

1

•

1/f (I;) dl;.

-1

Lemma 5.5.6. F(A) := supn nFn(lxl >A) < r::xJfor each A > 0 and F(A) tends

to zero when A --+

r::xJ.

Proof. We first prove this for the symmetrized distribution F!(dx) whose characteristic function is fl~n

=

(/;)

=

Fn(dx)

* Fn(-dx),

e-(2/n)Y,(/;}.

We apply the technique of (5.5.6) with the interval [ -1, 1] replaced by the interval [-2/A, 2/A] and use the inequality I - e-' ::5:. y to write

F:(l/;1

>A) ::5:. 2A { 111;1<1/A

::::::. 4A n

1

(I - /,~n(/;)) dl; IRe 1/f(l;)l dl;.

1<1<2/A

Therefore sup nF:(Ixl >A) ::5:. 4A { n

IRe 1/f(/;)1 dl;.

111<1<2/A

But 1/f is continuous at I; = 0, hence the right side of the final inequality tends to zero when A - oo. To handle the general (nonsymmetri c) case, note the following "symmetrizat ion inequality" for any two independent random variables X, Y: P[IX- Yl

~A] ~

P[IXI

~

2A]P[IYI ::5:. A]

(draw the picture). Applying this to X, Y distributed as Fn, we obtain

F

2A

n(lxl

>

) ::5:.

But the denominator tends to 1 when n so that

F!A) Fn(lxl ::5:. A). oo, from the basic hypothesis that P, -

lim sup nFn ( lxl > 2A) ::5:. lim sup n F: (I xi > A) ::5:. 4A { n

which tends to zero when A -

n

oo.

80 ,

IRe 1/f (/;) l dl;,

111;1<2/A

•

APPLICATI ONS TO PROBABILI TY THEORY

275

Having done this, we now tum to Lemma 5.5.7. The measure s Gn are tight: in detail

lim sup sup A---+CXJ

n

1

lxi>A

nx 2 1 +X2

Fn (dx) = 0.

Proof. We write nx 2 - - , F,(dx) ::; nF,(ixl >A)

f

I<J~A I

+X"

and we have shown above that the supremum over n tends to zero when A --+ oo.

•

Proof of Theorem 5.5.4. Now we prove the represent ation theorem. From the definition

1/r(~) =lim /r(~) r~o

=lim n

-

I

t

{(en~j,R

I)nF,(dx ).

For any n we can write (5.5.7) where J.l-, = JD?. sin xF, (dx). The integrand in (5.5.7) is defined by continuit y at x = 0. Note that Gn({O}) = 0 for all n. From Lemma 5.5.5, we may take a subseque nce for which the total masses G,(IR) converge to a limit. Since the measures Gn are tight from Lemma 5.5.7, we may take a further subseque nce that converge s weakly to a limit measure G. All of the limits below will be taken through this new subseque nce. The integral term on the right side of (5.5.7) converge s to

1

e•x~ -

I - il; sin x

- - - -- - - ( 1 2 D?. X

+

,

x") G(dx)

while the left side converge s to 1/r (i;). Letting ~ = I and taking the imaginar y part shows that J.l-n converge s to Im 1/r(l ). We have obtained the Levy-Khi ntchine represent ation in Feller's form: (5.5.8)

1/r(i;) =

1

e"' - I - ii; sinx

L<

X

2

(1

+ x 2 ) G(dx) + iJ.l-1;.

It remains to discuss the uniquene ss of the pair (J.l-, G). The number J1- is uniquely determin ed by J1- = Im 1/r( 1 ). To identify G, we form the convoluti on of 1/r with the kernel 1e-l
+

.!_ {

2 JD?.

e-li;-ryJl/r( TJ)dTJ =

{

la

e'x~G(dx),

which shows that 1/r uniquely determine s the character istic function of the finite measure G. But the uniquene ss theorem for character istic functions shows that G is thereby uniquely determin ed from its character istic function, which complete s the proof. •

276

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

5.6

THE BERRY-ESSEEN THEOREM

Fourier analysis is extremely effective in obtaining sharp estimates on the rate of convergence in the central limit theorem. The following discussion is presented for the case of convolution powers of a single probability distribution, but it can be easily extended to the case of different distributions , under suitable hypotheses. Let m be a probability measure on lR with (5.6.1)

l

xdm = 0,

l

x 2 dm =I,

3

llxl dm = m3 < oo.

From the central limit theorem, we know that the normalized convolution of the distribution function F converges to the normal distribution (5.6.2)

li~(F * F * ... * F)(x.../fi) = (x) =

X

e-u 2 j2

-oo

v'2Ji

!

du.

The Berry-Essee n theorem gives the rate of convergence .

Theorem 5.6.1. We have uniformly for -oo < x < oo

(5.6.3)

i(F * F * ... * F)(x.../fi)- (x)l

where C is a universal constant.

The idea of the proof is to regularize the given distribution by convolution with a smooth density. Then we can apply the inversion formula directly to the regularized convolution, whose Fourier transform is the product of the given Fourier transform with the smoothing factor. For the smoothing density we choose the Fejer kernel (5.6.4)

Kr(x) =

I - cos Tx , rrTx 2

with

Lemma 5.6.2. Let G be a probability measure on lR and set

(5.6.5)

d(x) := G(x)- (x),

rJ = supld(x)l,

'f/T

= supj(d *Kr)(x)l.

X

X

Then

where m = sup <1>' (x) = X

1

v'2ii"

Proof. Since ~(x) vanishes at ±oo, the supremum occurs at some pointx where we have 0 ~(xo ± 0) = ±ry. If ~(xo ± 0) = +11, we propose to estimate the convolution

(~ * KT )(xo + o) =

(is+ 1 -8

lxl::-8

)~(xo + [,-

x)KT(x) dx

where 8 is to be chosen. If -8 :::;:: x :::;:: 8, then ~(Xo

+o-x)= G(xo +o-x)- cl>(xo + [,- x)

:=:: G(xo ± 0)- [cl>(x0 ) = '7

+ m(x -

o)

0

+ m(o- x)]

APPLICATIONS TO PROBABILITY THEORY

277

We use the estimate on the Fejer kernel

1

l
1

Kr(x) dx < -

l
2

4

rrTx

rrT8

- -2 dx = - -

and the fact that J~ 8 xKr(x) dx = 0; thus

1:

On the interval\x\

:=:

~(x0 + 8- x)Kr(x) dx :=: 8 we use the bound ~(x)

j
~(x)Kr(x) dx :=:

(77- m8) ( l -

:=:

JT~ 8 ) .

-r, to write

:=:

-1711?co Kr(x) d>::

-77

JT~ 8 .

Adding the two, we obtain

(1 +j ) ~(xo + :=: (1- ~)- T/~-

(~ * Kr)(xo + 8) =

6

m8)

(T/-

8 -x)Kr(x)dx

lxl?:b

-b

rrT8

rrT8

The proof is completed by choosing 8 = 77/2m to obtain (~ * KT )(x0 ) :=: r,/2 - 12mjrrT, which immediately gives the required result. If ~(x0 ± 0) = -17, then we apply the above argument to - ~ (x). • Proof of the Theorem. Let G(x) = F,(x) = (F* · · · *F)(x..jii), whose Fourier transform is F(l; I ..jii)''. The Fourier transform of F, * Kr is obtained by taking the product with (l - \I; \/T) lro. TJ (\/; \). in particular the product is absolutely integrable on R Hence we can apply the inversion formula to write

1 2JT

(F, *Kr)([a,x]) = -

Applying the inversion formula to

1 (F,*KT-*Kr)([a,x ])=2rr

1T

e-,~x _ e-d;a ( \/;\) ~ ( 1; )" l-- F dl; -il; T ..jii .

-T

* KT

and subtracting, we have

1T e-'~' -e-i~a( . -tl;

-T

1 -\/;\)(~ F

T

( 1;r.:: )" ...;n

2

-e-~1 2 ) dl;.

For each fixed T > 0, n < oo the integrand is an integrable function, so that we can apply the Riemann-Leb esgue lemma to let a--+ -oo and obtain (5.6.6) (F,

* KT-



1T e-'~"'

, ') dl;. - . - ( 1 -\/;-\) ( F~ ( - 1; )" - e-~-;~

1 * Kr)((-oo, x]) = 2rr

_ 7 -tl;

T

..jii

It remains to estimate the integrand and apply Letnma 5.6.2. From the definition of the Fourier transform,

~~= 2

IF(!;)- 1

+

IL

: :; i

(e'~x- I - il;x + \/;:3\ F(dx)

\I; 13 =m36·

2 l;:x ) F(dx)l

278

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

We apply the same estimates to e-< 1 12 and use Holder's inequality to obtain 13 le-< 212 -1 +e/21:::: : 1 ~ 6

f

Jfil:.

lxl 3e-x 212

__:!!__ ~

21~1'

- 3./2ii <

2m,l~l 3

3-v'27r I~ 13

<mJ-.

-

2

Thus

Furtherm ore ~2 m3ll;l 3 <1--+ ---'-,- -

-

2n

6nj

~2

~2

2n

6n

<1--+-

-

~2

= 1--

3n

We now estimate the integrand in (5.6.6) by the telescopi ng sum

Applying Lemma 5.6.2 with T

2

7r

= .fo/4m 3 ,

we have

IFn (x-vn '-)-( )I < 2 !T (1 x -T <

m3

{

- .;n JIR

!!J.) m31~12 T

-t;2;3 dl:: 2 -vn '- e "

1~12e-<2!3 dl; +

+~ T 3/2 1r

12m3 Jnrr3/2

which is of the required form with 2rrC = (3/2) 3121.Jiii + 12/rr 312.

•

Remark . We can obtain a lower bound on the best possible constant by recalling the DeMoiv re-Lapla ce limit theorem from Chapter 1: In this case the explicit asympto tic analysis shows that Fn(O + 0)- Fn(O- 0)'""'"' 1/vfnii = .,J277i1j .Ji"n. Since m = 1, 3 we see that the best constant in (5.6.3) satisfies C > .../2f7i'""'"' .79.

APPLICATIONS TO PROBABILITY THEORY

5.6.1

279

Extension to Different Distributio ns

The Berry-Essee n theorem extends naturally to a sequence of probability measures (Fd that satisfy

(5.6.7)

l

xdFk = 0,

and the additional conditions that

A~:= a~+···+ a~--+

(5.6.8)

CXJ

ak

---+ 0

Ak

mk <)..a~

for some).. > 0. Under these hypotheses we claim that (5.6.9) for a suitable constant C. To prove (5.6.9), we apply Lemma (5.6.2) with G = F, * · · · * Fn· The Fourier transform of the left side of (5.6.9) is F 1 (~/An)··· F 1 (~/An)- e-~ 1 2 • Now if (g) = e-~ 1 2 is the Fourier transform of the standard normal distribution, we can write (g) = 1 (g /An)··· n(l; /An) where ci>1 (g) = e-~ a}/ 2 is the Fourier transform of a normal distribution with mean zero and variance so that we can write 2

2

2

aJ,

L (fr,, (g /An)) n ;(g /An) nF;(g /An). n

=

)=1

i<j

i>j

Arguing as before, we have for sufficiently large n,

so that

IFt(;/An) · · ·Fn(l;/An)- e-1;

2 /

2

1 :S:

2

e-1; /

n

3

L j=l

Now

IFJ(;/An)- ,(;/An)l.

280

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

so that

Applying Lemma (5.6.2) we have

AfT( 1 - 1~1) -

sup IG(x)- (x)l ::: xEIR 3An

T

-T

1~1 2 e-~ 2 13 d~

C + -.

T

Choosing T = An balances the two error terms and leads to the desired result.

5.7

THE LAW OF THE ITERATED LOGARITHM

The central limit theorem, which gives the limiting distribution of normalized sums of independent random variables, has an almost-everywhere counterpart, as follows: (5.7.1)

. hmsup k-.oo

Sk(t)

J2A~ log logAk

= +1

a.e. t.

The proof of (5.7.1) depends on careful bounds for the distribution of Sk(t)/Ak and a simple estimate for the distribution of the maximum of S 1 (t), ... , Sk(t). We will prove the following result: Theorem 5.7 .1. Suppose that {Xk (t) hEz+ are independent functions on 'II' = [0, 1] with f1fXk(t)dt = 0, f1fXk(t) 2 dt =a~, f1r 1Xk(t)l 3 dt < oo and that the conditions (5.6.8) are satisfied. Suppose further thatfor each x E Rand k E z+

(5.7.2)

l{t : Xk(t) < -x}l = l{t: Xk(t) > x}l.

Then ( 5. 7.1) holds. The distribution of the maximum is estimated as follows. Lemma 5.7.2. Let SZ(t) = max{SI (t), ... , Sk(t)} where (5.7.2) is satisfied. Then

(5.7.3)

j{t: s;(t)

> x}l < 21{t: Sk(t) > x}l.

Proof. Let A} = {t : sl (t) .:::X, ... 'sj-1 (t) .:::X, SJ(t) > x}. Then the event {t : s;(t) > x} is written as the disjoint union U~=IAr From the symmetry hypothesis (5.7.2) we have for any·n1 _::: n2, i{t: I:;,!n 1 Xj(l) 3::- 0}1 ::=:: Now

4·

SJ(t) >X, XJ+l (t) + ... + Xk(t) :::: 0 ==> Sk(t) >X hence A 1 n {X1 +1

+ · · · + Xk

:::: 0} C A1

n {Sk

> x}.

APPLICATIONS TO PROBABILITY THEORY

281

Then k

l{t: Sk(t) > x}l ::=::

L lA, n {t: Sk(t) >

x}l

j=l

k

: : :. L IA

1

n {t: X 1 +1(t)

+ ··· +

Xk(r):::::.

Oll

j=l

k

=

L

lA, II{!:

x,+1(t)

+ · · · +Xk(t)::::: Oll

;=I

1

k

: : : ;zLIA1

1

j=l

•

1 S*k(t) > x}l. = ;zl{t:

Lemma 5.7.3. lfxk--+ oo so that xU2 -logAk--+ -oo, then t: I{

I

Sk(t) > x } = exp[-xf(l +o(l))]. ~

Proof. From (5.6.9) we have

ll{t: s~~) > x}l- (1- (x))l ~ ~ where C is a constant. When x -+ oo we have 1 - (x) = exp [ -x /2( 1 + o( 1))]. The 2

hypothesis on xk is equivalent to 1/Ak = o(e-xif ) and therefore the error term can be absorbed into the Gaussian term when this is satisfied. • 2

We also need the first and second Borel-Cantelli lemmas, as follows. Lemma 5.7.4. Suppose that (Bd are measurable sets with L~! IBkl <

oo. Then

{ t:f:ls,(t)
Proof. The assertion is that, almost surely, only a finite number of the events Bk occur. The proof comes from applying the monotone convergence theorem to write

if

Is,(t)dt

=

11' k=l

f

iBki < oo.

k=l

•

Hence I:~ 1 18 , (t) < oo for almost all t.

Lemma 5.7.5. Suppose that (Bk) are measurable sets with L:~ 1 IBkl = oo and that (Bd are mutually independent: I B'f= I = 1 1B'f= I for any choice of±, where B+ =Band B- =Be. Then

n7=,

{ t:

f

k=l

18 .(t) =

n7=

+oo}

-I.

282

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. The assertion is that, almost surely, infinitely many of the events occur. To prove this. use the monotone convergence theorem with a = log 2 to write

l

exp( -a

'j;;

18 , (t)) dt = f r o T

=

18 •
dt

0(1- ~IBki)

~o

exp(

-~IBkl)

= exp( -a t.IBkl) =0. Thusexp(-a1::~ 1 1 8 .(t)) =Ofora.e.t,hence l{t: 1::~ 1 1 8 ,(1) < oo}l =0.

Proof of (5.7. 1). Given 0

>

•

1, define a sequence of integers (nk) by the recipe that nk = max{n :An ~ Ok}.

Thus An• ~ ()k < An*+' and it follows that An• . . .__ Ok and Ank+I /An• Now from Lemma 5.7.3 we have for any 8 > 0, l{t: Sn,(t) > (1

+ 8)An.J2loglogAn.}l

0 when k -

= exp[-(1

+ 8) 2 loglogAn.O + o(1))]

= (

)(1+~) (1+o(l))

1

00.

2

logAn•

. . .__ (

1

)(1+~)2(l+o(l))

log A,.. Since 8 > 0, this is the general term of a convergent series, so that by the first Borel-Cantelli Lemma 5.7.4, for a.e. t only a finite numberoftheseevents occur. Thus fork > k(t) we have (5.7.4) Now by Lemma 5.7.2, given 8 > 0, choose 0 > 1 so that /; 2 j(fJ - 1) > 1. Then the last series of terms converges and the first Borel-Cantelli Lemma 5.7.4 shows that for a.e. t we have fork > k(t) (5.7.5) Adding (5.7.4) and (5.7.5) we have fork> k(t) and S1 (t) = (S1 (t) - Sn• (t))

+

~ 8An• J2log logAn• ~ (1

nk ~j

<

nk+l

Sn• (t)

+

(1

+ 8)An• J21og log A,..

+ 2l;)A1 j2log logAj.

Thus lim sup1 S1 I A 1 J21og log A 1 ~ ( 1 + 28). But 8 > 0 was arbitrary therefore the limsup in equation (5.7.1) is less than or equal to 1. Applying the same reasoning to the sequence -Sj shows that the liminf is greater than or equal to -1.

APPLICATIONS TO PROBABILITY THEORY

283

To prove the lower bound, we consider the independent events B, := {t : S,, (t) - S,,

1

(t) :::-_ (1 -

8)A,, j2Iog log A,,}.

Then

IB~.i

=exp(-(1-8) 2 -

( k I~g

e)

I

(5.7.6)

I

n,

n,- n,_

1-o >'H' I

loglogA 11,(I

+o(l))

1 1

(II' --11'-

)(1-I>J'f(l -II"

1

)(I +nl II)

)li+o(i)l

( k loge

Given 8 > 0, we choose e > 1 so large that I - e- 1 > (I- of. Then the last term in (5.7.6) is the general term of a divergent series, to which we can apply the second Borel-Cantelli Lemma 5.7.5 to conclude that for infinitely many indices k --+ = (5.7.7)

Sn, (t) - S,,

1

(t) 2:: (1 -

8)A"* j2Iog log A,,.

But since we have shown in the first part that the Iiminf is greater than -1, we have for all sufficiently large k, (5.7.8) Adding (5.7.7) and (5.7.8) and dividing by the right side, we have lim inf k

s,, (t) A,k j2Iog log A,,

2:: (1 - 8)-

2

~ne-

v u

Now we rechoose e so that I -8-2/../0 > 1 -28. This proves that for any 8 > 0 there is a subsequence j --+ so that S1 (t)/A1 j2Iog IogA1 :::-_ (1 - 28). Hence the limsup of this ratio is greater than or equal to 1, which was to be proved. •

=

CHAPTER

6 INTRODUCTION TO WAVELETS

6.1

MOTIVATION AND HEURISTICS

Classical Fourier analysis may be viewed as the problem of reconstructing a function f from dilations of a fixed sinusoidal function x ~ e 2rrix by writing f(x) = JR e 2rcit;xj(l:,) dl;. The Fourier transform}(!;.) may be thought of as the amount of the sinusoidal oscillation e 2rrit;x present in the function f. The Fourier representation is instrumental in analyzing translation-invariant operators such as convolution operators and linear differential operators with constant coefficients, where we can write l j ( x - y)K(y) dy = l P

(~)f(x)

K(l;)e2nit;xj(l;) dl;,

= l.p(2JTil;)e2nit;xj(l;,)dl;.

However classical Fourier analysis suffers from the defect of nonlocality: The behavior of a function in an open set, no matter how small, influences the global behavior of the Fourier transform. We have also remarked on the simultaneous nonlocalizability in connection with the uncertainty principle. The theory of wavelets is concemed with the representation of a function in terms of a two-parameter family of dilates and translates of a fixed function that, in general, is not sinusoidal, for example: f(x) =

l.

\a\-11/1 (x a b) W..pf(a, b)dadb 2

where W ..pf is a suitably defined transform off. Altematively one may envision a series expansion f(x)

=

L j,k

284

c1 .k2j12 1/1(2jx- k)

INTRODUCTION TO WAVELETS

285

where we sum over the dilates in geometric progression. The factors of lai-I/ 2 and 2j 12 are inserted to preserve the L 2 -norm of the basic wavelet 1jr. In this chapter we will describe the properties of wavelets in one dimension, making full use of the tools of Fourier analysis.

6.1.1

Heuristic Treatment of the Wavelet Transform

The wavelet transform off with respect to 1/f is defined by the integral W>!ff(a,

b)= J~[

j(y){/f

(Y-a b) ~lal %.

It is straightforward to compute this transform and the inverse transform on the Fourier

exponentialsj(x) = e 2rrit;x; from the definition of the Fourier transform, we have W>!ff(a,

b) = =

b) -JjQf

{ e2rr'b{/J ( y -

JR

l

viJQT

dy

a

e2rnt;(h+a~l{/J(z) dz

= Me2rrit;bVr(a~).

Now we form the adjoint operator

w; W>!ff(x) = = =

{ (W>!ff)(a, b)ljr

J~

(x- b) vial ~ (X- b) db a

MVr(a~)

{ e2rrit;bVr

JR

MVr(a~)M

L

a

-JjQf

e 2rrd;(x-az)1/r(Z) dz

= lall~(a.;)/2e2rrit;x { W* W

}JR.

1/f

A

f( ) da = 1/1

x a2

2rrit;x {

JTR.

e

11/r(a.;)i Ia!

2

d a.

The final integral is independent of~, which is seen by making the substitution v = a~, from which we obtain the inversion formula ·~ jlR(W.~, W>!ff ja 2 )da f(x) = e 2 rr,.,x = A"' . 11/r(v)l 2 /I vi dv

JR

This leads us to impose the normalization fiR I~ ( v) 12 j 1v 1d v = 1, in order to obtain the wavelet representation f =

[

JR

w; W"'fda a 2

valid when f(x) = e 2rr,t;x. It now remains to investigate this inversion procedure for arbitrary f E L 2 (1R).

286

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

6.2

WAVELET TRANSFO RM

Let 1fr

E

L 2 (lR). The dilated-trans lated function is defined by

(6.2.1)

Vra,b(x) = lal -1/2 Vr

(X- b) -a-

,

0=/=aElR ,bER

This function is obtained from 1fr by first dilating by the factor a and then translating by b. Clearly lll/Fa,bll2 = llo/112·

Definition 6.2.1. 1fr

E L 2 (lR)

is a continuum wavelet

if

(6.2.2) The wavelet transform off E L 2 (1R) by 1/r is defined by

(6.2.3)

WlfJf(a, b) =

l

{/fa,b(x)f(x) dx.

From the Cauchy-Sch warz inequality, we see that W lfJf is a bounded function with iWlfJf(a, b)i < ll1/rll2llfll2· The intuitive meaning of WlfJf(a, b) is the amount of the dilated-trans lated waveform VFa.b that is present in the functionf.

Remark. If, in addition, 1/r

L 1 (lR), then the integrability condition (6.2.2) implies that JR lfr(x) dx = 0. Indeed, -(/;- is continuous at I; = 0 with -(/;-(0) = Jff?. lfr. If this is nonzero, then the integral (6.2.2) is divergent. The following is a form of Parseval's theorem for the wavelet transform: E

Proposition 6.2.2. Suppose that 1/r is a continuum wavelet with (1/r, 1/r)w Thenforan yf, g E L 2 (1R), we have

(6.2.4)

1

f(x)g(x)dx =

lR

11

IRIR

1.

-

dadb WlfJf(a, b)WlfJg(a, b ) -- .

a2

Proof. Let lfr(x) lfr(-~). Then Wy,f(a, b) is the convolution off with lfra.O• ~hose Fourier transform is v'JQT~(a~). Hence the Fourier transform of Wy,f is](/;)v'JQT~(a/;), and similarly for Wy,g. Therefore from Parseval's theorem for the Fourier transform we have (6.2.5)

We integrate both sides with respect to dafiai 2 • apply the Fubini theorem to the right side, and use the definition of (1/F, 1/F)w to remove this constant factor. The remaining integral is transformed by another application ofParseval's theorem in the form fr~. fg = JJR which completes the proof. •

Jg.

INTRODU CTION TO WAVELET S

287

This proposi tion can be interpre ted as the stateme nt that/ -+ W y,f is an isometr y from L2(JR; dx) to L 2 (JR 2 ; da dbfiai 2 ), where the inner produc t is defined as ((F, G)) :=

1 JR2

-

dadb F(a, b)G(a, b ) -2- . a

Theore m 6.2.3. Suppos e that 1/F is a continu um wavele t with (1/r, 1/r)w = 1. Then 2

forany f E L (1R), we have the L 2 inversi onform ula f(x) =

1 JR2

(6.2.6) =

dadb Wy,f(a, b)1/ra ,b(x)-a2

lim <'---+O,A,B ---+oo

i

<'
dadb W.pf(a, b)l/Fa .b(x)--- . a2

Proof. Writing S(E, A, B)f for the integral in (6.2.6), we note that this integral is absolute ly converg ent for each 0 0, since each factor is in L 2 (1R 2 ; dadbja 2 ). To prove the required converg ence, we first note that

=

llf- S(€, A, B)fli2

sup l(f- S(E,A, B)f, g)j. l/g112=I

Applyin g Fubini 's theorem , we see that (S(€,A, B)f, g)=

1- (! f g(x)

R

=

dadb) Wy,f(a, b)l/fa.b (x)dx 2 a dadb Wy,f(a, b)Wy,g( a, b ) -2 a

<'
<'
-

so that by (6.2.4) and Cauchy- Schwarz ,

l(f- S(E,A, B)f, g) I=

if.

f<'
~

(!. (!.

f•
=

{<<{a}
_

I a dadb)I / (1

Wy,f(a, b)Wy,g( a, b dadb ) -22

iWy,f(a, b)l 2 -

a

2

R2

dadb)I / 1Wy,gl 2 a2

2

2

dadb)I / iWy,f(a, b)l 2 - 2118112· a

When € ~ 0 and A, B ~ oo the region of integrat ion decrease s to the empty set, hence the integral tends to zero by the dominat ed converg ence theorem . This complet es the proof that IIS(E, A, B)f- fll2 ~ 0. •

We now give some exampl es of continu um wavelet s.

Examp le 6.2.4. The standar d Haar functio n is defined by 1/r(x) x < 1/2, l/r(x) = -1/or 1/2 < x < 1 and 'f/r(x) = 0 otherwi se.

+1 for 0 <

288

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

The Fourier transform is computed as .(f(x) =

1/2

{

e-2rrti;x

lo

dx-

11

e-2:ni;'

dx

112

-27Ti~

-2nil;

2ni~

(1 _

e-trr/;)2

2ni~

Clearly the integrabili ty condition (6.2.2) is satisfied, so that lfr is a continuum wavelet. In the next example, we specify the continuum wavelet in terms of its Fourier transform .

Example 6.2.5. Let .(f(~) = lr~-~ 1 (1~1). Clearly the integrabili ty condition (6.2.2) is satisfied. The continuum wavelet is

1/F(x) =

1/2

1

e2rrnl;

d~

+

~-1/4

1/4

e2rral;

d~

-1/2

- -------- -- + -------- ---2n ix

sin n x JTX

2n ix

sinnx/2 7TX

However this continuum wavelet is not integrable , since fiR llfr(x)llh = +oo. The following two examples include the normaliza tion (1/F, lfr).,... = 1.

Example 6.2.6. A Gaussian wavelet is defined by lfr(x) = Cxe-rrx 2 • The Fourier transform is computed as .(f(~) = -iC~e-rr/; • Clearly the integrability condition (6.2.2) is satisfied. The normaliza tion is computed from the integral fiR l.(f(~)l 2 /1~1 d~ = 2C2 fo= ~e- 2 "1< d~ = C 2 j2n, thus C = ~2

2

Example 6.2.7. The Mexican hat wavelet is defined through its Fourier transform by .(f(~) = cee-rr~< 2 • The continuum wavelet 1/F can be computed directly as the derivative of the previous example.A Thus 1/F(x) = C(l/2n - x 2 )e-rrx 2 • The normaliza tion is obtained by noting that fiR lo/(01 2 /1~1 d~ = 2C 2 fo= ~ 3 e- 2 "t;ld~ = C 2 j4n 2 , thus C = 2n.

INTRODUCTION TO WAVELETS

289

Figures 6.1.1 and 6.1.2 illustrate Examples 6.2.6 and 6.2. 7. y

FIGURE 6.1.1

Gaussian wavelet.

y

FIGURE 6.1.2

Mexican hat wavelet.

It is instructive to examine the inverse wavelet transform in the particular case of the 2 Gaussian example, where 1/f(x) = .Jiiixe-JTx • Define the partial inverse transform by

(6.2.7)

S,f(x) =

11 lR

Ia!><

dadb WyJ(a, b)l/ra.b(x)-2- . a

290

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

This can be computed through the Fourier transform by writing

s;j(~)

=

l

S€f(x)e-2rrt/;X dx

= ( {

d~ lall~(a~)l 2 da)J(~)

Jlal>€ a

But this is the Fourier transform of the convolution/* g€, where g€ is the Gaussian density 2 2 g€ (x) = e-rrx / 2 € 1.../2Ei, which is an approximate identity in the sense of Chapter 2, from which we conclude that for any homogeneous Banach space B, we have S€ f ~fin norm. In particular, iff is bounded and uniformly continuous we have liS€/- flloo ~ 0 when E ~ 0. Iff E LP(IR), 1 < p < oo, then liS€/ -fliP~ OwhenE ~ OandS€f(x) ~ j(x) for almost every x E JR.. This follows from the results on Gaussian summability in Chapter 2. These desirable properties are not shared by the partial inversion of the Fourier transform, for example.

Exercise 6.2.8. For the Mexican hat wavelet, define the partial inversion by (6.2.7) and explicitly compute S€f as the convolution with an integrable function, in particular verify that liS€!- flip ~ 0 in case off E LP(JR.), 1 < p < oo or f E BucOR). Exercise 6.2.9. Formulate the wavelet transform in n dimensions, beginning

with 1fr E L 2 (1Rn) satisfying fn~.· 1~(~)1 2 d~ /I~ In < oo and defining 'tfra.b(x) = lfr((x- b)ja)flaln/ 2 forb E lR.n and 0 =I= a E JR.

6.2.0. 1

Wavelet characterizatio n of smoothness

We can use the wavelet transform to characterize the smoothness of measured by the Sobolev norm

f

E

L 2 (JR.) as

Exercise 6.2.10. lfllfll2.s < oo,provethatfha skcontinuousde rivatives, where k < s- ~· Hint: Apply Cauchy-Schwarz to the Fourier integral representation of J<*>.

The next proposition applies to continuum wavelets that possess a certain number of vanishing moments: JR xklfr(x) dx = 0. The result states that for this class of wavelet expansions, the Sobolev norm is equivalent to a weighted L 2 norm of the wavelet transform.

INTRODUCTION TO WAVELETS

291

Proposition 6.2 .11 . Suppose that 1/F is a continuum wavelet with ( 1/F, 1/F) w = 1 and C

·- { 1/f.s . -

}JR

1~(~)12 ~~~1+2s

de < oo 5

for some s > 0. Then [

(6.2.8)

[

JJR JJR

2

=

Proof. Returning to (6.2.5) with/ respect to a E .IR. Thus {

{ IW /(

JrR JL'ii

"'

a,

dadb

2

IWwf(a, b)l lal 2 + 2 , = Cw.sll/llz..,·

lal2+ 2s and integrate with

g, we divide both sides by

h)l7 dadb = { { 1](.;)121~(a.;)l2 dad.; - lal2+ 2 ' JR JR lal 1+ 2s

= { lf(.;)l2 ( { l~l(v)l

JrR

=

JrR

c1/f.s

i

2

lvll+ls

dv) l~l2s d.;

2

1.; 1 ' 11<.;)1 2 d.;.

•

which completes the proof.

6.3

HAAR WAVELET EXPANSIO N

In this section we develop the properties of the Haar wavelet expansion, which is the oldest and most basic example of an orthonormal wavelet (to be defined in the next section). The Haar series expansion can be naturally motivated by the search for an orthogonal series representatio n of Lebesgue's differentiatio n theorem for a locally integrable function/ E ~1oc: f(x) =

lim b-.x,a---+x

-1b - a

1b

f(y) dy

a

a. e.

X E

JR.

We will systematical ly describe this connection in the following subsections.

6.3.1

Haar Functions and Haar Series

We begin with the basic Haar function 1/F (x) = 1

if 0 < x <

!,

1/F (x) = -1

if

!

:S x < 1,

and 1/F (x) = 0

otherwise.

1

Clearly / 0 1/F(x) d.x = 0 and f 01 1/f(x) 2 d.x = 1. A doubly indexed family of Haar functions is defined by writing (6.3.1)

11/f1 k(x)

= 2JI 2 1/F(2!x- k),

j, k

= 0, ±1, ±2, ....

l

We will prove that {1/FJdJ'k'"Z2 form an orthonormal basis of L 2 (1R).

292

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

Proof. The first task is to prove orthonorm ality, namely

if (j, k) =I= (/' k') if (j, k) = (/, k'). A direct proof of orthogon ality if j

L

= j'

is seen from the fact that if k =I= k',

1/f(2!x _ k)ljf(21 x _ k') dx =

r

1

L

1/f( y)ljf(y +

k - k') dy =

o

since the integrand is identicall y zero. Otherwis e we can assume that} < j', and write

r 1/f(2!x- k)ljf(2!' x-k')dx= r 1/f(y)1/f(2',_ y+2/-J k-k')dy J~ 1

JR

But both of these integrals are zero, since f 1 1/f = 0. Finally the normaliz ation in case 0 (j, k) = (/, k') is establish ed by computin g

L

2

1/f1k(x) dx

=

2

2! lljf(21 x - k) dx

The Fourier/ Haar coefficie nts off (6.3.2)

E L 2 (JR)

CJk = CJk(f) :=

i

= i_1fr(y) 2 dy =

1.

•

are defined by

f(x)l/rjk( X) dx,

leading to the Haar series (6.3.3)

j(x) . . .__

L

c1 kVr1k(x).

j.k

From the orthonor mality of {1/F1d and Bessel's inequalit y, we conclude that LJk ic ki 2 < 1 CXJ. The complet eness of L 2 (1R) further assures that the series (6.3.3) converge s in L 2 (1R). It now remains to identify the sum of the series with the given function / E L 2 (1R).

6.3.2

Haar Sums and Dyadic Project ions

In order to identify the sum of the Haar series, we introduc e the dyadic projection operator Pn as follows. Conside r the dyadic partition Fn, consistin g of the sets hn := ((k- l)/2n, kj2n], where n = 0, ±1, ±2, ... and k = 0, ±1, ±2, .... Pn is the projectio n off E L 2 (1R) onto the space L 2 (1R., Fn. dx), which consists of L 2 function s that are constant on each of the intervals hn. In detail, we have (6.3.4)

Pnf(x) = 2n

!..

j(y) dy

if

X E

hn·

Formula ( 6.3 .4) can be written explicitl y in terms of the scaling function (6.3.5) ¢(x) = I if 0 < x :S 1 and ¢(x) = 0 otherwis e (6.3.6)

Pnf(x) = 2n

L ( { ¢(2ny- k)j(y) dy) ¢(2nx- k). kEZ

J!R

INTRODUCTION TO WAVELETS

Indeed, the function x succinct, we have

->

¢(2"x- k)

=

P,J(x) =

l

(6.3.7)

0 unless x

E

+

(k/2", (k

293

1)/2"]. To be more

K,(x, y)j(y) dy

where (6.3.8)

K,(x, y) = 2"

L

¢(2"x- k)¢(2"y- k)

kEZ

if x, y E h, for some k otherwise. The projection operators are (i) increasing and (ii) converge to the identity: (n

~

<Xl)

in the sense that (i) Pnf = f implies P,+lf = f and (ii) lim, ..... =P,f =f, a.e. and in L 2 (1R) (to be proved in the next subsection). We now seek a representatio n of the operator P,+ 1 - P,, which is the projection onto the orthogonal complement L 2 (Fn+ 1 ) e L 2 (F,,). To do this, note that any square of the form ((k- 1)/2", k/2"] x ((k- 1)/2", k/2"] (where K, = 2") is decomposed into four smaller squares; on these smaller squares we have Kn+l (x, y) = 2"+ 1 on each of the smaller squares ((k - 1)/2", (k - 1/2)/2"] x ((k - 1)/2", (k - 1/2)/2"] and ((k- l/2)/2",k/2 "] x ((k- l/2)/2",k/2 "] whereas K,+ 1 (x,y) = 0 on the smaller squares ((k- 1)/2", (k- 1/2)/2"] x ((k- 1/2)/2", k/2"] and ((k- 1/2)/2", k/2"1 x ((k- 1)/2", (k- 1 /2)/2"]. Hence L,(x, y) := K,+ 1 (x, y)- K,(x, y) = 2"+ 1 = 2, + 1

-

_

2"

on ( k - 1 k - 1/2 2" ' 2"

2,

on ( k - 1 /2 !..._ 2" , 2"

Jx

on ( k - 1/2 !..._ 2" . 2"

2"

Jx

on ( k - 1 k - 1 /2 2" , 2"

= 0- 2"

Jx

( k - I

( k - 1/2 2"

J

X

(

k -

,

I /2

2"

J

!..._ J

, 2"

k - 1 /2

2"

!..._ J

, 2"

( k - 1 k - 1/2 2" . 2"

J.

This can be conveniently represented in terms of the Haar function by writing L,(x, y) =

L

2"1/f(2"x- k)l/f(2"y- k).

kEZ

Thus (6.3.9)

L,(x, y) =

L

1/fnk(x)l/f,t..(y ).

kEZ

For a fixed value of n, the functions {1/f,k} kEZ form an orthonormal basis of the space 2 e L 2 (F,). For each x E IR, the series (6.3.9) contains exactly one nonzero term, hence convergence is trivial.

L (Fn+l)

294

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

This provides the desired representatio n of Pn+l - Pn m terms of orthogonal functions, namely (6.3.10)

Hence we can write the original projection operator in the form n

Pn+d = Pof

+

L(P1 +if- P1 f) J=O

= Pof

+

i = L (1f(y)l/f1k(y)dy) 1/Fjdx) J=O kEZ

= Pof

+

IR

i~ (t; l/f k ® l/f k )f. 1

1

and thus the one-sided Haar series representatio n (6.3.11)

f = Pof

+

f:

L

J=O kEZ

(1f(y)l/f1 k(y) dy)

Vr1 k(x).

IR

In the following sections, we will abstract this to a more general setting, noting that the subspaces Vn := L 2 (1R, Fn. dx) have the following properties: (i) U~ 0 Vn is dense in L 2 (1R) (ii) f E Vn ifandonlyi ff(2-"·) E Vo (iii) {¢(x- k)hEZ is an orthonormal basis of Vo. This can also be extended to a bilateral family of subspaces by considering Vn for n < 0, namely larger and larger dyadic intervals. As above, the orthogonal projection on the subspace L 2 (1R, Fn+l· dx) e L 2 (1R., Fn. qx) continues to be represented by the formula (6.3.10), and the above nesting properties of the subspaces can be modified to 00

(i')

U n=-oo

n 00

Vnisdensei nL 2 (1R),

Vn = {0}.

n=-CX)

The latter property is evident from the fact that iff E Vn for all n < 0, thenf is constant on each interval [0, 21nl), hence must be identically constant for x > 0; butf E L 2 (JR) means that the constant must be zero. Before passing on to more general wavelet expansions, we note that the Haar function 1/f and the scaling function ¢ are related by the identity (6.3.12)

1/f(x) = ¢(2x) - ¢(2x- 1),

INTRODUCTION TO WAVELETS

295

whereas ¢ satisfies the identity (6.3.13)

¢(x) = ¢(2x)

+ ¢(2x-

1).

Much of the challenge of constructing more general wavelets will reduce to the suitable generalizatio n of these simple relations.

6.3.3

Completen ess of the Haar Functions

To prove the validity of the two-sided Haar representatio n (6.3.3), we go back to (6.3.10) and write n

(6.3.14)

P,+ rf(x)- P_mf(x) =

L L

Cjkl/f1 k(x).

J=-m !..E'Z.

It remains to prove that P_mf -+ 0 and P,+ 1 f -+ f when m, n -+ oo. First we prove that the operators P,, have uniformly bounded operator norms. Lemma 6.3.1. For any f E L 2 (!R.) and n E ::Z, we have IIPnfll2 < llfll2Proof. From the definition of Pnf, we apply Cauchy-Schw arz: X

E

h,. ==* IP,J(x)l 2 ::=: 2" {

}hn

IJ(y)l 2 dy

11P,J(x)l 2 dx ::=: 11f(x)l 2 dx IA.n

l1P,.j(x)l 2 dx ::=:

lA.,,

i

lf(x)l 2 dx.

•

We use the notation Co(!R.) to denote continuous functions vanishing at infinity and Coo (JR.) to denote continuous functions of compact support. Lemma 6.3.2. (i) If g E Co(IR.), we have IIP-mgll 00 -+ 0 when m-+ 00. (ii) Iff E L 2 (!R.), we have IIP-mfll2 -+ 0 when m -+ 00. Proof. If g

E

C 00 (1Fk) has support in [ -K, K], we can write

and similarly for -2"' ::S x ::S 0. Hence IIP-,.,gll"" ---+ 0. But these functions are dense in Co(IR); given/ E Co(IR.) and E > 0, there exists g E C 00 (1Fk), h E C (1R) so thatf = g + h 0 with llhlloo < E. Then lim SUPm-oo IIP-mfllx ::=: lim SUPm~oo IIP-,.,hlloc < E, which proves the required convergence.

296

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Iff E L 2 (JR), for any E > O,f = g + h, where g is continuous and has support in [ -K, K] for some K > 0, and llhll2 <E. Then for 2m > K, we have -2"' ::::0

X ::::0

K

2"'

===::;.

IP-mg(x)l =2-m

j_K

lgl ::::0 2-mh/(llgll2

11P-mgll2 ::::0 v'4KT"' 12 11gll2 IIP.,JII2 :::: liP-mgll2 + IIP-mhll2 ::::0

lim sup IIP_,JII2 ::::0

IIP-m8ll2 +

E

E

where we have used Lemma 6.3.1 in the last line, in the form IIP-mhll2 < llhlb- Since this holds for every E > 0, we conclude that P. mf ---+ 0 when m ---+ 00. • The above proof contains the following general principle: If a sequence of bounded linear operators has uniformly bounded operator norms and converges to a bounded operator on a dense subset of a Banach space, then it converges on the entire space. To prove that P,J -+ f when n -+ oo, we first prove that this holds on the dense set of continuous functions with compact support. Lemma 6.3.3. Iff

E

C 00 (lR), then P,,f-+ f uniformly and in L 2 (JR), when n -+

00.

Proof. Let f be supported in [- K, K], where we may suppose that K :=: 1. Given E > 0, from the uniform continuity off, there exists 8 > 0 so that If( y) - f(x)l < E I K whenever lx- yl < 8. If 2 " < 8, we have IPnf(x) - f(x) I ::::0 E I hK ::::0 E for all x, which proves the uniform convergence. Integrating over the support off, we have

which proves that IIP"f- fll 2

---+

0 when n

---+ oo.

•

We have thus proved the following theorem. Theorem 6.3.4. The normalized Haar functions {..PJk }j.kEZ form an orthonormal basis of L 2 (IR), in particular we have the L 2 convergent expansion

(6.3.15)

6.3.3. 1 Haar series in Co and Lp spaces The Haar series is well defined for any locally integrable function, hence it makes sense to study the convergence in other spaces of functions. We have treated theL 2 convergence of the Haar series by relating the partial sum to the fundamental theorem of calculus. These ideas can also be used to discuss the uniform convergence of the Haar series in spaces of continuous functions, as well as the norm convergence in LP(JR), 1 < p < oo.

INTRODUC TION TO WAVELETS

297

We first treat the converg ence in the space Co(lR), consistin g of continuo us functions with limlxl->< x,f(x) = 0. This Banach space contains as a dense subspace the set of continuo us function s with compact support, on which we have proved Lemma 6.3.3. It remains to prove that the operator s Pn are uniform ly bounded . We prove a more general estimate on the (larger) space of bounded continuo us function s.

Lemma 6.3.5. For any f

E

Be (lR), we have IP,,f(x)l < llfll=·

Proof.

• From Lemma 6.3 .2, we have for any continuo us function g with compact support, IIP-mgll ---+ 0 when m ---+ CXJ. Since these are dense in C 0 (1R), from the uniform boundedness of IIPn II we obtain IIP-mfll= ---+ 0 when m ---+ CXJ. Meanwh ile, Lemmas 6.3.3 and 6.3.5 show that P,,f ---+ f in the supremu m norm when n ---+ CXJ. This leads to the followin g general proposit ion on uniform converge nce. Proposi tion 6.3.6. Iff E C 0 (1R), then the Haar series (6.3.15) converge s uniformly on the entire real line.

Iff is merely bounded and uniform ly continuo us, we cannot expect a uniforml y converge nt expansio n on the entire real line, as shown by the followin g.

=

Exercise 6.3.7. Let f(x) 1. Prove that the Haar series expansio n (6.3.15) is identica lly zero, especial ly not converge nt to f. Exercise 6.3.8. Suppose that f E Buc (JR), the space of bounded and uniforml y continuo us function s. Prove directly that the one-side d Haar series (6.3.1 1) converge s uniforml y to f.

To treat converg ence in LP(JR), we first prove uniform bounded ness.

Lemma 6.3.9. Letf

E

LP(JR), 1

=:: p

<

CXJ.

Then IIPnfllp < llfllpfor all n

E

/Z.

Proof. Setp' =p/(p- 1) ifp > l. Then Holder's inequality gives X

E

hn ==> IPnf(x) I ::0 2n IPnf<xW ::: 2np

(fn 1/( (f" If<

y) lp dy)

2-n/p'

y) I~' dy) rnp!p'

11Pnf(x )i" dx ::0 rn2nprnp fp' {

~

I/p

hb

lf(y)IP dy

= 11/(y) l" dy. lkn

Summing on k E Z gives the result. This proof also applies in case p _ l I oo = 0 whenever p' appears.

I , by setting

•

298

INTRODIJC'TION TO I·Ul •RIF.R ANALYSI'i AI"D WAYEl ETS

To prove U' convergence, we must check that P,f ~ f and P_,J ~ 0. Lemma 6.3.10. Let

1 ::: p

<

CXJ.

Then II~J- fliP~ 0 when n ~

CXJ.

Proof. The space of continuous functions wnh compact support is dense in L''(lR). From Lemma 63.3 we have uniform convergence on this space. In particular iff is supported in [ -K, K], then for n > N(<=)

~

l-a

IP,,f(x) -f(x)l"dx.:::: 2K<= 1',

•

which shows that IIP,J -Ill, < ~(2K) 1 ' 1 '. It remains to consider P_mf, m ~

Lemma 6.3.11. Let

1

< p <

<Xl.

This puts a new restriction onp.

<Xl.

Then IIP-,JIIP

~

0 when m

~ CXJ.

Proof. It suffices to check this for g continuous with compact support in [-K, K]. If > K, then P "'g is constant on (-2"'. 0). (0, 2"') and zero elsewhere, so that from Holder'~ equality, 2m

K

0 .::=X.::= 2"' ==> IP

mg(A)I

n I' lc--mg(x)l

= 2 "'I {

Jo

=

2

mp

g(y)dyl

iK lg(y)l~'dy

(2K)f'lp'

0

{""' IP-mg(x)ll' dx

Jo

which tends to zero when m the same fashion.

= 2"'2-"'~'ljK lg(y)li' dy! (2K)PIP', -K

-+ <XJ.

The contribution from the negative axis is estimated in •

Hence we conclude the following. Proposition 6.3.12. Let 1 < p < converges in the norm of U'(.IR).

CXJ.

Foranyf

E U'(lR),

theHaarseries(6 .3.15)

In the case p = 1 there is a simple example to show that this proposition is sharp. Exercise 6.3.13. Letf = 1[O.I J· Prove that the Haar series (6.3. 1 5) is not convergent in L 1 (lR).

However this anomaly is not present for the one-sided Haar series. Exercise 6.3.14. Let 1 ::: p < CXJ and let f E U' (lR). Prove that the one-sided Haar series (6.3.11) converges in the norm of LP(JR). 6.3.3.2 Pointwise convergence of Haar series Since the projection operator P, agrees with the average over dyadic intervals, it follows from Lebesgue's theorem that P,f(t) -+ f(t) for almost every t E JR. In particular if/ is continuous at t, then we have lim,_.= ~J(t) = f(t). Iff has a jump discontinuity at a

INTRODUCTION TO WAVEUoTS

299

dyadic rational t, then we note that K, (t. y) = 0 for y < t and sufficiently large n. Then we can write P,,j(t) - f(t

+

0) =

lex. lf(

y) - f(t

+

0)]K11 ( t, y) dy-----+ 0,

n----"' CXJ

to conclude that P,J(t) -

f(t

+

t =

0)

k/2".

n----"' CXJ.

One can also con finn the absence of a possible Gibbs phenomenon for Haar series. Indeed, the kernel K 11 (x. y) ;::: 0 with fo. K 11 (.>.:. y) dy = I. Therefore iff E U'"(lR)

-llflk ::C P,,f(x) ::C llflk. which implies that for any sequence

X

11

----+

x, we must have

-II f lk ::=: lim inf P,,f (x,) ::=: lim sup P,,f (xn) ::=: II f II x. II

II

Exercise 6.3.15. Suppose that f(t) = I for 0 ::=: t < I /3 and that f(t) = 0 for 1/3 < t < I. Show that we have lim inf, P,,f(l/3) < lim sup, Pnf0/3), so that the Haar series diverges at t = I j3.

6.3.4

*Construction of Standard Brownian Motion

The Haar wavelet expan~ion can be used to make an effective construction of the standard Brownian motion process. By definition, this is an indexed family of real-valued functions X 1 (w) where 0 ::=: t ::=: I and cv E Q, where (Q, :F, P) is a measure space of total measure I. In this context, the functions w-----+ X 1 (w) are called random variables. They are assumed to have the following properties:

1. For each 0 S variance

t -

s < t S s: in detail

I, X 1

-

X, has a normal distribution with mean zero and 1

P[w: X,(w)- X,(w) < v] = · --/2rr(t- s)

~-'

2

e-u

/

2 lr-,)

du.

-oo

2. For any subdivision 0 = to < t 1 < · · · < t,v S I, the random variables X,, Xro, ... , x,N - Xr, are independent. 3. For a.e. w, the function t - X 1 (w) is continuous, with Xo = 0. I

From properties (I), (2), it follows that the random variables X 1 ,, . . • , X 1, have a joint normal distribution with mean values zero and covariance matrix defined by E[X1,X1) = min(t,, t 1 ).

Exercise 6.3.16. Prove this. The Haar functions !fr,k are not continuous, so we would not expect to be able to construct the Brownian motion as a Haar series. But the functions t - J~ 1/r k(s) ds are 1 continuous and can be used to construct the Brownian motion. In order to prove the

300

INTRODU CTION TO FOURIER ANALYSI S AND WAVELET S

distribu tional propert ies ( 1) and (2), we will first conside r a general orthono rmal basis of the space L 2 [0, I]. The Browni an motion will be constru cted as the infinite series

(6.3.I6 ) Here ( n) is an orthono rmal basis of the space L 2 [0, I] and (Zn) is a sequen ce of independen t standar d normal random variable s; in detail P[Zn <

yl

=

1 ~

v 2JT

fy

2 e-u 12 du

yER

n = 0, I, 2, ....

-00

Lemma 6.3.17. Suppos e that (n) is an orthono rmal basis ofL 2 [0, I] and (Zn) is a sequen ce of indepen dent standar d normal random variabl es. Then (6.3.16 ) converg es in L 2 (0.) to a limit X 1 (w), which satisfie s propert ies ( 1) and (2). Proof. Let the inner product in the space L 2 [0, 1] be denoted by (,) and let 1rs.rJ be the indicato r function of the interval [s, t]. With these notation s, we can write J,' c/>,(u) du (1rs.rl• ¢,)so that we can compute the variance of the sum (6.3.16) as

=f.( ['

{(X,- X,) 2 dP

Jr:z

n=O

2

c/>,(u) du)

s

00

= Z:::::(lr,.rJ·¢,>2 n=O

=t-s where we have used Parseva l's identity for the orthono rmal basis (¢,).Th is proves that the series (6.3.16) converg es in L 2 (Q); the partial sums of the series are normall y distribu ted with mean zero, so that the limit is also a normall y distribu ted random with mean zero and the asserted variance , proving (1 ). To prove (2), we note that the partial sums of the series define a Gaussia n distribut ion on JRN so that the indepen dence can be inferred from the covarian ce function by showing that the increme nts are orthogo nal in pairs. Now if s < t ::S u < v, we have

00

=

L

(1 r-.rl' ¢,) (I [u.v]' ¢,)

n=O

= Ors.t]• lru.vJ)

,o where we have used the bilinear version of Parseva l's identity and the disjointn ess of the interval [s, t] from the interval [u, v]. This proves the pairwise orthogon ality. Since the vector is multivar iate normal, the indepen dence is thereby proved. •

301

INTRODUCTION TO WAVELETS

6.3.5

*Haar Function Represent ation of Brownian Motion

The normalized Haar functions l/f1k(t) together with the constant function provide a convenient orthonormal basis of the Hilbert space L 2 (0, I). We find it convenient to relabel them as follows: 1/fo(t) = I,

If n :::: I, then we can write n we set

=

1/f,(t)

2!

=

+

k for j

l/l"jk(t)

=

=

0, 1, 2, ... and k

=

0, I, ... , 21 -

I and

2112 l/f(21 t - k).

From the one-sided Haar series representatio n (6.3.1 1), we see that {1/f, (t) }n=O.I.2 .... is an orthonormal basis of £ 2 (0, 1). To display the Haar series representatio n of Brownian motion, we introduce a sequence of independent standard normal random variables Z,, n :::: 0 with (6.3.17)

P[Z, .:5 x] =

Jx e- u v 2n -·= ~

2

1

2

du.

These may be defined on a probability space, denoted (Q, .F, P). The Brownian motion is sought in the form

(6.3.I8)

It is immediate from the orthonormal basis properties of 1/f, that for each t E [0, 1], the series (6.3.18) converges in L 2 (Q, .F, P). From this it is immediate from the proofs of (I), (2) above that X 1 has the required distributiona l properties of Brownian motion.

6.3.6

*Proof of Continuity

We will now prove property (3) of Brownian motion, by showing that the series (6.3. I 8) converges uniformly for almost all w E n. Lemma 6.3.18. There exists M = M(w)

P

[

w: sup n

<: CXJ

so that

/Z,(w)/ ~ < M(w) log n

J

= I.

y'

Proof. From the normal distribution (6.3.17), we have the bound

n

=

0, I, 2, ... ,x > 0.

Setting x = 2~. we have the bound P[IZ,.I ::: 2~] :5 n- 2 , which is the general term of a convergent series. Therefore by the first Borel-Cantell i lemma there exists no(w) < oo a.e. so that n ::: n 0 (w) implies jZ,.(w)l :5 2~. Hence we can set M(w) = max{2, (IZn(w)l)/~, n :5 no(w)}. •

302

INTRODUCTI ON TO HJURIER ANALYSIS AND WAVELETS

Lemma 6.3.19. M 1 (uJ) := 2M(w)vf2 Jog 2.

ds are polygonal functions supported on

Proof. For fixed), the functions S,k (t) := fr.: 1{/ , (s) 1 disjointint ervalsofle ngth2-1 withO:::: sj,(t):::: 2 I and LosAoo2f-l IS}dt)- s,k(s)i :::: I t - sl2112 Thus

I ~-~<J 1

2

21

,..

,.(w)

1'

1{/ 2 r+k(s)

X

dsl < o·~~~-

1

2112 . Hence Losf..s2J-I S;,(t):::: z-;/ 2

L

122'+/.. (w)l

O:_k2~

S,dt) 1

I

•

as required.

From this it follows that the jth dyadic block of the series (6.3.18) is bounded by a constant multiple of --/]2-11 2 , the general term of a convergen t numerical series. Therefore by the Weierstras s M test, this series of dyadic blocks converges uniformly to a continuou s function, denoted t ----+ X 1 (w). This proves property (3), hence we have proved the existence of the Brownian motion process. •

6.3.7

*Levy's Modulus of Continu ity

The method used to prove continuity can be easily extended to obtain a modulus of continuity , first establishe d by Paul Levy ( 1948). This is encapsula ted in the following theorem.

Theorem 6.3.20. There exists M 1 = M 1 (w) so that

if it

-

sl ::::: 8

< ~, then

Furthermo re there exist intervals (sn. t 11 ) with tn - Sn ----+ 0 such that [X1, (tn - Sn) log (I/ (t, - Sn)) is bounded below by a positive constant.

vf

-

Proof. We write the increment of (6.3.18) in two parts: X,(w)- X,(w)

= Zo(w)(t-

s)

+

(t + ,?;;) ~

22

,+, (w)

!

1

1{/2 ;+1- (u) du

where L will be chosen in terms of 8. Now L

Is- tiM• (w)

L ;=0

.jj211 ".

XsJI

INTRODUCTION TO WAVELETS

303

The two sums are estimated by the elementary inequalities L

L

(6.3.19)

,/j2';z s c,.Ji2L;2.

jdl

so that if it- sl ::::: 8, then

IX,- X, I ::::: 1Zo(w)l8

+ M 2 (w)(8.Ji2L 12 + .JirL 12 ).

The final two terms are balanced by taking 82L ~ I. or specifically, L = [log 2 (1/8)], with the result

which completes the proof of the upper bound, since 8 ::::: ../8 for 0 :S:: 8 ::::: I. To prove the second statement, we consider the independent events

A~=

{ w:

Xkz-n -

Xck-1)2-"

:5:

c~} ,

where c is to be chosen. We use independence , the tail of the normal distribution, and the elementary inequality 1 - x ::;: e-x to write p

[

J n (1 -!

2" nA~ = k=l

2"

=

k=l

<Jn

e-u2;2_:!!:!._

)

-./2ii

::::: exp(-2ne-nc 2 /2(1 + o(l)))

= exp(-(2e-c 212 )n(l

+ o(l))).

It suffices to choose 0 < c < J2log 2 so that 2e-c 212 > I, and we have the general term of a convergent series, and by the first Borel-Cantell i lemma the series In;::,,Az < <Xl for almost all w. Therefore for n sufficiently large ni:,A~ fails to occur, in particular for some k, Xk; 2 n - Xck-l >/ 2" ::: c ./ii72", which proves that the Levy modulus is a sharp lower bound also. •

I:;:,

Exercise 6.3.21. Prove that the elementary estimates (6.3.19) hold with the constant

c, = c2 =

1/(1 - 2- 112 ).

Hint: Compare a sum with an integral, which can be estimated by partial integration.

6.4

MULTIRES OLUTION ANALYSIS

In this section we return to the construction of general wavelets. The main features of the Haar wavelet expansion can be abstracted as follows.

Definition 6.4.1. An orthonorma l wavelet is a function \II doubly indexed set

{2il 2

\lf(2 1 t -

L 2 (lR) such that the k)}J,k.eZ is an orthonorma l basis ofL 2 (.1R). E

304

INTRODlJCTI ON TO FOURIER ANALYSIS AND WAVELETS

We have already seen that the Haar function provides an example of an orthonormal wavelet. To develop a systematic method for producing orthonorm al wavelets, we introduce another notion. which generalize s the Haar constructi on.

Definitio n 6.4.2. A multiresol ution analysis ( MRA) is an increasing sequence of

subspaces { V,}

C

L 2 (JR.) defined for n

··· c together with a function

E

V~1

E

Z with

C Vo

c

V,

c ···

L 2 (JR.) such that

(i) U~~oo V, is dense in L 2 (1R.), n~~= V, (ii) f E V, if and only iff(2~"-) E Vo

=

{0}

(iii) {(x- k)}kE:Z is an orthonorm al basis of Vo. is called the scaling function of the MRA. Clearly Vo is uniquely defined by through (iii), and V, is further uniquely determine d through (ii). However we do not require that be unique; a given family {Vn} may have several different possible choices of . The job of the theory is to show that there exist other nontrivial examples of multiresol ution analyses, to construct the correspon ding orthonorm al wavelet bases and to discuss their properties .

Example 6.4.3. Let V, be the set off intervals hn = [(k- 1)/2", k/2").

E L 2 (1R.),

which are constant on the dyadic

Clearly all of the properties are satisfied, with the Haar scaling function (x) 1ro.l) (x).

Example 6.4.4. Let V, be the setoff on each dyadic interval hn-

E L 2 (JR.), which are continuou s and linear

It is straightfo rward to see that properties (i) and (ii) of Definition 6.4.2 are satisfied. The choice of a scaling function is less obvious and will be obtained in this section. This example is related to piecewise linear spline approxima tion. In order to develop scaling functions for more general MRA systems, we first develop the necessary properties of orthonorm al systems and Riesz systems.

6.4. 1

Orthono rmal Systems and Riesz Systems

Let H be a Hilbert space with inner product(, ). A set of vectors (xn) is an orthononn al system, by definition , if (Xn, Xm) = Omn·

INTRODU CTION TO WAVELET S

Lemma 6.4.5. The set (x,) is orthono rmal comple x number s (a,), we have

305

if and only if for every finite set of

(6.4.1) Proof. If (x,) is orthono rmal, then the left side of (6.4.1) is the finite sum

Lama, (x",x,) = La,a, = L la,l 2 . tn,n

n

n

Convers ely, if (6.4.1) holds, first we choose a, = 8,N to obtain (xN, XN) = I. Then choosin g a,= 8,M- 8,N with M =1= M gives 2 llxN- xMII 2 2 - (xN,xM )- (xM,XN) hence 0 (XN, XM) (XM, XN). Replaci ng XM by iXM We Obtain 0 (XN, XM) - (XM, XN ), from

=

+

=

=

=

which the result follows.

•

This leads us to formula te a more general concep t.

Definit ion 6.4.6. Let H be a Hilbert space. A set of vectors (x,) is, by definiti on, a Riesz system, if there exist constan ts 0 < c < C < oo such that for any finite set of comple x number s (a,) (6.4.2)

c

L

la,l

2

:::

n

II

L:a,x,f < cL: ja,j n

2

.

n

Clearly any orthono rmal system is a Riesz system , where c = C = 1. If (xn) is a Riesz system , then the vectors (xn) are linearly indepen dent: Ln anXn = 0, implies that an = 0 for all n. Examp le 6.4.7. Let H

=e

(ll~.) and Xn (t)

=

A (t- n) where A is the tent functio n

i\(t) = (1- itl)l[-I. I)(t).

To verify the Riesz propert y, we note that the linear combin ation A(t) Ln ani\(t - n) is piecew ise linear with A(n) =a, for all n. Hence

Jm{ IA(t)i 2 dt = IR

ln+l i(n + 1 -

L nEZ

We use the Cauchy inequal ity i2abl < lai 2

i

2

IA(t)i dt:::

IR

+

t)a,

n

+

( t - n)an+ti 2 dt

ibi 2 to obtain the upper bound

~ L(lan l 2 + lan+tl 2 )

L

=

nEZ

ianl 2

nEZ

and the lower bound

1IA(t )l 2 dt

>

~ L(ian l 2 +

R

Therefo re (6.4.2) is satisfie d with c

lan+Ii

nEZ

=

~, C

=

1.

2

)

=

!_ 3

L nEZ

ja,j 2 .

306

INTRODUCTI ON TO FOL;RIER ANALYSIS AND WAVELET<;

The next propositio n characteri zes Riesz systems and orthonorm al systems in terms of the Fourier transform . Propositi on 6.4.8. Let condition s are equivalen t.

E

L 2 (JR) and 0 < c < C < oo. The following two

(i) The periodize d square of the Fourier transfonn satisfies the double inequality

(6.4.3)

I c" L>"(~ IEZ

+I) I'

"c

a.e . .;:

E

lR

(ii) {(t- m)},,E:z is a Riesz system with constants (c, C).

Recalling that orthonorm ality is characteri zed by c = C = l, we obtain the following useful characteri zation. Corollary 6.4.9. The translates {(t- m)},E;z of E L 2 (lR) are orthonorm al and only ij ;;z I cP(l; + 1) 12 = l for almost every E JR.

Ltc

s

if

The sum in (6.4.3) is well defined a.e., since we may compute the integral over the unit interval as < 00

hence the integrand is finite a.e. and defines a !-periodic function. Before giving the proof of Propositio n 6.4. 8, we give some examples of the computational power of these relations. Example 6.4.10. Haarscal ingfuncti on: Let (t) = 1!0.n(t).

Clearly the translates (t- m) are orthonorm al, hence c = C = 1. The Fourier transform is computed explicitly as (s) = e-irr~sin(ns)/ns. so that (ii) gives for a.e . .;:

which is equivalen t to the partial fraction expansion of the function csc 2 7r.;:. Noting that the series on the right converges uniformly on each finite interval, we infer that the series defines a continuou s function, hence the equality holds for every .;: . Example 6.4.11. Shannon scaling function: Let (t) = sin(nt)/n t fort =1= 0, with (0) = I .

The Fourier transform is ($) = 1 1 _~.~ (s). thus we have LiEZ l(s + 1)1 2 = 1 a.e., 1 since for I; ¢. Z, all terms in the se-ries are zero except for one which = 1. Hence {(t- n)}nEZ is an orthonorm al system in L 2 (1R). This is the orthonorm al system that

INTRODUCTI ON TO WAVE-.LETS

307

occurs in the Shannon sampling formula, studied in Chapter 4 and to be redone in the context of wavelets. Proof of Propositio n 6.4.8. We first establish an identity for the~ Riesz sum in terms of the Fourier transform. The Fourier transform of

(!;)whe re A is the 1-periodic function

From Parseval's identity for Fourier series. we have j 1 \A(i;) 1 2 dt; 0 Parseval's identity for Fourier transforms, it follows that

LnEC£:

\a, 1 2 . From

(6.4.4)

=

~ 1'~

=

~

11

=

1

IA(l;)l

1

2

2

IA
+

/)12 dl;

(.2: l(~

+ /)\ 2 )

\A(l;)\2\(t ;

1

2

0

i
d~.

To prove that (i) implies (ii) in Proposition 6.4.8, we simply note that the integrand in parenthese s in ( 6.4 .4) is bounded above by C and thus the integral is bounded above by 1 C .fr1 \A(.!;)\ 2 = C L,. Ia, I!. similarly for the lower bound, which proves (ii). To prove that (ii) implies (i). we use the abov~ transforma tions to rewrite the Riesz condition (6.4.2) in the form

(6.4.5)

cs

fol

\A(l;)\2

(L,Ez: l
\A(l;)\2dl;

S C.

This holds for every trigonomet ric polynomia l A(l;). Taking a sequence AN that converges boundedly to the indicator function of the interval [a, b] c (0, 1) (the partial sums of the Fourier series of 1 1" ;, 1 will suffice for this purpose), we obtain

This holds for every interval (a, b). Taking a sequence with (a, b) ---+ {x} and applying Lebesgue's differentiat ion theorem, we obtain (i).

•

Example 6.4.12. Returning to the tent function A (t), we have sin lT~ )4 (L \A(~+ /)12) = L ( 7r(~ + l) IE''l'.

fEZ

308

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

This sum can be evaluated by repeated differentiation of the series from Example 6.4.10, namely rr 2 esc 2 rr~ = """' ~ (~ IEZ

1

+

l)2

to obtain the identity . sm rr 5t-

"""'

fa

(

JT(~

+

l)

)4

I

-

-(1

3

+

2cos 2 rr~).

which again reaffirms that c = 1/3, C = I for this Riesz system. Exercise 6.4.1 3. Check the details of this computation.

We can use Corollary 6.4.9 to estimate the support of the Fourier transform of a scaling function. Corollary 6.4.14. Suppose that E L 2 (JR) and that { (t - k) hEZ is an orthonormal set. Then Isupp I > 1, with equality if and only if = 1K for some measurable set K with iKi = 1. Proof. From Corollary 6.4.9 we have Lta lei>(~ From Parseval's identity

+ /)1 2

-

I a.e., hence lci>(~)l < I a.e.

which proves that 1supp ci> I ~ l. If equality holds, then the middle terms give

But the integrand is nonnegative a.e., hence I -lci>(~)l 2 means that ci> = IK a.e., where IKI = lsupp 1 = l.

=

0 a.e. on the support of ci>, which •

Proposition 6.4.8 allows us to obtain the following orthogonalizatio n procedure to generate scaling functions from a Riesz sequence. Proposition 6.4.15. Let EL 2 (JR) be such that {(t - m)}mEZ is a Riesz sequence. Then there exist complex numbers bn with LnEZ ibn 12 < oo such that {r (t- m)}mEZ is an orthonormal sequence, where <1> 1 (t) := LnE:Z bn(t- n). Furthermore, the span of {r ( t - n)}nEZ equals the span of {(t- n)}nEZ· Proof. From Proposition 6.4.8, it suffices to find bn such that LtEZ 1 ci> 1 (~ From the definition of <1> 1 , we have

+ I) 2 = 1

1 a.e.

INTRODUC TION TO WAVELETS

:= L lcPI (~ +

309

B (!;) cP (!;)

/)1 2 =LIB(~+ 01 2 lcP(~

feZ.

+ 01

2

IE:...

=

IB(l;)l 2

L

lcP(~

+

1)1

2

.

IEZ

Therefore we must choose the constants bn so that

1

IB(~)I2:=1I;:bne-hm
nEZ

lcP(l;+/) 12. /e2

Clearly there are many possible solutions . The simplest one is to take the positive square root, leading to (6.4.6)

This is clearly the Fourier transform of an L 2 function, since the denomina tor is bounded above and below by the Riesz condition . To prove the last statement , we need to study the equation (6.4.7)

L nEZ

an(t- n)

=

L

Cnl ( f - n)

neZ

and to show that, given (a,) E / 2 (';!,), we can solve for (c of Fourier transform s, this is written

11 )

E

l 2 (Z) and conversel y. In terms

Recalling the relation between <1> 1 and is a special case, with this substituti on we see that (6.4.7) is implied by the identity

an

= b,,

Cn

= 8,o. Making

(6.4.8) := B(l;)C(l;) .

But we have shown above that c-J ::::: I LnEZ bne-Zrrm<. I ::::: c- 1 from the Riesz property. Hence, given (a 11 ) E / 2 (Z), we may solve (6.4.8) uniquely by taking en as the Fourier coefficien ts of the !-periodi c function A(l;)/B(l;). Converse ly, given C , one simply refers 11 to (6.4.8) and chooses a 11 as the Fourier coefficien ts of the right side. •

The above proof shows that the set of function s describe d by the left side of (6.4.7) when (a,) E / 2 (2) is identical to the set of function s describe d by the right side when (c,) E / 2 (2). But the set on the right is a closed subspace of L 2 (lR), from the orthonor mality of { 1 (t - n)} nEZ. Hence the set on the left is also a closed subspace , which is the closed linear span of {(t- n)},a.

310

INTRODUCTION TO I Ol'RIER At'.AI YSI'-; ,'\NO \V/\Vl-1 I'T<;

Example 6.4.16. In the case of the tent function <1> Fourier transform is obtained as ~

<1>,(.;)=

(sin rr.; jrr.;-> 2

.J(T +

2cos 2 rr~)/3

A, the orthogonalized

.

This corresponds to the MRA of Example 6.4.4, where the functions are continuous and piecewise linear on each dyadic interval. We write <1> 1 (t) = Ln bnAU- n), which is clearly a piecewise linear continuous function with <1> 1 (n) = b 11 • The coefficients are obtained from the Fourier expansion 1

(6.4.9) ./ (I

+

2 cns 2 rr E.) /3

-_ "" L..,; b ne 2:r ml; . nco:·.

Since the left side is a real analytic function, the Fourier coefficients have an exponential decay. Exercise 6.4.17. Show that there exist constants K > 0, Ke-fllnl and obtain an estimate for {3.

6.4.2

f3

>

0 so that Ibn I <

Scaling Equations and Structure Constants

The axioms describing an MRA system are not completely independent of one another, as we will show. First we note a simple consequence of properties (ii) and (iii) from Definition 6.4.2. Proposition 6.4.18. For each j basis ofV1•

E

Z. {21 12 (2 1t - k)h<=£7. is an orthonormal

Proof. From property (ii). ~ and \1( 1 are isomorphic by virtue of the map x --+ 2· 1x. The indicated functions are clearly orthonormal. Now we pull back to V 0 and use property (iii) .

•

sum

In order to proceed further, we discuss the consequences of the inclusion Vo c V 1 • Since V 1 is spanned by translates of { (2t - n)} 11 ,~::_:, we have the L 2 convergent

(6.4.10)

(f)=

L

a 11 <1>(2t- n)

nEG

where the structure constants satisfy LnE:L la 11 12 < CXJ. Relation (6.4.10) is called the scaling equation and will be instrumental in the sequel. Example 6.4.19. If (t) = lro.ll(t), then clearly (t) = <1>(21) + <1>(2t- 1) is the scaling equation, with structure constants a 0 = 1, a 1 = 1 and a, = 0 otherwise. Exercise 6.4.20. Suppose that E L 1 (lR) n L 2 (1R) satisfies the equation (2t) + <1>(2t- l ). Prove that for some c, (t) = c1 10 . 11 (t) a.e.

INTRODUCTION TO WAVELETS

Hint: Take the Fourier transform of both sides and iterate to solve for

311

.P.

The next exercise shows that a similar two-scale difference equation can have radically different solution behavior. L 1 (lR) satisfies the functional equation 3<1>(3t- 1). Prove that (t)- 0.

Exercise 6.4.21. Suppose that 2<1>(t) = 3<1>(3t) Hint: Show that

+

E

¥ 0 must agree with the Fourier transform of the Cantor measure.

The next example shows that the existence of a scaling equation does not follow from the orthogonalit y properties.

Example 6.4.22. Let(t) = (an)

E / 2 (Z)

1r-~.~J(t).

Wesuppose that(6.4.JO )holdsforso me

and obtain a contradictio n.

To see this, we have from the orthogonalit y of { (2t - n) }nE:Z

a, =

2l

(t)(2t- n) dt.

But this integral is nonzero unless n = 0, ± 1, in which case we obtain a 0 = 1, a± 1 = But this leads to a contradictio n on the interval ~ < t _::::: ~, where the left side of (6.4.1 0) is zero but the right side is nonzero. We record some properties of the structure constants.

!.

Proposition 6.4.23. The structure constants obey the following properties: (6.4.11)

ak =

2l

(t)(2t- k) dt,

L

(6.4.12)

lakl

2

kEZ

= 2

kE:Z

L

(6.4.13)

ak'a2k+k' = 28k0

(Kronecker delta).

k'E:Z

/falso

E

L 1 (lR), JTJ?. # Oand(6.4.1 0) converges inL 1 (lR), then

(6.4.14) Proof. Since a, I h are the Fourier coefficients of <1> E V 1 with respect to the orthonormal basis vtz(2t- k), we have ad-viz = JTI{ (t)h¢(2 t- k) dt, as required. Parseval's theorem gives Lk<,z !a, 12 /2 = II <1> 11 2 = 1. To prove ( 6.4.13 ), we begin with (iii) of definition 6.4.2, in the form

l

(t- k)(t) dt

=

00k·

312

INTRODUC TION TO I"OURIER ANALYSIS AND WAVELETS

Substitut e (6.4.10) and use Parseval' s identity and orthogon ality to write 8ok

=

L 1

2

f

ak'ak"

k'.k''

z=

JTR

(2t- 2k- k')(2t - k") dt

ak·ak"·

2k+k'=k"

which is the same as (6.4.13). In particular , taking k = 0 we retrieve a new proof of (6.4.12). If, in addition, we have E L 1 with JrR -=1= 0, then we integrate (6.4.1 0) term-byterm to obtain

r <~> dt = z:::: r ¢<2r- k> dr JJR ak

JTR

which we divide by

kEZ

•

JrR to obtain (6.4.14 ).

It is often useful to work with (6.4.1 0) in the Fourier domain. The Fourier transform of (2t - n) is easily obtained :

l

(2t-

l

n)e- 2 rra~ dt = & =

(u) exp[

-2rri~ ( n; u) Jdu

&e-inrr~~(~)

so that ( 6.4.1 0) is written (6.4.15) where the scaling filter is defined by (6.4.16)

mo(O :=

~ L ane-2rrin~. nEZ

The existenc e of a scaling equation can be formulat ed in the Fourier domain as follows where L 2 (1R/Z) denotes 1-period ic function s that are square-i ntegrabl e on any period. Proposi tion 6.4.24. E L 2 (1R) satisfies a scaling equation (6.4.10) with (an) E l 2 (Z) if and only if there exists m 0 E L 2 (1R/Z) so that (6.4.15) holds, in which case (6.4.16) holds. In particul ar an = 2 ~~~~ m 0 (~)e 2 rrin~ d~. 2 Proof. If satisfies (6.4.10), then we can take the Fourier transform of both sides to obtain (6.4.15). Converse ly, if (6.4.15) holds with mo E L 2 (lR./Z), we define an = 2 J~~~ 2 mo(5)e 2 rcm< d~, so that (6.4.16) holds. The Planchere l theorem ensures that the map -+ is bijective. Since the right side of (6.4.15) is the Fourier transform of Ln an(2t - n), it follows that (t) = Ln an(2t - n), which was to be proved. •

INTRODUCTION TO WAVELETS

313

Example 6.4.25. The Shannon scaling function is I

sin(.rrt) = (t)= .rrt

!2

-~

e 2rrtti'; dl:: ., .

The scaling equation can be obtained at the level of Fourier transforms by solving (6.4.15) as follows: 1r-t.1J(/;) = mo

(~) 1 r-t.~J (~) ·

Therefore we need to choose the coefficients so that mo(l;) = 1 for II; I < ~and mo(/;) = 0 for ~ < II; I < The structure constants are obtained from ( 6.4.1 0) as the Fourier coefficients

!.

Thusao = 1 andan = (-2/n.rr)sin( n.rr/2). Exercise 6.4.26. Consider the spline function A(t) = (1 - lt1)1r- 1• 11 (t) with cP(/;) = (sin .rrl; /.rr/;) 2 . Show that A satisfies a scaling equation (6.4.10) and exhibit the scaling filter m 0 (1;). Use this to infer that its orthogonaliz ation, defined

by cP(/;) = A(/;)/ )Lt;EZ IA(/;)1 2 • also satisfies a scaling equation (6.4.10).

6.4.3

From Scaling Function to MRA

We now prove an important theorem, showing the existence of MRA systems under useful hypotheses. Theorem 6.4.27. Suppose that E L 2 (JR) is such that

(i) The translates {(t- m)}mEZ are orthonormal . (ii) (t) = LnEzan (2t-n), anL 2 -convergent sum, with LnEZ lanl 2 < (iii) The Fourier transform cP(I;) is continuous at I; = 0 with lcP(O)I = 1.

CXl.

Define\')= span {(2it- k)hEZ· Then{\')} defines an MRA. Proof. The scaling equation (ii) implies that Vj c V,+l· Now let ct> k(t) 1 and let P1 be the orthogonal projection on the space V,. In detail

PJ =

=

2JI 2 ct>(21t- k)

L (/' ct>jk) ct>jk. kEZ

These projection operators satisfy the bounds IIP1 /II ::=:: 11/11. To prove the MRA property, it suffices to show that limj~oo P1 / = f and lim1 ~-oc P1 f = 0 for allf E L 2 (~). This is done in two separate lemmas.

314

INTRODUCTION TO FOURIER ANALYSIS AND\,, ' :' : ·.,.,

Proof. Since IIP1 11 = 1, it suffices to prove the result on a dense set, e.g., L 2 functions with compact support. Iff has support in [ -R, R], then

fu 2 (f: = 11/112 fu f_~::J2~R

=

11111:

1

14>(2

1

s- k)l ds) 2

j
~, then these integrals are over disjoint intervals whose union is written U 1 UkEJ::(-k- 2 1 R, -k + 2 1 R), with n1 U1 = Z, which has Lebesgue measure zero. Therefore

If 2 1 R <

by Lebesgue's dominated convergence theorem.

•

To proceed further, we now tum to the Fourier domain and prove a useful identity. L 2 (JR) with a Fourier transform j that is bounded and supported in [-R, R]for some R > 0. Thenfor 2j-i > R we have Lemma 6.4.29. Let f

E

(6.4.17) Proof. We use Parseval's identity to write

L I(PJ, l

IIPJII 2 =

KE.Z

2

kEZ

where we have used the fact that the Fourier transform of the function k is explicitly 1 written 2-1 12 e- 27"kl;2 1 $(2-1 ~). Now if 21 - 1 > R, the last integral is equal to the integral on the interval L-21 -· 1 , 2'- 1 ], where the functions {2-1 1 2 e 2 ,.;k~r1 }kEZ form an orthonormal basis. Moreover,~ ---* j(~)(2-1 ~) E L 2 (-2i-I, 21- 1 ), so that by Parseval's theorem fm

1:--.:TRODUCTION TO WAVELETS

Fourier series, we have for 2'-

1

315

> R 1 '

2

IIPJII =

I

~~~ lf(~ll 2 d~

JR lf(r 0I 2

=

1

2

dt;.

- R

•

which completes the proof.

Corollary 6.4.30. Suppose that the scaling function satisfies the additional condition that (l;) is continuous at l; = 0 with l(O)I = 1. Then for any f E L 2 (JR), IIP1 f - !II ---+ 0 when} ---+ CXJ. Proof. Since P, is a contraction, it suffice~ to prove this on the dense set off whose Fourier transforms have compact support and are bounded. Furthermore, from the projection property, we have IIP,/11 2 = 11/11 2 - llf- PJII 2 • so we must show that IJP1 /II---->- 11/11. Using the hypothesis, we see that 1<1>(2-'/;)l converges uniformly to l on compact sets, so that (6.4.17) give;, for 21 1 > R

IIP,/11 2 = ---->-

1:

J:

llcO(r1 )/;l 2 dt; 2

1lct;)l dt;

= llfll".

•

which completes the proof.

Combining the above lemmas and corollaries completes the proof of the theorem. The following exercise provides a one-parameter generalization of the Shannon wavelet. Exercise 6.4.31. Let K = [a - 1, a] where 0 < a < 1 and set that is the scaling function of an MRA.

1K. Prove

Hint: First check that has orthonormal translates. To find the scaling relation, it suffices to find m 0 E L 2 (1R/Z), by solving <1>(2/;) = m 0 (t;)(t;).

Example 6.4.32. One should not infer from the continuity at I; = 0 that is continuous elsewhere, much less that E L 1 (lR). Consider the Shannon scaling function, where = 1 1_ ~. ~ 1, which is continuous at!; = 0 but is discontinuous at c-

5-

6.4.3. 1

±.!.

2'

..

Additional remarks

We note some additional relations between the above notions. • If we have an MRA, the condition I (O) I = 1 actually follows from the apparently weaker condition that (l;) is continuous at l; = 0. To see this, apply Lemma 6.4.29 to a function! whose Fourier transform is bounded with compact support. Since we

316

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

assume an MRA, it follows that P1 f ~ have for 2i > R,

=

llff

1<1>(0)1 2

/_:

f

when}~ oo. Taking limits in (6.4.17), we

I]Cl;)l 2 dl;

=

2

l(O)I 11Jf

by Parseval 's identity. Hence I (0) I = 1, as promised. • Theorem 6.4.27 remains true if one weakens the condition l(O)I = 1 to (O) =I= 0. To see this, suppose thatf is orthogonal to UJEZ~- Then P1f = 0 for all} E Z. Given E > 0, there exists g whose Fourier transform is bounded and supported in ( -R, R] for some R > 0 and so that II!- gil < E. Hence IIP1 gll = IIPJ(g- f) II < E. Applying Lemma 6.4.29 yields the estimate E

2

>

IIPJgll 2 = /_: lg(l;)l 2 1(2-1 01 2

~ l(O)I 2 11gll 2 2: l(O)I\11!1 1- £) 2 •

This holds for every

E

> 0, which is a contradictio n if E is sufficiently small.

• The continuity condition (iii) in Theorem 6.4.27 can be weakened to lim l(2-1 l;)l = 1 }---"> 00

a.e.

.;

E JR.,

and this condition is also necessary. Indeed, the sufficiency is apparent from application of Lemma 6.4.29 to f, whose Fourier transform is bounded with compact support. To see the necessity, we anticipate a result from the next section, that lmo(l;)l ::::: 1 a.e. From this it follows that

In addition 1 = LiE:Z l(l; + 1)1 2 > l(l;)l 2 , so that we have the existence of the limit g(l;) = 1im1 -">oo 1<1>(2-1 !;)1 and g(l;) < 1. Applying Lemma 6.4.29 toj with]= lr-I.IJ. we see that

where we have applied the Lebesgue dominated convergence theorem, thanks to the bound lg(l;)/ < 1. Hence f~ 1 (1 - /g(l;)l) dl; = 0 where the integrand is nonnegative , hence g(l;) = 1 a.e. • If the scaling function E L 1 (JR.) n L 2 (JR.), then the conditions of Theorem 6.4.27 are necessary as well as sufficient. To see this, note that E L 1 (JR.) implies that is continuous, especially at t; = 0. If generates an MRA, then we can apply Lemma 6.4.29 to f =1= 0 whose

INTRODUCTION TO WAVELETS

317

Fourier transform is bounded with compact support. Taking j ~- oo in (6.4.17), we obtain 11/11 = lcP(O)IIIfll. hence lcP(O)I = 1, as promised. • If the scaling function 1:1> E L 1 (ffi.) n L 2 (ffi.), then cP (l) = 0 for 0 =I= l E Z and LkEZ t:l>(t - k) = cP(O) a.e. Indeed, from the orthonormali ty relation, we have a.e. 1 > IcP (~) 12 + IcP (~ + l) l2 . But cP is continuous, hence we can take ~ ~ 0 avoiding the exceptional set to obtam lcP(l)l < 0, which was to be proved. From this it also follows from the Poisson summation formula that the periodized scaling function LkEZ t:l>(t- k) = cP(O) a.e., since its Fourier coefficients are all zero except for one term. • If the scaling function has compact support with JJ<: t:l>(x) dx = 1, then the Fourier transform argument of Lemma 6.4.29 can be avoided. This is formulated as follows.

Proposition 6.4.33. Suppose that 11> is the scaling function for a compact MRA with t:l>(x) dx = 1. Then u}EZ vi is dense in L 2 (ffi.).

JR

Proof. The orthogonal projection P1 onto 1-j is given by PJ = 2 1

L

<1>(2 1 x - y) (

yEZ

f

J'R

f(y)(2 1 y - y)dy)

and satisfies (6.4.18) Therefore, to show that P1 f --+ f in L 2 , it suffices to prove that (6.4.19)

IIP;/11

--+

llfll-

Since the operators P1 have norm 1, it suffices to prove (6.4.19) on a dense set, e.g., linear combinations off= /A, where A= [a, b]. If supp E [ -M, M]

= 2-;

L y<=Z

= 2 -; (

(1

(x-

y) dx)2

2JA

L yE[2la+M.2lb-MJ

+

L

+

Yit(21a-M.2lb+MJ

L

)

d!St(y.21A):SM

(1

A

(x-

1

y)dx)

2

In the first term the integral is one, while in the second term the integral is zero. In the third term the integral is less than II 11 1 and the number of such terms is less than 4M. Thus

2-J

~ (1JA (x- y) dx)

2

= 2-J

=

2Ja+M~2ib-M I + 0(2-j)

(b- a)+ 0(2-1 ).

Hence liP, lA II~ --+ b - a = lilA II~, which completes the proof.

•

318

INTRODUCTION TO f·Ol!RIER ANAl YS!S ·\ND WA\'l I l-'1 '>

6.4.4

Meyer Wavelets

The previous examples of scaling functions include the Haar wavelet-where the scaling function has compact support but is not smooth. At the other extreme we have the Shannon wavelet, where the scaling function is infinitely differentiable but has slow decay at infinity. Theorem 6.4.27 allows one to construct a large class of new scaling functions, including the Meyer wavelets, where the scaling function is infinitely differentiable and rapidly decreasing (Schwartz class S). This general class also includes the Shannon scaling function from Example 6.4.25 as an extreme case. We begin with a function 8 (!;) defined on the interval 0 :S !; :::; 1 satisfying the following properties: (6.4.20)

0.:::: 0)(.;).:::: 1,

(6.4.21) (6.4.22)

8(!;)

!;

---+

(6.4.23)

0)(/;)

+

0)(1 -.;) = I.

is monotone decreasing,

0)(/;) = 1

The symmetry condition (6.4.21) implies that 8(.!;) = 0 for~ .::::!; .:::: L C-9(~) = t while the monotonicity condition (6.4.22) shows that C-9(/;) ~ ~ for 0 _:s !; ~-We extendC0tothereallinebysettingC-)(/;) = (0(-l;)for-1.::::!;.:::: Oandsetting8(!;) = 0 for I.!; I > 1. The resulting function is even on the entire axis and satisfies 0.:::: 8(/;) < 1. Now we define

:i

(6.4.24)

t E JR.

Proposition 6.4.34. (t) is the scaling function of an MRA system satisfying the conditions of Theorem 6.4.27 and is of class ex with all derivatives bounded: lU>(t)l < C,. If, in addition, I; ---+ y'E>(l;) is of class Ck, then itkU>(t)i < Ckt for all real t and I E z+. In particular if.; ---+ ../8 (/;) is of class C 00 , then E S. Proof. We first check the conditions of Theorem 6.4.27. Since e is supported in an interval of length 2, the sum Lta I¢(~ + I) 12 = LtE:: 8(~ + /) consists of at most two nonzero terms, of the form 8<~>

+

8<~

+I>=

ec-~>

+eo+ n

=

I,

which proves the orthonormality of {(t- m)LnEZ· To prove the scaling equation, define

mo(~) = /8(2~) m 0 (~ + 1) = m 0 (~), especially =I= 0, so that the equation ¢(2~) =

and extend m 0 to the real line as a !-periodic function: moE L 2 (~/Z). In addition 8(~) = 1 whenever 8(2~)

mo(~)cP(~) holds for J~l _:: : ~- In addjtion m 0 (~) is zero for ~ _:::: ~ _::::j, so tha~ we have the scaling equation <1>(2~) = m 0 (~)<1>(~) for all~ E ~. since both (~) and <1>(2~) are zero when I~ I > ~, while m 0 is zero when < I~ I < ~. Finally, the condition (6.4.23) guarantees that <1> is continuous at~ = 0 with_ I¢(0) I = 1. Hence by Theorem 6.4.27 there exists an MRA corresponding to <1>. Since¢ has compact support, is of class c= and we have the bounds I
*

t

INTRODUCTION TO WAVELETS

319

derivatives, we can write

•

which completes the proof.

Example 6.4.35. /f8(~) = lfor -~ < ~ < ~·then 8(~) = Ofor I~ I> ~with 8(±~) =~-This gives the Shannon scaling function (t) = (sinnt)/nt.

6.4.5

From Scaling Function to Orthonormal Wavelet

Once we have a scaling function satisfying the hypotheses of Theorem 6.4.27, 1t 1s relatively straightforward to construct a corresponding orthonormal wavelet, namely a function \II E L 2 (1R.) so that {21 12 \IJ (2jt- k) hEZ is an orthonormal basis of the orthogonal complement Vj+ I e ~. This general construction will specialize to yield the Haar function in case = lro.IJ· · To describe the defining equations on \II, it suffices to take j = 0. Since \II E V 1 , which is spanned by {(2t- n)}nEZ• there exists an L 2 -convergent expansion (6.4.25)

\II(!)=

L

bn(2t- n).

nEZ

But \II must be orthogonal to V 0 , namely (6.4.26)

L

(t-

k)~(t) dt

=

o

Vk

E

Z.

These are translated into the Fourier domain as follows: (6.4.27)

L.: 1"'u + ~>1

2

= 1,

IEZ

(6.4.28) (6.4.29)

Vk

E

Z,

(6.4.30)

The !-periodic function mr is called the wavelet filter. It allows us to pass directly from the scaling function to the wavelet via (6.4.25). It remains to periodize these relations. At the same time we formulate the periodized version of (6.4.15) from the previous discussion. The idea is that the scaling equations allow us to rewrite the orthogonality relations as identities on the circle IR/Z in terms of the scaling filter and wavelet filter.

Proposition 6.4.36. Suppose that is a scaling function of an MRA with scaling .filtermo defined by (6.4.16). Then m 0 (0 satisfies (6.4.31) a.e.

320

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

( i)

if \11 is an orthonormal wavelet with respect to <1>. then the 1 -periodic functions m 0 (~). m 1 (~)satisfy the relations (6.4.32)-(6.4.33 ), a.e. lmo(01 2 + Jmo (.;

(6.4.31)

+ ~) J2

= 1,

lm,(~)l 2 + Jm, (~ + ~)J = mo(~)m, (~) + mo (~ + ~) m, (~ + 2

(6.4.32) (6.4.33)

1,

n

= 0.

(ii) Conversely, given m 1 E L 2 ("'1R./Z) satisfying (6.4.32) and (6.4.33), if we define \II by (6.4.28), then {\11(2lt- k)}kE;z is an orthonormal system in Vt e Vo.

The equations (6.4.31) and (6.4.32) suggest the term quadrature mirror filter for the functions m 0 (s), m 1 (~).since the point~+~ is the mirror image of~ in the circle of unit circumference, with respect to which the quadratic functional equations are satisfied. Proof. We apply Corollary 6.4.9 to formula (6.4.15) summing separately over the odd and even IE Z.

=

L:: 1<~><.; + 2k)l

2

kEZ

=

+

L:: 1<~><.; + 2k + 1)1

r

t; lmo ( + + 1) rI<~> ( + lmo ( ~) 1 ~ I<~> ( ~ + k) 1 + lmo ( ~ + 4) rt; I<~> ( ~ + 4+ k) 1

t; lmo (

l;

~ 2k) I<~> ( l; ~ 2k) 12 +

2

=

=

2

kEZ

I mo

(

~)

2;

2

2

1

l;

+

Imo

(

~ + 4)

l;

2;

+

1) 12

2

2

1

where we have applied Corollary 6.4.9 twice in the last line, thus proving (6.4.31). To prove (6.4.32), we replace by \II on the left and mo by the !-periodic function m on 1 the right. Applying (6.4.27) and (6.4.28), we see that with these replacements, all of the above computations apply and we obtain (6.4.32). To prove (6.4.33), we periodize (6.4.29) by writing

11 = 11 (r= =

L

(l;

IEZ

0

+ l)J,(l; + l)e-2rrm< dl;

0

(l;

+ I)J,(l; +I)) e-2rrin< dl;.

IEZ

The Fourier coefficients of the indicated !-periodic function are all zero, hence (6.4.34)

L IEZ

(t;

+ l)J,(l; +I)= 0

a.e. l;.

INTRODUCTION TO WAVELETS

321

Now we can apply the same transformatio ns that were used to prove (6.4.31) above: in detail

o

=I:¢<~+ 1)\i!<$

+ /)

IE'f:

=I: <~ + 2k)~<~ + 2k) +I: <~ + 2k + o~<~ + 2k + kE:f:

=

fu

l)

AEZ

~ + k) m ~ + k) /¢ ( ~ + k) / 2

mo (

1(

~ mo ( ~ + k + 4) m ~ + k + 4) I¢ ( ~ + k + 4) 1 2

+ =

1(

mo(~) m~(~) ~ j(~ +k )1

2

~ + ~) m1 ( ~ + ~) fu I¢ (~ + ~ + k) / 2

+ mo ( =

mo(~) m1 (~) + mo(~ + ~) m1 (~ + ~),

which completes the direct proof. Conversely, if (6.4.32) and (6.4.33) hold, then we can compute LtEZ 1¢(~ + 1)1 2 following the proofof(6.4.3 2) above, using the orthonormalit y of (t- m). Similarly, the proof of (6.4.33) demonstrates that

L!EZ

(~ + /)~(~ + /) =

0.

•

This lemma may be paraphrased by the statement that the matrix

is unitary. Example 6.4.37. Haar wavelets

We illustrate the above formulation in the case of the Haar wavelet, where (t) = 1,

\ll(t) =

{~1

O:::t
The structure constants are obtained from the scaling relations (t) = <1>(2t)

+

<1>(2t- 1),

\}1 (t)

= (2t) - (2t -

1).

Hence ao = I, a 1 = I , bo = I, b 1 = -1 and otherwise a = b = 0. The scaling filter 1 1 and wavelet filters are given by (6.4.16) and (6.4.30):

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INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The Fourier transform of the scaling function and wavelet are ~(~)

1_

1

=

1 (1!- h) e~2rritl; e~Z:rrirl;

dt =

.

=e

2:rr ~~

0

~(~) =

e~2rril;

1_

~irrl;

sin :rr~ Jf~

2e~irrl;

dt =

,

+ e~2rril;

(1 _

-

2:rr i~

e~irrl; )2

21f i~

~irrl; sin 2 (:rr ~ /2) = ze (n~ /2) . .

Equivalently

mo(~) = (~)

mt (~) =

"'(2~) ~

(~)

. ~ . = ze ml; sm :rr~.

Returning to the theory, we now solve for the wavelet \II by means of the function

m 1 • We give the first row of the matrix M and must find the second row. The orthogonality condition (6.4.33) requires that we have (mt (~). mt (~

~)) =a(~)

+

(mo

(~+~),-rna(~))

for some 1-periodic complex-valued function a(~). The normalizations (6.4.31) and (6.4.32) further require that Ia(~) I = 1. Finally, the substitution~ --+ ~ + ~ shows that a must satisfy the half period condition a(~ + ~) = -a(~). Thus we find the general solution (6.4.35)

m1

<~) =

mo (~ +

Da(~).

where a(~+

n

=-a<~). la<~)l

=

1.

It is immediately verified that this choice of m 1 satisfies the conditions of Proposition 6.4.36. Therefore the wavelet \II can be obtained through its Fourier transform as (6.4.36) Clearly we have infinitely many choices for a. A unique choice is dictated by the normalization that the Haar scaling function = lro.t) give the standard Haar function 1/r. ~us we choose a(~) = -e~Zrril;, which satisfies all of the conditions. Computing in detail, we have ml

(~)

=

-e~2rril; mo ( ~ + ~)

= - .!.e~2rril; ~ ii e2rrin(/;+l/2) 2 L- n nEZ

=

I

l

L nEZ

iin( -I)n+le2rri(n~l)l;

!NTRODL'CTION TO WAVELETS

323

Referring to (6.4.25 ), we have the explicit representatio n of \II (t) in terms of {<1>(2t- n)}nEZ• namely (6.4.37)

\ll(t) =

I:<-I)"a 1 -n<1>(2t- n). nE:S

This explicit formula displays the orthonormal wavelet in terms of the scaling function and the structure constants. In case of the Haar scaling function (a 0 = 1, a 1 = 1) we obtain the standard Haar function \ll(t) = <1>(2t) + (2t- 1). In the Fourier domain (6.4.37) is written \II (f;) = -e ~ "mo

(6.4.38)

-;T{

A

-

(

2~ + 21 )

A

(

2f;) .

Finally, we prove that the spaces vi+ I e vi are spanned by the set consisting of {\11(2 1 t - k)hEZ· By scaling, it suffices to prove this in casej = 0.

Proposition 6.4.38. Any f E V 1 8 Vo can he represented by its Fourier transform as

]<~) where v

E

=

mo (~; + ~) v(~)(~)

L 2 (1R/Z) satisfies v(~

+

f (t)

=

(6.4.39)

~) = -v(f;). In particular. we can write

L

c, \II ( t -

n)

11 E: ~~

Proof. Sincef E V,, we havef(t) = LnEZ. a,(2t- n) for some a E / 2 ('1?..). The orthogonality to Vo further requires J.~f(t)Ci>(t - k)dt = 0 for all k E Z. In terms of Fourier transforms, we have

'Vk

E

Z.

Now we can apply the same computations as those following (6.4.34) to conclude that

mo(~)C(~) + mo (~ +

4) C (~ + 4) =

0.

But we have already seen from the proof of Proposition 6.4.36 that the general solution of this equation is obtained as (6.4.35). Finally, this can be written as a multiple of (~ /2) defined in (6.4.38 ), expressed as

/<~)

=

mo (~ +

which translates into the t domain

4) v(,;)(~) a~

=

-a(~)e 2 'TI<(O.

(6.4.39).

In conclusion, we note some other basic properties of the scaling filter m . 0

•

324

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proposition 6.4.39. Jfmo is the scaling filter for an MRA, then lmo(~)l < 1 and = 1, mo (-4) = 0.

mo(O)

Proof. This follows from (6.4.31) and then setting Fourier domain (6.4.15).

6.4.5.1

Direct proof that Vt

e

~

0 in the scaling equation in the

•

Vo is spanned by {'li(t- k)}kEZ

Proposition 6.4.40. Any f E V 1 is equal to its projection on the set offunctions spanned by {(t- k)}kEz, {'li(t- k)hEZ· Proof. Since V 1 is spanned by {
L:::Ck

lo.EZ

kEZ

where ck. dk are the generalized Fourier coefficients of
.£.

Ck =

(t- k) dt

i

=

~

=

~ .£. cl> (~) e-mi;t

= =

where we have changed line. Similarly

c1> (

~) e-mi;t
i I¢ d~ 1' mo(~)e-2rrd;(2k+/) d~. ~ -+ 2~

in the next-to-last line and used periodization in the last

1' (~)e-2n,Wk+l> d~ -1' (I;+~) =( 1'

dk =

m,

=

e2"'"e-2"'<<2k+t> dl;

mo

-1) t-t-2k

mo(TJ)e-z"'ry dTJ.

Thus ck is the 2k + /th Fourier coefficient of m 0 and dk is ± the 2k +I- lth Fourier coefficient of m 0 . Hence by Parseval,

~ 2 ,L_.(icki

+

ldki 2 )

=

1'

kEZ

1 lmo(TJ)i 2 dTJ = -.

0

2

On the other hand, { i
JU{

which completes the proof.

~ 2

{ i
la

~. 2

•

INTRODUCTION TO WAVELETS

325

6.4.5.2 Null integrability of wavelets without scaling functions Any wavelet may be expected to have integral zero. Indeed, (6.4.38) shows that W(O) = m 0 (D ¢>(0) = 0, by Proposition 6.4.39. In particular, if \IJ E L 1 (lR). then we must have \IJ = W(0) = 0. Yet this may not be strictly true if the scaling function does not belong to the space L 1 (JR.). For example, the Shannon wavelet has (t) = (sin nt)jnt, which fails to be integrable, from which it follows that the associated wavelet defined by (6.4.38) also fails to be integrable. The following proposition generalizes the property of null integrability in two directions: (i) to wavelets that do not necessarily belong to an MRA and (ii) to higher dimensions. The proof assumes only orthonormality. We use the notation s1 for the cube of lattice points k defined by the inequalities -21- 1 _:::: k; < 21- 1 for i = I, ... , d. Thus card(S;) = 21d.

J'R

Proposition 6.4.41. Suppose that \II E L 1 (JR. d) n L 2 (JR. d) has the property that {\IJ1d := {21di 2 \IJ(2 1 t - k)h:s::::N.kE:Zd is an orthonormal set and that Q => [ -1, 1]d is a cube centered at 0 with 0 , I\IJ I < ~I \IJ I > 0. Then

J

..JN _:: : 2v'12QT/I JR.d

then

Jfii.d

J'Rd

\llf. In particular, if {\IJ1d;EZ•.kE:Zd is an orthonormal set,

\IJ = 0.

Proof. Let fJFid \II = Re'H with R > 0. Replacing \II by e -IIJ \II, we preserve the orthonormality of \111k while achieving fJRd \II = R > 0. Now let Q ::;, [- 1, I ]d be a cube centered at 0 so that 1f0 , \II I < R/2. Then for any set A with Q c A, we have I fA' \II I < R/2 and IRe fA \II - Re fJFid \II I .:::: R/2, so that Re fA \II :::: R - R/2 = R/2. Now let (6.4.40)

'YJ := L\11(2 1 x-k) ==> 1111 11~ = kES1

L:r

1

d

= l.

kES1

On the other hand,

But if Q = [ -M, M]" and x Q C 2 1 + 1 Q- k fork E S, and x

E E

(6.4.40

Q and j :::: 0, then l(x, Q. This means that Re

f

]2Q

11

::::

+ k,)j2i+ 1 I

<

M, so that

~. 2

On the other hand, the orthonormality of \111 ~_ implies that {1 1 } is an orthonormal sequence, so that for any N :::: I we have (6.4.42)

On the other hand, (6.4.41) shows that (6.4.43)

Re

{

]2Q

R

vN. -'N 2

326

INTRODUCTION TO FOURIER ANALYSIS AND WAVFI FT._

We combine these and apply the Cauchy-Schwarz inequality in the last line:

f!_.JN 2

< Re {

}2Q

-

from which the first statement follows. To prove the second, we take obtain a contradiction, hence R = 0.

6.5

./N

>

2v12Qr/R to •

WAVELETS WITH COMPACT SUPPORT

In this section we develop the tools to construct MRA wavelets whose scaling functions are differentiable and vanish outside of a finite interval. This will be done by a passage from the scaling filter to the scaling function. Clearly, a necessary condition for a compactly supported scaling function is that the scaling filter be a trigonometric polynomial. This is also sufficient, formalized as follows. Proposition 6.5.1. Let be the scaling function of an MRA satisfying the hypotheses of Theorem 6.4.27.

(i)

= 0 for ltl ;:::: M, then the structure constants satisfy a, = 0 for In I > 3M. In particular the scaling filter is a trigonometric polynomial:

If ct>(t) (6.5.1)

mo(~)

"'"' ake ~~rrikl; = -1 ~ .

2

Jkl<3M

(ii) Conversely, suppose that the scaling filter is a trigonometric polynomial (6.5.1 ). Then the scaling function can be obtained from the infinite product (6.5.2)

(~) =

n =

mo(~ /2 1 ),

j=l

and has compact support. Proof. From orthogonality, we have a, = 2 J!"M (t)¢(2t - n) dt. If In I =::: 3M then the support of (2t- n) is disjoint from [ -M, M] and the integrand is identically zero, hence a" = 0 for lnl =::: 3M. Conversely, suppose that m 0 is a trigonometric polynomial. Since mo(O) = I, the infinite product (6.5.2) converges and we have for any N

where the last factor tends to I when N ~

CXJ,

hence (6.5.2) holds.

INTRODUCTiON TO WAVELETS

327

From the construction,
where lr1 1 :::: M. This is the Fourier transform of a linear combination of 8-measures 1 concentrated at the points x = 1 2- r1 E [ -M, M]. Hence the scaling function is the weak limit of measures concentrated on [ -M, M], in particular of compact support. •

2::;:

6.5.1

From Scaling Filter to Scaling Function

Having obtained some conditions in terms of the Fourier transform of the scaling function and its scaling filter, we now attempt to go in the other direction. Beginning with the scaling filter m 0 (t;) we attempt to construct the scaling function. The next theorem does not assume that m 0 is a trigonometri c polynomial.

Theorem 6.5.2. Suppose that m 0 (t;) is a ]-periodic function on the line that satisfies the following conditions:

(6.5.3)

mo(O) = 1, lmo(/;)1:::

c >

0 for 11;1 <

1

4'

(6.5.4) (6.5.5)

c

11 - mo ( t;) I < - -g-=-2 - - - 1;-l-) 10 0 11

I

11;1 <

2"

Then the infinite product O'f"mo(l; /2k) converges and defines an L 2 function

(t- k) }kEz is orthonorma l and is the scaling function of an MRA.

We remark that although condition (6.5.3) can be weakened, it cannot be dispensed with entirely, as shown by the next example.

Example 6.5.3. Let m 0 (t;) = !O

+ e-

6

rri~).

Computing the infinite product explicitly, we find that for t; -=1=- 0, = ~ lto. , whose translates do 31 not form an orthonormal sequence.

Exercise 6.5.4. (a) Check this calculation. (b) Check that 11<1>11 2 < 1. (c) Show

that {(t- k)}kEz does not form a Riesz sequence.

We will see below that the lower bound in (6.5.3) can be replaced by a weaker condition, which is in fact necessary. Proof of the Theorem. We will break the proof into four distinct steps, which will reveal the points where the different assumptions are invoked. The first step is a general statement

328

INTRODU CTION TO FOURIER ANALYSI S AND WAVELET S

about L 2 function s. If

E

L 2 (1R), let

and

L

(t) (t- k) dt

=

0,

Vk E

z\

{0}}.

Step 1. M is a closed subset of L 2 (1R). To see this, we let <1> E M with <1> -+ E L 2 (1R). Using the notation Lk(t) 1 1 (t - k), a norm-pr eserving operator , we have (' Lt.. ) - (}' Lk I) = ( - }' Lk )

l(, Lk ) - ( <1>1 , Lk 1 )I

~

+ ( J' Lt.. - Lk }) II - <1>1 liz IlL* 112 + II 1 liz IILk - Lk J 112

= 11<1>- JII2IIll2

+

ll1 ll2ll- JIIz

-+ 0.

Taking k = 0 shows that II II 2 = 1 . Taking k =I= 0 proves that (. L~ ) = 0, i.e., E M. Step 2. We define an inductiv e process beginni ng with the Shannon wavelet. Let o(t) = (sinrrt)/ JTt, with 0 (~) = 1 _t·tl' and define for}~ 1. 1

J (0 =

<~>o(.L 2J ) mo (£) 2 ···mo(.L 2J ).

We claim that <1>1 EM for all}~ 0. To see this, we will use mathem atical inductio n to prove that {<1> (t- k) ha is ortho1 normal. For j = 0 we have L: .,z lo(~ + /)1 2 = 1, so that o(t- khez is 1 an orthono rmal set, i.e., o E M. Assumi ng that J--l EM we write J(~) = m 0 (~ /2)<1> _, (~ /2) and compute 1

.L;: 1<1>1<~ + /)1 le£.

2

= (. L + I even

=

L:) lmo (~;

I odd

1 )

<~>J-1 (~; 1) !

2

lmo (~)r ~~" 1<~>;-' ( ~; /)r +

lmo (~ + ~)12 ~ I<~>J-I (~; /)12

The first sum can be written Llez I J- 1 (~ /2 + I) 12 , which equals I by the inductio n hypothesis . The second sum can be written LJeZ 1<1> _, ((~ + 1)/2 + l)jl, which also equals I 1 by the inducti 1 = l, complet ing Step 2.

JR 1

Step_ 3. The infinite prod~ct (6.5.2) converg es uniform ly on compac t sets to E L 2 (JR) with 11<1>112 ~I andlim~--o(l;) =I.

INTRODUCTI ON TO WAVELETS

329

It suffices to prove uniform convergenc e on the interval I~ I ::: M with M > 1. This follows from the estimates

lmo ( ~1)

- II : : -lo-g,.--2(-~-2--1-)'

I~ I

:::

M, 2'- 1

:::

M

c

-

(log M- j log 2) 2

<

-----=-2

c

()log 2)

'

which is the general term of a convergent numerical series. To prove that E Lz(IR.), we use Fatou's lemma: II¢ ll2

=

II lim ¢>1 liz ::: lim inf II ¢>1 liz I

1

which completes Step 3. We define <J> as the Lz function such that¢ Step 4. 11<1>1

=

=

1,

¢.It remains to prove that EM.

<1>11 2 -+ 0 when)-+ oo. To see this we will first prove that for some C 1 > 0, 1¢ (~)1 ::: C1 ¢(~)for all~ E JR. 1 and all sufficiently large j. From Step 3, the infinite product is uniformly convergent on hence for some M we have nJ>Mim 0 (~2- 1 )1 > for all I~ I::: Hence -

l-1. 11.

1

-

j<J>(~)I:::

1

2

n lmo(~r )1::: M

1

1 2CM,

1-

~~~:::

I

2

)=1

where c = inf 1 ~; 1 <e1/4 lmo(~)l. Hence inf 1 ~; 1 ~1;21¢(~)1 > C > 0. From the definition of <1> 1 in Step 2, we see that on the interval 1~ 1 > 21- 1, ¢ is zero, whereas on I~ 1 ::: 21 - 1 we have 1 ¢(~) = ¢1 (~)¢(~/21). We can solve this to write for all), (6.5.6)

J<~>1 <~>J =I ¢(~ ¢<~> I::: c-l 1¢(~)11 1 _ :v-'.v-, 1 <~)::: c- 1 1¢(~)1 /21)

where C = inf1_ 112 . 1121 !¢(~)1, so we have proved the stated domination . Thus I¢;(~)­ ¢(~)1 tends to zero and is bounded by an L 2 function, which shows that II¢;- ¢11 -+ 0, by 2 the dominated convergenc e theorem. Parseval's identity shows further that II <1> - <J> 11 -+ 0 2 1 which completes Step 4. Combining these results, we see that <J> E M, namely the integer translates of <J> form an orthonorma l sequence. The uniform convergenc e from Step 2 shows that ¢ is continuous at~ = 0 with ¢(0) = 1, while the scaling identity follows from the definition of ci>:

Applying Theorem 6.4.27 completes the proof.

•

Exercise 6.5.5. Suppose that the ]-periodic fw:r:tion m 0 satisfies (6.5.4) and (6.5.5). Define an operator P on nonnegati ve measurab le functions by

330

INTRODUCTION TO f-OURIER ANALYSIS AND WAVELETS

(i) Check that Pl = l. If g satisfies the equation g(.;) - mo(~ j2)g(~ /2), define e(~) Ltez ig(l; + l)i 2 and prove that Pe =e. (iii) Define¢.(!;) := n~ m 0 (l;j2i).Assu mingthatthe uniquesolut ionofPf =f 1 is f = const, prove that { (t - k) hez is an orthogonal sequence. (ii)

Hint: For part (ii). copy Step 2 of the above proof.

6.5.2

Explicit Constructi on of Compact Wavelets

In order to construct wavelets with compact support, it remains to exhibit trigonometri c polynomials with the properties (6.5.3) and (6.5.4), since (6.5.5) is automaticall y satisfied for any trigonometri c polynomial. We begin with a trigonometri c polynomial C(~). which satisfies the following conditions (6.5.7) C(~):::

0, C(~) > 0 for II; I<

1

4'

C(O)=l,

For example C(l;) = cos 2 ;rl; satisfies these conditions.

Exercise 6.5.6. Show that any trigonometri c polynomial that satisfies (6.5.7) is of the form 1

(6.5.8)

C(l;) = -

2

N

+ 'L)ak cos(2k +

1)27r~

+ bk sin(2k +

1)2;rl;)

k=l

for some N and suitable values of the real constants ak, bk. Once we have found a solution of (6.5.7), we need to find the scaling filter mo as a suitable square root. This is accomplishe d by applying the following lemma on factorization of trigonometri c polynomials .

Lemma 6.5.7. Fejerand Riesz: Suppose that t(x) = L~n ckeikx is a nonnegative trigonometr ic polynomial. Then there exists a trigonometr ic polynomial q(x) so that t(x) = q(x)ij(x) for all x E R Proof. We first prove the lemma in case t(x) > 0 for all x. Since t(x) is real, the coefficients must be Hermitian-sym metric: c -k = ck for -n ::; k ::; n. Define a polynomial in the complex plane:

Clearly we have 2

z "P (

~) =

P(Z.).

This implies that, whenever a is a zero of P in the exterior of the unit circle, then lja is a zero in the interior of the unit circle, of the same multiplicity. Furthermore, there are no

INTRODUCTION TO WAVELETS

331

zeros on the unit circle, since t(x) > 0. Factoring out the zeros and simplifying, we have

P(z) = Czm

n

(z- a,) (z -

k

for suitable integers m, p. Since t(x) > 0, we must have p leads us to choose

q(x)

=

cnde" -ad.

c

==

-I:-) a, =

0 and

c;n,ci,

> 0. This

jn~ci" ·

which completes the proof in case t(x) > 0. In the general case when t(x) :::: 0, we apply the above construction to the trigonometric polynomial E + t(x), to obtain a trigonometric polynomial q, (x), which depends continuously on E. In particular, we can take q 0 (x) lim, q, (x) to obtain the required trigonometric polynomial. •

We present several concrete formulas for generating the required trigonometric polynomials

6.5.2. 1

Daubechies recipe

This begins with the identity cos 2 7r ~ 1

=

(cos2 JT~

+ sin 2 7r ~

+ sin2 7r~)2N+I =

= I, which is raised to an odd power:

't; (2Nk+ 1)

2N+l

sin2k(7r~) cos4N+2-2k(7r~).

This sum has an even number of terms. We take the first half of the terms and define (6.5.9)

C(~) :=

. L (2N + 1) sin2k(7r~) cos4N+2-2k(7r~). N

k=O

k

t

When we replace I; by ~ + the sines tum into cosines and the cosines tum into sines, yielding the remaining terms in the binomial expansion; thus C(~) + C(l; + = 1, as required. Clearly C(O) = 1 and C(/;) > 0 for all!;, so that the conditions (6.5.7) are satisfied. It is also useful to write this in the factored form

!>

(6.5.10)

The second factor can be rewritten as a polynomial in sin 2 JT/; with positive coefficients, which will be shown below. Therefore the maximum of the second factor is attained 2 when sin 7r I; = I, cos 2 7r ~ = 0, and we obtain the useful bound C(l;) :::: (cos2

JT/;)N+l

(2NN+I)

332

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

The last factor is the middle term of the binomial expansion of (1 + 1) 2 N+ 1 and can be conveniently expressed by Laplace's method as asymptotic to 2 2 N+I I v0VJT.

Exercise 6.5.8. Show that (6.5.9) can be written as a trigonometric sum in the form 1 C(O = 2

m

L

+

ak cos(2k

1)2rr~

+

k=l

for suitable values of the constants ak.

Exercise 6.5.9. Use Laplace's methodfor integrals to show that

(2N 1) ,. ._ N+

;.Jjj;;

22N+I

N~

oo.

Hint: Begin with the integral representation

_I_ 2 28 + 1

(2N

+

N

1)

_I_j"

=

2rr

e-'e cos2N+I

e de.

-T

We still need to show that the Daubechies recipe can represented as a polynomial with positive coefficients. In fact Lemma 6.5.10. Let PN(Y) = L~=O eN/ 1)yk(l- y)N-k for 0 < y < I. Then PN(Y) = L~=O (N;k)yk. In particular PN(Y) :::: PN(l) = eNN+ 1). Proof. From the binomial theorem, we have

} = (y + -

(J _

)'))2N-t-l

(2Nk+ 1)

2~1 ~ k~O

=

(l _ y)N+l

t

2N+I-k J.. y (1- y)

(2N

~

k

= (1 _ y)N+IpN()')

QN(y)

=

L N

k=O

which is the first N

+

= 1+

~k

(2N+ 1))'/..-N-I(l -y)::>N-t-1-J..

k=N+I

In particular (1 - y)N+I PN(Y)

1)yk(l _ y)N

k

k=O

+yN+I

+

+ )'N-!IpN(l

_ )').

O(yN+I ), y-+ 0. Now let

(N +k)

y' = (1- y)-IN+I)

+ O(yN+l),

k

)'-+ 0,

l terms of power series expansion of (1 - y)-
= 1 + O(v"'+ 1 ),

v-+ 0

hence the difference polynomial satisfies (1 - y)N+l[PN(y)- QN(Y)] = O(yN+l ),

y-+ 0.

333

INTRODU CTION TO WAVELET S

In particul ar, the first N derivativ es of the polynom ial P,v - Q,v evaluate d at y = 0 are all zero, hence the polynom ial is identica lly zero, and we have proved that P,v(y) - Q,v(y) 0, as required . Since the coeffici ents of QN are positive , it follows immedi ately that for 0 ~ y ~ I , p N ( y) ~ P,v ( 1) = ( 2N+-I) • " .

=

Exercis e 6.5.11 . Show that we have the followi ng values of PN ( y). P 1 (y) = 1 + 2y,

+ 1+

P2(y) = 1 P3(y) =

+ 4y + 3y

P, (1)

6y

2

= 3 =CD

P2(I) = 10 =

,

10y 2 + 20y-',

P3(1)

=

35

G)

=G)

P4(1) = 126 = (~)

+

Ps(Y) = 1

6y

+

2ly 2 + 56y 3

+

I26y

+

4

252y

5

P 5 (1) = 462

,

= (~ ) 1

P6(1) = 1716 = (~)

In the above Daubec hies recipe, we used an odd expone nt in the binomi al theorem so that we could conven iently group the terms into two separat e groups. The followi ng exercis e shows that one can make a parallel comput ation for even expone nts also.

Exercis e 6.5 .1 2. For N > 2, define N-l(2 1(2N) RN(y )={; k N). yk(l-y )N-k+ 2 N yN(l-y )N. Show that 1 = ( 1 - y )N RN ( y) + yN RN ( 1 - y) and deduce that RN is a polyno mial of degree N- I, specific ally, RN(y) = PN-l (y), defined above, for N > 2. 6.5.2.2 Hernan dez- Weiss recipe This begins with the functio n

(6.5.11 )

g,(~)

=

J1 sin

2m+

1

(2.rr t) dt

I

J0 2 sin 2 m+l (2.rrt) dt

-!

defined for gm(O) = 1, gm further that

S ~ S ~ and extende d periodi cally. Clearly 0 < gnr (~) < 1 with (!} = 0 and gm(~) > 0 for 0 :S I~ I < !·A direct comput ation shows

2 fo 2 sin 2 "'+ 1 (2.rrt) dtI

J1 sin

2 m+l

(2.rrt) dt-

~

fo +I sin 2m+l (2.rrt) dt I

l

j 0 2 sin 2 m+l (2.rrt) dt l

fr} sin 2"'+ 1 (2.rrt) dt- J ~ sin 2 m+l (2.rrt) dt0

I

1;+ 2

sin 2 m+l (2.rrt) dt

I

J0 2 sin 2"'+' (2.rrt) dt =1

since the integra nd is odd about t = ~·Thus the conditi ons (6.5.7) are satisfie d.

334

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Exercise 6.5.13. Show that (6.5. 11) can be written as a trigonometric sum in the form

for suitable values of the constants ak.

6.5.3

Smoothness of Wavelets

We now investigate the smoothness properties of the compact wavelets, which are constructed in terms of the scaling function <1>. The estimates obtained below, although not optimal, do show that one can obtain compactly supported wavelets of any degree of smoothness, beginning with a suitable trigonometric polynomial mo(/;). We begin with the function C(l;) = lmo(/;) 12 , which satisfies the properties (6.5.7). Following the Daubechies example from (6.5.10) we can write C(l;)

=

(cos 2 ni;)N+l PN(sin 2 n/;),

where PN is a polynomial of degree N that satisfies a bound IPN(y)l < KN for some constant KN. The scaling function satisfies the relation

I (I;) 12

=

n

c(

i=l

I;[ ) 2

s

fi c ( i=l

(j > 1).

I;[ ) 2

A pointwise estimate is obtained by using the identity sin() = 2 sin(() /2) cos(() /2) to write

l(l;)l

2

< )]

c(~~) nl;

< ( cos 2

nl; )2
sinnl; (

,..,;+ 1 • £/

Sin

J+l

· · · cos 2)+1

( rrl; ) 2'+'

)2
KN

J+l

KN

.

Now restrict I; to lie in the dyadic shell defined by the inequalities 2)- 1 < II; 1 :5 21. With this restriction, we have 7l" I; ;zi+ 1 < 7l" /2 and we can use the inequality sin() ~ 2() 1n to underestimate the denominator. At the same time we use the upper bound 1 sin () 1 < 1 in the numerator. Combining these with the bound for KN we have

l(l;)l

<

(_!__)(N+I) K(j+l)/2 21 N ,

INTRODUCTION TO WAVELETS

335

Therefore < (

2)!2

)

21/2 ( (Nn)l/4

2N+l/2 )

(

(Nn) 114

(Nn)i/ 4

)j (

2N+I/2 )

(Nn)I/4

log Nn - 2 4

2 a · - __:=-----

To complete the proof of smoothness, it remains to prove the following proposition from classical harmonic analysis, related to the Littlewood-Paley method. Proposition 6.5 .14. Suppose that f E L 2 (JR.) and that for some a > 0, C > 0, we have the system of inequalities j

(6.5.12)

=

0, 1, 2 ....

Then (i) JfO
In particular f is a.e. equal to a continuous function. To prove the Holder continuity, first. suppose that 0 < a < 1. Define the index j = j(h) so that (1/2)j+ 1 :::;: Jhl < 1/2j. From (6.5.12) we see that j E L 1 (1R) so that we can write the absolutely convergent integral j(x +h) - f(x)

=

l

e21rix{

(e21rih~

-

l)](~) d~

=I +II +Ill

where

11

= {

e2rnx< (e21r1h{ _

l)f(~) d~

Ji:5.l/;l:5_21

Ill= { ],1;1?:.21

e 2 1riX~

(ebr•h{

-l)](~)d~.

336

INTRODIJCfi ON TO FOURIER ANALYSIS AND WAVELETS

It suffices to prove the HOlder condition when Jhl <

k0

-

I. Making use .of the inenuality

IJ :::; (}, we have

:::: c

J

Jhl L2k r"k 1.=0

=

L J

C JhJ

2k(l-a)

k=O

=

C(a)JhJ (2(;+1)(1-a >-

1)

:::; C(a)Jhl"'

where we have used the definition of j = j(h) in the last line. Finally, we have

L

DO

::S 2C

2-ka

k=J+l

= 2C(a)T;a :::; 2C(a)Jhl"',

which completes the proof in case 0 < a < I. Now if a > 1 we see immediatel y that fa l~Jlf(~)l d~ < oo so that we have the absolutely convergent integral f'(x)

=

L2rrix~e "'' f(~) d~ 2

from which we can repeat the above reasoning to show that f' satisfies the appropriate Holder condition. The higher derivatives are handled in the same manner. •

We can summariz e the above results in the following form. Theorem 6.5.15. For any preassign ed integer s, there exists a compactly supported scaling function with continuou s derivative s up to orders, so that the correspon ding MRA generates a wavelet \II of compact support with continuou s derivative s of orders.

6.5.3. 1

A negative result

It is interesting to note the impossibi lity of constructi ng wavelets that are simultane ously infinitely differentia ble and of compact support. The following discussion includes the case of MRA wavelets, but also applies to wavelets which may be constructe d by other means.

INTRODUCTION TO WAVELETS

Proposition 6.5 .1 6. Suppose that 1/f E L 2 (~) {2' / 2 1/f (21 x - k) }J,kE'Z is an orthogonal set.

has

the

property

(i) lfl/f E L 1 n U"'' n C, then f'R 1/f = 0. (ii) If for some m E ;;z+, 8 > 0, we have 1/f, 1/f' ... , 1/f(m) E L 1 n V)O 1/f(m)(x) = 0(( )m+l+li), lxl -+ 00, then 0 = JR 1/f = JRxl/f 11 fw. xml/f.

!

337

that

n C with

= ··· =

= k ;21 E Q 2 be an arbitrary dyadic rational number, we have from the definition of orthogonality Proof. Letting a

i. i. = i.

0 =

tfr(x)1/F(2jx- k) dx

= 0

t/F(x)1/F(2j(x- a))dx 1/r(a

+

y2-j)1/F(y) dy

where we have made a change of variable in the integral. Letting j ~ oo with k = a2j (a fixed), we use the dominated convergence theorem to conclude thatO = 1/r(a) 1/F(y) dy. If fiR 1/F ::/= 0, then 1/r(a) = 0, Va E Q 2 hence 1/r 0, a contradiction, proving part (i). To prove part (ii), let

JR

=

O,(x)

= ~~ 1/r(y)dy, 02(x) = /_~ Ot(y)dy, ... , Om(X)

=

/_~ Om-I(y)dy.

Clearly O,(x) = 0(1/jxi)"'H,x ~ -oo. But since we have proved in (i) that fiR 1/r = 0, 00 we can write e, (x) = 1/r(y) dy, showing also that el (x) = O(l/lxl)m+8' X ~ +oo. Hence we can integrate by parts and write

-£

i.

xtfr(x) dx =

i.

xe; (x) dx = -

i.

el (x) dx.

Now we use orthogonality and partial integration to write

0

=

i.

tfr(a

.£. = - .£. =

+ y2-1 )1/F(y) dy

tfr
+ yrj)el
Lettingj ~ oo with a E Q2 fixed, we have 0 = 1/r'(a) fiR 01 (y) dy = -1/r'(a) fiR x1fr(x) dx. If fiR x1fr(x) dx ::/= 0, we conclude that 1/r'(a) 0, Va E Q2 , which proves that 1/r is a linem function, a contradiction. Proceeding inductively, suppose that we have shown that

=

0

=

i.

e,(y)dy

= ... =

.£.

em_J(y)dy.

From the hypothesis on 1/r. we obtain the bounds O,(y) = 0 (

y1 ) m+o , ... , Om(Y) = 0 ( y1 )

1 +o

,

IYI

~ 00.

338

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Partial integration shows that

i

x"'l/f(x)dx

=

-m

i

x"'- 10l(x)dx

From the orthogonality relation we have for any a

o= Letting}

---+

£

1/f(a

+

yrj)lfr(y) dy

=

E

= ... = Qz,}

(-1)"'

E

(-l)"'m!

i

e,(x)dx.

Z

1. 1 -

1/f<m>e,(y) dy.

oo with a fixed and using the hypothesis on 1/f(mJ proves that 0

=

1/f(m)(a)

1

-

e,(y) dy

lR

=

1/f(m) (a) ' m.

y"'l/f(y) dy,

lR

from which it follows that if frrt y"' 1/f =I= 0. then 1/f is a polynomial, a contradiction.

Corollary 6.5.17. /f'tfr 1/r(x) = 0.

E

•

C 00 and{2!1 2 1fr(2ix-k)}J .kezisanorth ogonalset, then

Proof. From the previous proposition, we have fiR x"' 1/f (x) dx = 0 for all m E z+. Since 1/f has compact support, we can find a sequence ofpolynomial spn(x) that converge uniformly to lfr(x) on the bounded interval of support. From this it follows that fa 11/f(x)l 2 dx = 0, which shows that 1/f 0, as required. •

=

We can obtain the same conclusion if we have a suitable decay rate. Corollary 6.5.18. Suppose that 1/r

E

S and that for each A > 0

/f{2i1 2 1fr(2ix- k)}J.keZ is an orthogonal set, then 1/r(x) = 0. Proof. From the hypotheses we can apply the previous proposition to conclude that = 0 for all n. Now we can estimate {jr by applying Taylor's theorem with remainder to the complex exponential function. Thus

fiR x"l/f(x) dx

I (21r:;)N

le-21r•<x _ ~ (-2::~x)" ~ Multiplying by 1/f and integrating, we obtain

j{jr(~)l ~ 12:~IN

1_1?111/f(x)j dx.

Applying the hypothesis shows that the right side tends to zero, hence {jr (I;) the result follows from the uniqueness of the Fourier transform.

6.5.4

= 0, so that •

Cohen's Extension of Theorem 6.5.1

We have seen by an example that the positivity condition on the scaling filter in (6.5.3) cannot be totally omitted. Nevertheless it can be relaxed to a condition that is both necessary and sufficient for the existence of a scaling function of an MRA system.

INTRODUCTION TO WAVELETS

To motivate Cohen's condition, note that the set K = [- 4-, properties:

4-]

339

has the following

(6.5.13) n=l

L

(6.5.14)

1K(~ + l) = 1

a.e.

IEZ

Furthermore if (6.5.3) holds, then (6.5.15)

j > 1,

~ E

K.

Condition (6.5.14) was instrumental in proving that the sequence of approximants 1 satisfies the orthonormality condition for eachj, while (6.5.13) was necessary to identify lim1 ¢>1 with the infinite product¢> on the entire real line. Condition (6.5.14) states that K is congruent to [modulo 1. More precisely, for a.e. ~ E R, there is precisely one l E Z so that~+ l E K. As a simple example, consider K = [0, U [~. 2]. This can be transformed into [by translating the second interval by 2 to the left. More general examples can be obtained by further cutting and pasting. Now suppose that m 0 is one-periodic and satisfies (6.5.4) and (6.5.5) of Theorem 6.5.2. Then the infinite product (6.5.2) converges uniformly on compact sets. Further, suppose that K C R is a compact set that satisfies the three conditions (6.5.13)(6.5.15). Defining \II so that ¢> 0 = lK, we see that 0 has orthonormal translates, since LtEZ lo(~ + 1)1 2 = LtEZ lK(~ + l) = I a.e. Hence we can begin the inductive process of Theorem 6.5.2 and define ¢>1 (~) = ¢> 0 (~2-i)mo(~ /2) · · · m0 (~ j2i), which also satisfies the orthonormality condition. The limit function¢> E L 2 (JR) by Fatou's lemma. If ~2-i ¢. K then ¢>1 (~) = 0. Otherwise we may solve for ¢> (n = ¢>(~)/¢>(~2-i) and 1 copy the argument in the proof of Theorem 6.5.2 to conclude that we have an estimate on the real line: 1¢>1 (~)1 < Cl¢>(~)1, from which we can apply dominated convergence to deduce that II ll2 = I, and that has orthonormal translates. Thus we have proved half of the following theorem.

4, 4],

4, 4]

4J

Theorem 6.5.19. Suppose that mo is ]-periodic and satisfies (6.5.4) and (6.5.5). Suppose further that there exists a compact set K c R such that (6.5.13)-(6.5.15 ) hold. Then , defined by the infinite product (6.5.2), defines the scaling function ofanMRA. Conversely, suppose that E L 1 (lR) n L 2 (1R) is the scaling function of an MRA for which LtEZ 1¢>(~ + 1)1 2 = 1, V~ E R, and that the scaling filter m 0 satisfies (6.5.5). Then there exists a set K C R, which is a finite union of closed intervals such that (6.5.13)-(6.5.15 ) hold. Proof. The direct assertion has been proved in the above discussion. To prove the converse, we begin with the orthonormality condition of <J>:

L:: I<~><~+ /)12 = 1, /EZ

340

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

In particular, for each ~ E [ - ~, ~ ], there exists k - k(~) E Z so that I <1>(~ + k) I > 0. But <1> E L 1 (IR) implies that <1> is continuous. Hence there exists 8 = 8~; > 0 so that 1<1>(~' + k)l :::: C(~) > 0 for W- ~I < 8. Since 1<1>(0)1 = I, we can take k(O) = 0. From the Heine-Borel theorem, we can extract a finite system of these open intervals which covers [- &, ~ ]. Now we form a finite union u;~o V, that covers [- t, beginning with V 0 = [ -80 , 80 ]; replacing V, by V,e(V0 U· · ·U"J- 1 )n[- ~, we may assume that int V, are disjoint. Now define K := U;"= 0 (V, + k(~1 )). Since 0 E V0 c K, we see that U;;':._ 1 2"K = IR, as required. Since {V1 } is a finite partition of [- t, t ], we have I::_;"=o 1 v, (~) = 1 a. e. on &] . Then we compute the !-periodic function

t],

t ],

(- ! ,

~ IK(~ + [) = ~ (t lv,+k(l;1 l(~ +I))

t (L: lv,+k<<,l(~ n) +

=

J=O

IEZ

N

L

=

lv,(~)

j=O

=

I

a.e. ~ E [ -

t, t]

where we have used the fact that ~ +I E V1 + k(~1 ) precisely once, when I = k(~1 ) (otherwise we would contradict V, c But this sum is !-periodic, so that it is 1 almost everywhere on JR. Finally to prove (6.5.15) we note that from the above construction 1<1>(~)1 :::: c) when~ E "), hence for all~ E K, we have 1<1>(~ + ~J)I :::: c min{C0 , ... ,CN}. Referringto(6.5.2), weseethatforanyJ E z+,~ E K, 0 < C < 1<1>(~)1 ~ n;= 1 m0 (~2-1 ), so that we must have m0 (~2-1 ) ¥: 0 for all~ E K andj 2: 1, completing the proof. •

[-!, t]).

Remark. The sum LIEZ l(~ +I )1 2 = 1 under mild additional conditions, especially whenever this series is uniformly convergent on compact sets. This will hold if, for example, we have an estimate of the form c:i>(~) = 0(1~ 1-f:l), 1~1 ---+ <x:>, with ,8 > This will be satisfied whenever ' E L 1 (lR). Cohen's theorem yields an important corollary that gives an optimal estimate for the interval on which m 0 must remain positive.

"i·

Corollary 6.5.20. Suppose that m 0 satisfies (6.5.4) and (6.5.5). (i)

If

lmo(~)l > c >for I~ I <

~. then the infinite product (6.5.2) generates

anMRA. (ii) There exists mofor which lmo(~) I > Ojor I~ I < ~for which (6.5.2) does not

generate an MRA.

[-t.

Proof. It suffices to apply Cohen's theorem to the set K = ~]. Clearly (6.5.13) and (6.5.15) are satisfied so that the direct part of the theorem applies. To see that this is optimal, consider the example m 0 (~) = (I+ e- 6 rr'<)j2. From Exercise 6.5.4 we see that the infinite product (6.5.2) is the Fourier transform of 110 . 3 J13, which does not have orthonormal translates. •

INTRODUCTION TO WAVELETS

6.6

341

CONVERGENCE PROPERTIES OF WAVELET EXPANSIONS

If the scaling function satisfies additional regularity properties, we can expect that the series expansio~f(t) = Ljk cJkVrJk(t) will converge uniformly whenf is continuous or in IJ'(lR) whenf E £P(JR), 1 < p < CXJ. Indeed, this has already been demonstrated in detail for the Haar series expansion in Section 6.3. In the present section we will describe a class of scaling functions for which one has these extended convergence properties, as well as the properties of a.e. convergence. Going beyond the qualitative fact of convergence, one can also discuss the speed of convergence, when f has additional regularity. Indeed, in the case of Fourier series in Chapter 1, we saw that the L 2 -Holder continuity of jis equivalent to a rate of convergence result in the L 2 norm. The corresponding results for continuous functions in the supremum norm are attributed to Jackson and Bernstein in the case of trigonometric series. We will see below that corresponding results apply to a class of MRA systems associated with a suitably regular scaling function .

6.6.1

Wavelet Series in LP Spaces

We begin with a scaling function that satisfies the estimate (6.6.1)

l(t)l ::S K(2jtl),

where K : [0, CXJ) -+ lR is a monotone decreasing integrable function. In particular, E L' (lR) and I fiR (t) dtl = 1, by the remarks following the proof of Theorem 6.4.27. The projection operator PJ is represented by the series (6.6.2)

PJf(t) =

L

jk(t) ( { f(s)¢,jk(s) ds)

JIR

kEZ

= 21

L

<1>(2Jt- k)

kEZ

= 2J

l

f

JR

j(s)¢,(2Js- k) ds

<1>(2 1 t, 2Js)f(s) ds

where the wavelet kernel (6.6.3)

(t, s) :=

L

(t- k)¢,(s- k).

kEZ

Proposition 6.6.1. The wavelet kernel (t, s) enjoys the following properties: (6.6.4) (6.6.5) (6.6.6) (6.6.7)



i

E L 1 ~(JR 2 ),

(t, _s) = (s, t), l(t, s)l ds < C <

CXJ

and

l(t, s)l < CK(It- sl).

f

J)R

(t, s) ds = 1,

342

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Proof. From (6.6.3). we have l(t, s)l :::

L

l(t- k)ll(s- I )I <

CXJ,

k.l

where the right side is the product of two periodic, locally integrable functions, which proves (6.6.4). From the definition (6.6.3), (6.6.5) is immediate. Now { l(t, s)l ds

JIR

~

L

l(t- k)l { l(s- k)l ds

]fill.

kEJ'.

=

llll1

L

l(t- k)l

~

C

1-E'ii:

and

=

{ (t, s) ds

]fill.

(1~ ) L

(t- k).

kEZ

But (l) = 0 for I E Z, so that the Poisson summation formula shows that the last sum has the constant value (O) = Jll l. , which proves (6.6.6). To prove (6.6.7) we write

L

(t, s) ::: (

~ K(ls- tl)

L

+

111-11":1'-11/2

L

) K(21t- II)K(21s- /1)

11/-;l:o:l;·-11/2

K(21s- /1) + K(is- ti)

IEJ'.

L

K(2it- /1)

IEZ

::: CK(It- sl),

•

which completes the proof.

The main purpose of this section is to prove that P1 f ---+ f when j ---+ oo and PJf ---+ 0 when j ---+ -oo. We can reduce this to the study of Po by introducing the dilation operator Jr, defined by (6.6.8)

r E Z.

Proposition 6.6.2. We have the following properties:

(i) For any r E Z, we have the commutation relation PjJr = JrPj-r and the norm relation IIJJfiiP = 2-J!PIIfiiP, 1 < p :S oo. (ii) There exists a constant Cp such that for each j E Z and for each f E LP(JR), 1 ::: p::: 00,

Proof. From (6.6.2), we make the substitution 2 1 s P1 j(t) =

=

u to obtain

{ (2 1 t, u)j(u2-1 ) du

IR

= Po(J J)(2 1 t) = J PoJ- j(t), 1

1

INTRODUCTION TO WAVELETS

343

which is equivalent to the stated commutation relation. The norm relation follows from the identity II!,! liP= { lf(2 1 tW dt = z-1

la lf(s)IP ds.

Ja

P

{

To prove (ii), we first do the case j = 0. Then IPof(t)l = l i (t, s)f(s) dsl

~

C i

K(lt- sl)lf(s)l ds

~

CIIKIIt llflloo

IIPofllp ~ IlK *flip ~ IIKIIJII/IIp. where we have used Young's inequality for convolutions. To treat the case j =I= 0, we use part (i) to write IIPJIIP = 11-f,(PoJ-1 )/IIp = z-J!PIIPoLJIIP

~ CIIKII1T11PIIJ-JIIp

•

= CIIKII,IIfllp·

These bounds allow us to formulate and prove a general theorem on the convergence of the small-scale projection operators. Theorem 6.6.3. Suppose that is the scaling function of an MRA and satisfies the bound ( 6. 6.1 ). (i) Iff E BucOR), then IIPjf- /lloo ---+ 0 when}---+ 00. (ii) Iff E LP(IR)for 1 < p < oo, then IIP1 f - /lip---+ 0 when}---+ oo.

Proof. First we note that P11 = 1, which follows from JR (t, s) ds = 1. This allows one to writef(t)- Pjf(t) = 21 JR <1>(21t, 21s)[f(t)- f(s)] ds. Sincef is uniformly continuous, given € > O,let 8 > 0 be chosen so that 1/(t) -f(s)l < €j2C for Is_, tl < o."We write

f(t)- P1f(t) = 2 1

(1

+

1•-rlso

1 )

[f(t)- f(s)](2jt, 2 1s)ds.

1.<-tl>o

We apply the bound fa i(t, s)l ds ~ C in the first integral to conclude that this term is less than €/2, for allj. To estimate the second integral, we use the boundedness to obtain the upper bound 211/lloo

1

2 1 1<1>(2 11, 2 1s)l ds

1•-tl>o

~

Cllfll""

= Cllflloo

1

ls-tl >o

1 1

2 1 K(2 1 Is - tl) ds K(iu- 2 1tl) du

lu-21tl>218

=

Cllfll""

K(v) dv,

lvl>21o

which tends to zero when) -+ oo, by the dominated convergence theorem. This is a uniform bound, independent of t E JR., from which we obtain the asserted uniform convergence.

344

INTRODUC TION TO FOURIER ANALYSIS AND WAVELETS

To prove the V' converge nce, we first discuss the case p = 1. From the uniform boundedn ess IIP fll 1 ::: 11!11 1 , it suffices to prove the L 1 converge nce on the dense set of 1 continuou s functions with compact support in [ -R, R]. For such anf, we have

II/- P1 /ll1 =

1/U)I dt

{ J1r1 <e2R

+ {

IP1 f(t)l dt.

Jlrl >2R

The first integral tends to zero by virtue of the uniform converge nce already proved. To estimate the second integral, we write

cf { :sc[ f :S

2 1 K(2 1 Is- tl)dtds

III2R

2 1 K(2 1 Is-tl)dt ds

I•IR

::: 2CR {

K(v) dv -+ 0

j-+

00,

}!...,1>2 1 8

which complete s the proof of L 1 converge nce. To treat the case 1 < p < oo, it again suffices to deal with continuou s functions with compact support. In this case we have the bounds

i

lf(t)- P;f(t)IP :S llf- P;/11~ 1 lf(t)- P;f(t)J lf(t)- P1 f(t)l" :S llf-

P;/11~ 1

i

lf(t)- P;f(t)l dt

= llf-P;/11~ 1 IIf-P1 fiii,

which tends to zero, by the converge nce in case p = the theorem.

1, already proved. This proves

•

Remark . An alternati ve approach to the proof of Theorem 6.6.3 is to utilize theorem s on approxim ate identitie s, which were proved in Chapter 2. We begin with the bounds IPJf(t) - f(t)l < C2 1 { lf(t)- f(s)JK(2 1 lt- sJ)I ds,

}'*-

which shows that the left side is majorize d by K *f - f where K is a family of kernels 1 1 defined by K1 (s) = 2 1 K(2is), to which we can apply Theorem 2.2.21 from Chapter 2. It is interesti ng to compare the relative simplici ty of the above proofs of I f convergence with the difficulty in proving LP converge nce of trigonom etric series. For trigonom etric series we have two main obstacle s: (i) converge nce does not hold in case p = 1 and (ii) it is nontrivia l to prove uniform LP bounded ness of the operator s P . In the 1 case of MRA wavelets , the basic hypothe sis ( 6.6.1) on the decay of the scaling function easily provides the necessar y uniform bounds, as we have seen.

INTROD UCTION TO WAVEL ETS

6.6.1 .1

345

Large scale analy sis

To comp lete the analy sis of LP conve rgenc e of gener al wavel et series , it remai ns to pro~e that PJf ~ 0 when j ~ -oo. As in the case of Haar series . we expec t only that thts will take place for LP(JR.), 1 < p < oo and in the space Co(IR. ).

Propo sition 6.6.4 . (i) Iff f

E

E Co(IR.), then IIPJfll oo ~ 0 when j ~ -oo. (ii) If LP(IR.), 1 < p < oo, then IIPJfll p ~ 0 when j ~ -oo.

Proof. We begin with/ (6.6.9)

E

C 00 (1R). Ifj(t) = 0 for itl > R, we can write

P_mf( t) = 2-mc

JR j(s)¢>(2-mt, 2-ms) ds -R

:::; rmc /_: i/(s)J K(2-m it- sl)i ds

:::; 2-mCi l/lloo2 RK(0) --+ 0,

m --+ oo.

But C 00 (.1R) is dense in C 0 (1R) where we have the estima te IJP 1 /IIoo ::= Cllfllo o· To prove the LP conver gence, it suffice s to take/ E Coo(IR ). For

ltl :::; R the estima te (6.6.9)showsthatf~R lP-mf( t)JPdt --+ O.For t > Rwem akethe substi tution v = 2-m(t -s)

to write

JP-m/( t)J :::; 11/lloo

j_

2-m(t-+RJ

_m

2

JK(v)J dv

:::; llflloo 2-mR K(Tm (t- R)) loo JP-mf(t)JP dt:::;

11/11~2-mp ioo

= llfll~2-mp2m =

ll/11~2m
JK(2- m(t- R))JP dt

1

00

IK(s)JP ds

IlK II~--+ 0

with a simila r estima te fort < -R.

•

In exact parall el with the case of Haar series , the large scale projec tion opera tors to do not behav e well on L 1 (JR.).

Exerc ise 6.6.5 . Letf

E L 1 (JR.).

Prove that fiR P1 f(t) ~ fiR f(t) dt when}~ oo.

This mean s that we must restric t the range of p when formu lating a gener al LP conve rgenc e theore m for wavel et series . Simila rly, we must restric t to C 0 (1R.), since the identi ty Pj 1 = 1 shows that PJf ~ 0 is false in gener al when f E Buc(lR .) for j ~ -oo. Comb ining Propo sition 6.6.4 with Theor em 6.6.3, gives a comp lete pictur e of the conve rgenc e of one-d imens ional wavel et series in the space s C 0 (IR.) and LP(IR.), 1 < p < oo. This can be restat ed as a separa te theore m.

Theo rem 6.6.6 . Suppo se that the scalin g functi on satisfi es (6.6.1 ). (i) Iff E Co(IR.), then the s~m

unifor mly to f when m, n

~

"L,J=-m oo.

LkEZ Jdt) fiRf(s) ¢>jk(s ) ds conve rges

346

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

(ii) Iff E LP(lR), 1 < p < oo, thenthesum"£;=-m LkeZ ct>jk(t) JJR.f(s)¢>jk(s)ds converges to f in U' (lR) when m, n ---. oo. 6.6.1.2

Almost-everywhere convergence We can prove that the projection operators P1f converge almost everywhere when}---.

oo, by using the techniques appropriate to monotone kernels from Chapter 2. Theorem 6.6.7. Suppose that the scaling function satisfies (6.6.1) and letf E L 1 (JR). Then P1 f(t) ---. f(t) for every t in the Lebesgue setoff, in particular almost everywhere. Recall that the Lebesgue set off, Leb (f), is the set oft for which lima~o a- 1 j::_a if(t)f(t + u)idu = 0. Proof. From the previous computations. we write 1/(t)- PJ(t)l ::: 2 1 i_lf(t)- f(s)IK(2 1 1t- sl) ds

=

(1

00

+ /_:)

1/(t)- f(t

+ 2-1 v)IK(v) dv.

To treat the first integral, we define G(u) :=

Jor 1/(t)- /(t + w)l dw.

If t E Leb (/),then G(u)ju --* 0 when u all u. Then we can set E = 2-; and obtain

1

--*

0 and we have the bound IG(u)l ::; Clul for

00

(6.6.10)

1 1

00

K(v)lf(t)- f(t

+

Ev)l dv

=

K(v)G'(Ev) dv

00

=

K(v)d (G(Ev)) --

o

E 00

=-

1

G(Ev)

vdK(v)-- dv.

0

The monotonicity of K implies that vK(v)

-1

--*

00

EV

0 and a partial integration shows that

1

00

vdK(v) dv =

K(v) dv < oo

so that we can apply the dominated convergence theorem to conclude that (6.6.10) tends to zero with E. A similar analysis applied to the integral for -oo < v < 0 proves that PJ(t) --* f(t) when t --* oo. •

Corollary 6.6.8. Suppose that f PJf(t) ---. f(t) when t---. oo.

E L 1 (lR)

and that f is continuous at t. Then

Indeed, iff is a continuity point, then it is also a Lebesgue point off. Exercise 6.6.9. Extend Theorem 6.6.3 to f

E

U', 1 < p < oo.

INTRODUCTION TO WAVELETS

We also note that iff from the estimate

E

L 1 (JR.), then P1 J(t)

IP1 f(t)l ::: 2 1

l

---+

0 when j

---+

347

-oo. This follows

K(2 1 1s- tl)lf(s)l ds

::: 2 1 K(0)11/III ---+ 0

j ---+

-00.

This observation can be combined with Theorem 6.6.3 to obtain a statement on the almost-every where convergence of the bilateral wavelet series.

Corollary 6.6.1 0. Suppose that the scaling function satisfies (6.6.1 ). Then for L 1 (JR.). the wavelet sum m, n---+ oo.

any f

6.6.1.3

E

L-7=-m c1k \I!Jk (t)

converges for a. e. t, when

Convergenc e at a preassigned point

It is natural to expect that the wavelet series will converge to the normalized value off

in case f has a simple discontinuity . In the case of Fourier series one needs additional regularity conditions in order to ensure this convergence . In the case of MRA wavelets it is sufficient that the wavelet kernel satisfy the mild normalizatio n condition

00 (t, s) ds =

1

(6.6.11)

I

1 - =

!I

2

(t, s) ds.

-00

Proposition 6.6.11. Suppose that the scaling function satisfies (6.6.1) and that the wavelet kernel satisfies (6.6.11 ). Suppose thatf E LP(JR.)for some 1 ::: p < oo and that there exist the one-sided limits f(t + 0) = limsJ- 1 /(s) and f(t - 0) = limstd(s). Then lim P1 f(t) = .!. [f(t 2

J~=

+ 0) + f(t-

0)].

Proof. Using (6.6.11 ), we can write 1 z[f(t

+ 0) + f(t-

0)]- PJ(t) = 2 1 ~~ , <1>(2 1 t, 2 1 s)(f(t

+2 1

+ 0)- f(s)] ds

/_~ <1>(2jt,21 s)(f(t+O)-j (s)]ds.

From the proof of Theorem 6.6.3 we see that both of these integrals tend to zero when

j

6.6.2

--l> 00.

•

Jackson and Bernstein Approxim ation Theorems

In this subsection we formulate results that relate the speed of convergence of wavelet series to the smoothness off. We focus attention on the rate of decay of IIPJf- flip·

348

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

In order to measure the smoothness of a function, we introduce the LP modulus of continuity: (6.6.12)

wp(f; 8) := sup

lifO- f(·- h) lip·

0
This is certainly defined iff E LP(JR), but may also be defined more generally, e.g., if f = 1. The elementary properties are detailed as follows: Proposition 6.6.12. The LP modulus of continuity satisfies the following conditions:

(i) (ii) (iii) (iv) (v)

8 ~ wp(f; 8) is monotone increasing. Iff E LP(JR), then wp(f; 8) ~ 0 when 8 Wp(f; 81 + 82) < Wp(/; 81) + Wp(f; 82) Wp(/1 + f2; 8) :5 Wp(/1; 8) + Wp(f2; 8) Wp(Jaf; 8) = 2-afpwp(f; 2a8)

~

0.

Proof. Property (i) comes directly from the definition. Property (ii) is immediate for continuous functions with compact support, which are dense in LP(lll), hence the general case. Properties (iii) and (iv) follow from the triangle inequality for the U norm. Property (v) is a direct computation when we recall the defining properties of JJ from (6.6.8). •

We define the space (6.6.13)

for all

8 > 0}

Exercise 6.6.13. Iff E MCp(IR), prove that lflP E ~~(IR).

In order to prove suitable approximation theorems, we need to consider a smaller class of scaling functions, defined by an estimate of the form (6.6.14)

c

I<J>(t)j < (1

+

jti)B

B > 2.

Lemma 6.6.14. If satisfies (6.6.14), then the kernel function (t, s) satisfies the estimate

(6.6.15)

c

l(t, s)l :5 (1

+It-

si)B

Proof. The proof is identical to the proof of (6.6.7) in Proposition 6.6.1.

•

The direct approximation (Jackson's estimate) is the following statement. Theorem 6.6.15. Suppose that the scaling function satsifies (6.6.14). Then there exists C = Cp such thatforallf E MCp(IR)

(6.6.16)

INTRODUCTI ON TO WAVELETS

Corollary 6.6.16. If wp(f; 8) < C8a for some fliP ::0: C2-;a.

a with

0 <

a ::0:

349

1, then IIP1 f -

This is a direct counterpa rt of the correspon ding result for trigonome tric approxima tion, proved in Chapter 1 in case p = 2 or p = CX). Proof of the theorem. First we do the case j = 0. We pick a > 0, b > 0 so that B = a + b and ap > p + 1, bp' > 1. Applying (6.6.15), Holder's inequality, and the Fubini theorem, we obtain

II!- Pofll~;

~ <

-

l

(lif(t) -f(s)ll(t , s)ldsr dt

C { ( { if(t)- j(t + u)i du)P dt Jli<. JJR (1 + iui) 8

< C {

-

< C {

-

lf(t) - f(t + u)IP du) ( { (1 + iui)"P la (1

( {

Ja JIR

JIR

du

+ iui)bP

'du)p/p' dt

wP(f; lui)" du. (1 + iui)aP

On the interval lui ~ 1 we apply the monotonici ty of w, to majorize this contributio n to the integral by Cw,(1). On the interval lui > 1 we apply the subadditivi ty property (iv) tO write Wp(/; u) ~ 2iuiw,(j; 1), hence this contributio n tO the integral iS majorized by Cwp(/; 1) ~U/>1 lui" j(l + iui)"Pdu < 00, since ap > p + 1. To prove the estimate for j ~ 1, we use the scaling properties of JJ·

PJ- f liP1f

-/!!,

=

J1 (Po -I)J_;f

= T 11Pi!(Po

-/)J_JII,

~ C211Pw,(J_ ;J; 1)

= Cw,(f; T

1

),

where we have used the scaling property (iv) in the last line. The proof is complete.

•

Remark. This proof can be rendered more transparen t by replacing the Holder inequality with Minkowsk i's integral estimate: beginning with IPif(t)- f(t)l ::0:

we have IIPif- flip ::0:

l

L

lf(t

+

u)- f(t)IK(u)d u,

wp(f. u)K(u) du

::0: wp(f; 1)

1

iul;::l

K(u) du

+ 2wp(f;

1)

1

luiK(u) du,

iul~l

which provides the required bound. This result can be reformula ted in terms of an estimate of the best approxima tion by elements in the space vi. Let si(f) = infgEV, llf - gllp. which is the distance from f to the subspace Vi. If p = 2, then Pif provides the best approxima tion, so that

350

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVELETS

s1(j) = II/- P1!11 2 . In general, we have II/- PJfiiP > s1(f). On the other hand, given € > 0, let g E Vj so that llg- fliP < s (f) + €. Then we can write 1 f - P1 f = ( f - g)+ (g- PJf) = ( f - g)+ PJ(g- f)

llf- P1fllp < llf- gllp

+

IIPJIIp.pll g- flip

< Cllg -fliP < C(s1 (f) +E)

where we have used the uniform LP boundedn ess of the projection operators P . But 1 € > 0 was arbitrary so that we have the two-sided bound sJ
leading to the following restateme nt of Theorem 6.6.15. Corollary 6.6.1 7. Suppose that the scaling function satisfies the estimate (6.6.14). Then the distance from f E MCp(lR) to the subspace ~ satisfies the bound

We now formulate the Bernstein inequality , for which we impose a condition of smoothnes s, namely that the scaling function have a continuou s derivative ' (t) which satisfies an estimate (6.6.17)

l'(t)l < Cl(t)l

for some C > 0. From this it follows that the wavelet kernel is estimated by l(t

+ h, s)

- (t, s)l =

L((t

+ h- /)-

(t- l))(s- l)

/eZ

< ClhiK(It - sl).

Theorem 6.6.18. Suppose that the scaling function satisfies (6.6.1) and (6.6.17). Then there exists a constant Cp so that for any f E Vj n LP, we have

(6.6.18) Proof. First we do the casej = 0. Then/= P 0 f

(i 1/(t+h)

1

-f(t)!Pdt y

P

and we can write

i

[(t + h, s) - (t, s)]f(s) ds

~

lhl

i

~

lhiiiKIIIil/1 1"'

f(t +h) - f(t) =

1/(t +h)- f(t)l

E V0

K(lt- sl)lf(s)lds = lhiK

* 1/1

INTRODUCTION TO WAVELETS

351

whichprovestheresultinca selhl:::: l.Iflhl > l,thenwp(f;h):::: 211/llr,whichcompletes the proof in case j = 0. If j =I- 0, thenj e Yj iff J_1f e V 0 and we can use the scaling properties of wP to write for f E Yj

Wp(/;

8) =

r

11Pwp(J_J; 2 1 8)

:::: Cz-1 !r min(2 1 8, l)IIJ-JIIr

= Cmin(21 8, 1)11/llr-

•

The Bernstein inequality (6.6.18) can be combined with the Jackson inequality to obtain a characterization of the LP Holder continuity in terms of the speed of convergence to zero of IIPJf- fliP" The method follows closely the corresponding proofs in Chapter 1.

Proposition 6.6.19. Suppose that the scaling function satisfies (6.6.1) and (6.6.17). Then f E LP(lR) satisfies IIPJf -fliP < c2-Ja for some 0 < a < 1, if and only iff satisfies the LP Holder condition wp(f; 8) < C8a. Proof. The direct statement that IIPJf - fllr :::: cz-Ja is an immediate consequence of Jackson's estimate, as previously noted. To prove the converse, we define QJf = PJ+ 1 - PJ f, hence IIQJfllr :::: cz-Ja. Given h > o, let m = m(h) be defined by the inequalities zm-l :::: 1/h < zm. Then we write m-J

f(t) = Pof(t)

+

L

oo

QJ(t)

+

j=O

L

QJ(t)

J=m m-l

j(t +h) - j(t) = Pof(t +h) - Pof(t)

+

L[QJ(t +h) - QJ(t)] J=O

00

+ L[Q,J
- Q,J].

j=m

The first term is Lipschitz continuous, from (6.6.17). To estimate the first sum we have from (6.6.18) m-l

L

m-l

IIQJ(- +h)- QJIIr ~

j=O

L

zjhiiQJfllr

j=O m-1

~ Ch .L:zJz-ai j=O ~

Ch2m(l-a)

= Ch"'.

352

INTRODUCTI ON TO FOURIER ANALYSIS AND WAVF:LETS

To estimate the second sum we write DC

L

"X-

IIQJ(t +h)- QJ(t)il, :::: 2

L

IIQJII,

J=rrl

•

which completes the proof.

The alert reader will note that the converse Propositio n 6.6.19 has been stated only for 0 < a < 1, whereas the direct result following Jackson's inequality is valid for 0 < a < 1. Indeed, we see that the proof of the converse propositio n breaks down if a=l.

Exercise 6.6.20. Suppose that liP;/ -/lip < C8log(lj 8)for 0 < h < ~.

6.7

c2-i. Prove that Wp(/; 8)

<

WAVELE TS IN SEVERAL VARIABL ES

We conclude this introducti on to wavelets with a glimpse of the multidime nsional theory. The one major change in passing from one to several variables is the need for a system ofwavelet s, in contrast with the single wavelet \11, which suffices in the one-dimen sional case. In order to motivate this, we consider two important generaliza tions of the onedimension al Haar expansion .

6.7 .1

Two Importan t Example s

Without leaving the one-dimen sional setting, we can already see the need for several wavelets when we try to generalize the Haar expansion to describe an orthogona l expansion associated with a b-adic subdivisio n of the real line, where b E {2, 3, ... }. Consider the projection operator (k+l)jbl

P1 f(x) := b 1

1

kjbl

f(y) dy,

k bJ

<x<

k+1 bJ

and the limit relation limJ--= PJf(x) =f(x) a.e. where/ E L 2 (JR). In order to transform this into a wavelet framework , we begin with the Haar scaling function (t) = 1 . l(t) 10 1 in the context of the scaling equation (t) = (bt) + (bt- 1) + ... + (bt- (b- 1)).

Defining Vj = P1 (L 2 (JR)), Wj = VJ+l e V;. we need to exhibit a basis of the space WJ. the orthogona l compleme nt of VJ in VJ+t· It is clear that on any interval [kjbl, (k + l)jbi) the space WJ has dimension b- 1. Beginning with any basis, we can apply the Gram-Sch midt

INTRODUCTION TO WAVELETS

353

procedure to obtain an orthonormal basis. For example, in case b = 3, we define ifO < t < ~ if l3 -< t < ~3 if~1 -< t < 1

\112(t) =

Jd (t)\11 1(t) dt =

{1

-2

if 0 ::::: t < ~ if£1 -< t < 1

1

0 = f 0 (t)\11 2 (t) dt and that f 01 \11 1(t)\11 2 (t) dt = 0. Therefore we have an orthogonal basis of V 1 e V 0 on [0, 1 J. Normalizing these functions we obtain an orthonormal basis of L 2 (lR) by {\11 1 (3 1 t - k) }jEZ:.kEZ:• {\11 2 (3i t - k) } EZ:,kEZ:· 1 This demonstrate s the necessity of two different wavelets. In general, we will need b - 1 different wavelets in order to generate L 2 (lR). These can be obtained by applying the Gram-Schm idt orthogonaliz ation to the Haar functions {H(bt- k)/2}~;:~. For example in case b = 3 we can take \11 1 (t) = H(3t/2), \112(t) = 2H[(3t- 1)/21 + H[(3t)/2]. As a second example, we consider the problem of generalizing the Haar series to 2 2 L (JR ). We begin with the projection operator It is clear that

2.l
P1 j(x,y):=2 J

k/21

1/21

f(s,t)dsdt,

(x,y)E

1) x [-

k+ -.,-.1 2 2J

[ k

I

2J

I+

1)

,-.2J

and the limit relation lim1 __. 00 P1 j(x, y) = f(x, y), a.e. (x, y) E JR 2 . As above, we let Vi = Pj(L 2 (1R)) and Wj = \'J+I e Vj. The associated scaling function is (t 1 , t ) 2 1 ro. o (t 1 ) 1ro. o (t2) with scaling equation

Letting H(t) be the standard Haar function, we consider the products \11

10

(!,, t2) = H(ti)lro.n(t2 )

1 \11° (t,. t2) = 1[0. I) (t, )H (t2) 11 \11 (t" t2) = H(t,)H(t2).

It is clear that these functions are orthonormal and orthogonal to the scaling function (t,, t2). We obtain an orthonormal basis of L 2 (JR 2 ) by considering the three sets of

functions

This is the two-dimensi onal Haar wavelet basis, which is constructed in a canonical manner beginning with the one-dimensi onal Haar wavelet basis. If we perform the corresponding construction in JRd, we would need a system of 2d - 1 basic products. This general construction is described in the following subsection.

354

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

6.7. 1. 1

Tensor product of wavelets

In order to generalize the previous example, we let (EI, ... , Eo) be a multiindex, where E E {0, 1}. If {, \II} is an MRA wavelet, the symbol \11€ is interpreted as forE = 0 and is interpreted as \II is E = 1.

Proposition 6.7 .1. Let { a, Wa } 1:sa:sd be a set of one-dimensiona l MRA wavelet systems. Let~ be the span of 1(2jtl - y)2(2jt2- y) · · · d(2jtd- y) where y E zd andj E 'JL.. Then Wj := ~+I 8 Vj is spanned by {w~'

where ( E 1 ,

6.7 .2

••• ,

c2jt1 - y) ... w;dc2jtd- y)}

Ed) ranges over all2d - 1 multiindices with 0 <

E1

+ · · · +Ed

< d.

General Formulation of MRA and Wavelets in JR.d

We now abstract these examples to formulate the MRA concept in JRd.

Definition 6.7 .2. A d x d matrix A is called a dilation matrix if it has integer entries with eigenvalues larger than 1 in absolute value.

In cased = 1, a dilation matrix is defined by an integer b E {2, 3, ... }. The two-dimensiona l Haar system is associated to the dilation matrix

A=(~ ~). Definition 6.7.3. A wavelet set with respect to \11 1 ,

••• ,

\11 5

E

a

L 2 (1Rd) so that

dilation matrix A is a set

j {I detAV"/2 \II r (Axy)}ye:Z.dJeZ.I:sr:ss

is an orthonormal basis of L 2 (JR. d).

Note that the factor of I detAVI 2 is inserted in order to preserve the norm of the functions wr.

Definition 6.7 .4. A d-dimensional MRA with respect to a dilation matrix A is an increasing sequence ofsubspaces {Vj} c L 2 (1R) de.finedfor j E Z with · · · C V-1 C Vo C V1 C · · ·

together with a scaling function E L 2 (JR.) such that (i) U~-CXl ~ is dense in L 2 (1R),

f

n~-CXl ~ = {0}.

E ~if

and only iff(A-j·) E Vo. (iii) {(x- y)}ye:Z.d is an orthonormal basis of V 0 • (ii)

In the previous section we have exhibited two examples of such MRA systems.

INTRODUCTION TO WAVELETS

355

6.7.2. 1 Notations for subgroups and cosets If A is a dilation matrix, the image of zd under A is an additive subgroup of zd, denoted A(Zd). The quotient group zd /A(Zd) is obtained by applying the equivalence relation: v = w iff v- w E A(Zd) to the points of zd to obtain a collection of cosets. These cosets may be labelled by coset representatives k 1, ... , km E zd with k 1 = 0. Thus we have the disjoint union m

(6.7.1)

m

U
zd =

U U (k, + A(y)).

=

1=1

Note that A maps zd to a proper subset of zd, but A is invertible on JR.d, since q I det A I > 1. In particular, for any two sets C 1, C 2 C JR.d, A(C1 U C2) = A(CI) U A(C2), A- 1(Cr U C2) = A- 1(CJ) UA- 1 (C2),

A(C1 n C2) = A(CJ) n A(C2), A- 1(C1 nC2) =A- 1 (CI) nA- 1(C2).

Proposition 6.7.5. The number of elements in the quotient group zd /A(Zd) is equal to I detAI. This will be proved using the following simple lemma on sets whose integer translates cover JR.d. Lemma 6.7.6. Suppose that Q C JR.d has integer translates that cover JR.d: JR.d. Then IQI > 1 with equality iff these integer translates are a.e. disjoint: Uyezd(Q + y) -

iff

IQI =I

IQn(Q+y)I=O,

VyeZd,

y-:/=0.

Proof. Let Qo := [0, I]d,f(x) := Lyezd IQ(x- y), an integer-valued periodic"function on JRd withf(x) 2: 1, by hypothesis. Clearly (6.7.2)

IQI

=

1

lQ(X) dx

JRd

=

L YEZd

1

lQ(x) dx

Qo+Y

=

1

j(x) dx 2: l.

Qo

If IQI = 1, then (6.7.2) shows that J,.!o (j(x)- 1) dx = 0, hencej(x) = 1 a.e. x E IR_d, hence for a.e. x E JRd the series definingj(x) contains only one nonzero term, in particular for any y =f. 0 we have a.e. 1 =f(x) 2: lQ(x) + lQ(x- y), which proves that IQ n (Q + y)l = 0. Conversely, if IQ n (Q + y)l = 0 for ally =I- 0, then the series definingj(x) contains a.e. at most one nonzero term, since x - y 1 E Q and x - y 2 E Q contradicts the hypothesis with Y = Y1 - Yz and x replaced by x - Yl· Thusf(x) ~ I a.e., which implies thatj(x) = I a.e., hence (6.7.2) implies that IQI = l. • Proof of Proposition 6.7.5. Using the disjoint union ( 6. 7.1 ), we can write JR.d as the a.e. disjoint union: m

IR.d=

U
i=l

yezd

(Qo+k,+A(y)).

356

INTRODU CTION TO FOURIER ANALYSIS AND WAVELET S

Let Q be the a.e. disjoint union Q := U7~ A- 1 (Qo 1 m ldet(A- 1 )1 = mjq. Then for y E 1'./ we can write Q

+

"'

y = uA- 1 (Qo

+ k,

+

k,), a set of measure IQI-

+A(y))

t=l

m

u

(Q

+

y)

=

yt::.Zd

u

uA-I(Q o

+ k,

+A(y))

YE'Zd ,=1

~A

'('d" Q
=A - I (J!~d)

=

.IR.d

so that the integer translate s of Q cover JR.d. But IQ

]A (Q n(Q + y)) j =

n (Q +

jA(Q) n(A(Q )

= )Q(Qo

+ k,) n

y) I = 0 for y =I= 0, since

+ A(y))j Q(Qo

+ ki +A(y)) l

=0. Therefo re we can apply Lemma 6.7.6 to conclud e that 1 proved.

6.7.2.2

IQI = m/q, which was to be

•

Riesz system s and orthon ormal system s in .!Rd

It is straight forward to general ize the notion of Riesz system to JRd.

Definit ion 6.7.7. A set of functio ns {
E

L 2 (1Rd) is a Riesz system iff there

exist constan ts 0 < c :S C < oo such that for any finite set of comple x number s (ad

In particu lar we have an orthono rmal sequen ce if and only if c = C = 1.

Propos ition 6.7.8. Let Fe L 2 (1Rd). • {F(x- y) }yciR" is a Riesz system iff c :S LiEZ" IF(~ - /)1 2 :S C a.e. ~ E JRd. • The sequen ce is orthono rmal iff L!EZ" IF(~ - l) f = 1 a. e. • lfG E L 2 (1Rd), the.p the setAoft ranslat es {G(x- y)}yElR" is orthogo nal to F and only ifL 1a" G(~- l)F(~- /)I = 0 a.e. ~ E .!Rd.

if

INTRODUCTION TO WAVELETS

357

Proof. The Fourier transform of Lk akF(x - k) is Lk ake-z:rr,, t; F(l;) . - A(;)F(;). Therefore from the Plancherel theorem and periodization, we have

=

z= lro.r

iA
IELd

= If c ~ I.::,Ezd

iF(; +

L

l) i 2 ~

f lro. 11J

J]d

iA(,;)i 2

(z= ih.;

+ l)i 2 ) d;.

IEZd

C, then Parseval's theorem for Fourier series yields

iiakF(x- k)ii 2

~

iA(0i 2 d;

C {

lro

1-

=

C

L

iaki

2

kEZd

Jjd

and

which proves the Riesz property. On the other hand, if {F(x- k)hEz" is a Riesz system, then for any trigonometric polynomial A. we have

fro. iA<.;W (I.::,Ezd ih.; + tW) d~ fr diA(;)I2d;

c <

11 d

. 0 11


-

First we take a sequence of trigonometric polynomials that converge boundedly to the indicator function n~= 1 lra,.b,J• then we let b, --+ a, to conclude that for a.e. ~ E ][~d. c ~ 2 LiEZd iF (I; + l) i ~ C. The orthogonality statement is proved by writing

=

=

L

IEZd

1

1

[O,J]d

A(;)F(l;)G(~- l) d;

[O,J]d

A
(z= f<;)G(;- t)) d;. J,c]d

The inner sum is a.e. zero if and only if all inner products on the left are zero.

6.7.2.3

•

Scaling equation and structure COTf~tants

Since V, is spanned by translates of { (Ax - y) }yEZd, there exist constants ay so that we have the L 2 convergent sum (6.7.3)

(x) =

Lay cf>(Ax- y) yEZd

where the structure constants satisfy LyEZd layl 2 < oo. Relation (6.7.3) is the scaling equation for wavelets in JRd and plays the same role here as for d = 1. The scaling filter

358

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

is defined by

m 0 (.,t:) :=

(6.7.4)

1

""'"' ~aye -

2rr

i~. Y.

I detAI yEa:.'"""

The Fourier transformed equation, proved below, is written (6.7.5)

Then we have the counterpart of Theorem 6.4.27.

Theorem 6.7.9. Suppose that

E

L 2 (.1R) is such that

(i) The translates {(x- y)}yEZd are orthonormal. (ii) (x) = L y EZd ay(Ax- y), an L 2 -convergent sum. (iii) The Fourier transform (~) is continuous at~ = 0 with 1<1>(0)1 = 1. Define Yj =span {(Aix- y)}yEZd· Then {V1} defines an MRA system with respect to the dilation matrix A. A

A

The proof follows the one-dimensional development and is left to the following exercises. As before, the MRA property reduces to the behavior of the projection operators P1, which satisfy the uniform boundedness property in L 2 : IIP1 112. 2 = 1.

Exercise 6.7 .1 0. For any f

E

L 2 (.IR), lim1._ -oo P1 f = 0.

Hint: First prove this on the dense set of continuous function with compact support.

L 2 (.1Rd) with a Fourier transform} that is bounded and supponed in [-R, R]d for some R > 0. Thenfor allj sufficiently large, we have

Exercise 6.7.11. Let f

E

(6.7.6) Hint: Imitate the proof of Lemma 6.4.29.

When we combine the continuity hypothesis (iii) with the identity in Exercise 6.7.11, we see that for a dense class off E L 2 (.1Rd), limJ.-oo IIP1fllz = llfll2. which implies that P1f ~ f for allf E L 2 (.!Rd). 6.7.2.4

Existence of the wavelet set

It now remains to constlllct the wavelet system {\II 1 , ••• , ws} as previously announced. To do this, we first need to describe the orthogonal complement W 1 = V 1 e V 0 •

INTRODUCTION TO WAVELETS

359

Lemma 6.7.12. Suppose that cl> is the scaling function of an MRA with respect to the dilation matrix A. Let f E VI. Then the Fourier transform satisfies

(6.7.7)

1

(6.7.8)

lmJ(~)I2 d~

llfll~

=

I detAI

[O.IJ"

LyEZd !ay!z

-

I detA1 2

Proof. Any f E V 1 has the L 2 convergent expansion

(6.7.9)

f(x) =

L

ay(Ax- y).

yEZd

We must compute the Fourier transform of (Ax - y). We make the change of variable y = Ax - y to write

{

}Rd

(Ax

-

y )e_z,.,{

x

dx =

l

I detAI

{

}Rd

= e-2Jr•i; A- 1(y)

( y)e-Zrr•~ A_, (y+y) {

J[td

(y)e-2"''

dy

1 A- (v)

d~

We define (6.7.10) Taking the Fourier transform of (6.7.9), we have (6.7.11) Letting~

= A*1J, we obtain (6.7.7) as written. Now we compute the L 2 norm in two different ways. From (6.7.9) we have

On the other hand, from (6.7.11), we have {

l~&d

lf(A*~)I 2 d~ = JfRd { lmt(~)l 2 icP(~)I 2 d~

360

INTRODIJCTION TO FOURH:R ANALYSIS AND WAVELETS

where we have used the orthonormality of {(x- y)} in the last step. Combining these two calculations, we have

id

1 I de tAI

2

lf(x)l dx

Ly.:::zd lay 12 I detAI 2

•

The orthonormality properties of and the wavelet basis are translated into unitary properties of the functions m1 , as follows. We rewrite the scaling equation (6.7.5) as (6.7.12) We denote by r I, . . . , r q a set of representatives for A* (Zd) and their antecedents by k1, ... , kq. defined by A*(k,) = r,. The following identity is useful for dealing with orthogonality. Lemma 6.7 .13. Let be the scaling function of an MRA, with scaling filter m 0 . Letf, g E V1 with}(A*l;) = mf(/;)<1>(1;), g(A*l;) = m 8 (l;)(l;). Then q

(6.7.13)

L:i
+ l)g(A*l; + l)

L

=

IEZd

I=

m1 (l;

+ k;)mg(l; + k;).

I

Proof. We break up the sum (6.7.13) according to the cosets defined by A*('£/). Each point of 'lld is uniquely represented as I =A • y + 1, where I :::=:: i :::=:: q and y E 'lld. Thus

}(A*~+ A*y

+

1,)g(A*~

+ A*y + 1,) =

mf(~

= mJ(~

+ y + k,)m8 (~ + y + k;)l(~ + y + k;)l 2 + k,)m 8 (~ + k,)l(~ + y + k,)l 2

q

Lf(A*~

+ l)g(A*~ +I)=

IEZd

L

L

t=

mf(~

I yEZd

+ k,)m8 (~ + k,)l(~ +

y

+ k,)l 2

q

=

L: mJ<~ + k;)m~~<~ + k,> L: I<~><~+ y + k,) 1 2

<=I

yEZd

q

=

L

mf(~

+ k,)m 11 (~ + k,)

where we have used the orthonormality of (x- k) in the last step.

•

This identity is used repeatedly in what follows. Proposition 6.7 .14. Let be the scaling function of an MRA with scaling filter E JRd, we have

mo. Then the scaling filter satisfies the condition that for a. e. l; q-1

(6.7.14)

L

r=O

lmo(l;

+

kr)l

2

=

1.

INTRODUCTION TO WAVELETS

361

If m 1, ... , mq-l correspond to the wavelet basis {\II 1 , ••• , wq-l }, then the row vectors (m,(~ + k, ), ...• m,(~ + kq)) are mutually orthogonal unit vectors in cq fori= 1, ... , q - 1 and are orthogonal to the vector (mo(~ + k,), ... , mo(~ + kq)). Conversely, if we have ]-periodic functions m, (0 .... , mq-1 (0 with the aforementioned orthogonality properties, then defining f,, ... ,fq-1 by (6. 7. 7), we obtain a wavelet set. Hence a d-dimensional wavelet is described by a q x q unitary matrix just as in the one-dimensional case, where q = 2. Proof. We take/ = g = in (6.7.13), noting that the left side is a.e. equal to I, by orthonormality, hence (6.7.14) follows. Now if we have a wavelet set \f/ 1 , ••• , \flq- 1 , then we take f = \II', g = to deduce the orthogonality of the first row with the other row vectors. Taking f = \II', g = \111 proves the mutual orthonormality of the remaining row vectors. Conversely, if we are given 1-periodic functions m 1 , ••• , mq-l with the aforementioned orthogonality properties, we apply (6.7.13) repeatedly to deduce that the functions {f; (x - k) h ~· ~q-l ,h z:d constitute an orthonormal family. •

Remark. It is noteworthy that the number of wavelets is equal to q = I detAI, independent of any details of the scaling equation. 6.7.2.5

Proof that the wavelet set spans V 1

e

Vo

Proof. It remains to prove that any f E V1 e V 0 is an L 2 convergent sum of linear combinations of integer translates of \11 1, ... , \llq-l_ Since V 1 is spanned by {(Ax- y)}yezd, it is sufficient to prove that functions of this form are equal to their projection on the integer translates of , \11 1 , ••• , wq-l. The following computation is carried out in the case y = 0. Using the notation B =A*, the projection of (Ax) is written

L

ck(x- k)

L

+

kEZd

diw'(x- k)

+ ··· +

kEZd

L

d%-'wq- 1 (x- k).

kEZd

Applying Parseval's identity, the scaling equation and periodization, we obtain ck =

{ (Ax)(x- k) dx J.ad = !. { cP(B- 1 ~)¢>(~)e2n-•k< d~ q

]Jll.d

= _!_ { q ]Jll.d = =

{

]Jll.d

mo(B- 1 ~)icP(B- 1 ~)i 2 e 2 'l'lk·~ d~

mo(7]) IcP(7]) 12e2rr•k Bry d7]

{

mo(7])e2rr1Ak ry d7]

lro.IJd

= _!_ { q JB[O,J]d

mo(B-I~)e2rr•l;kd~.

362

INTRODUCTION TO FOURIER ANALYSIS AND WAVFLFTS

But B[O, I]d is the disjoint union of q sets each of which is congruent to [0, I]d. Hence we can write (6.7.15) so that by Parseval's identity we have (6.7.16)

L led=~ q~

,E;zd

{

JIO.lld

~~mo(B a~ll

1~

2

+

ka)l

d!;.

We compute the other sums in the same fashion, replacing mo by m 13 , 1 :::0: {3 :::0: q - 1, thus obtaining (6.7.17) When we add the terms of (6.7.16) to those of (6.7.17) and apply the unitary properties of the matrix m 13 (t; + k"), we find q terms, each equal to 1jq 2 , hence the sum is Ijq. On the other hand, fril.d I (Ax) l2 dx = 1 I q, and the proof is complete. •

Exercise 6.7.15. Carry out the computatonfor (Ax- y) with y =I= 0.

6.7.2.6 Cohen's theorem in JR.d In order to construct a scaling function from the scaling filter mo, we formulate the extension of Cohen's Theorem 6.5.19 to the multidimensiona l case. As before, the main task is to prove that the infinite product n~ 1 mo((A*)-1~) has orthonormal translates. We restrict attention to the case of trigonometric polynomials, which is equivalent to scaling functions of compact support. Theorem 6.7.16. Let A be a dilation matrix and suppose that m 0 is a trigonometric polynomial that satisfies m 0 (0) = 1 and lmo(~ + kr)l 2 = 1. Suppose that there exists a set K C JR.d with the properties that K contains a neighborhood ofO, LyeZd lK(~ + y) = 1 a.e. and mo((A*)-1~) # Ofor j = 1, 2, .... Define

2:;:6

o11 -t (mo((A*)- 1 ~). Then <1> converges 1 in L 2 (JRd) to , which is the scaling function of an MRA.

The proof follows exactly the steps I through 4 in the proof of Theorem 6.5.2, as modified in the proof of Cohen's theorem in one dimension. The details are left to the reader.

Exercise 6.7 .17. Prove that the scaling function has compact support only if the scaling filter m 0 is a trigonometric polynomial.

6.7 .3

if and

Examples of Wavelets in JRd

We can construct an interesting class of d-dimensional wavelets by considering scaling functions that are the indicator functions of suitable measurable sets. If <1> = IQ is a scaling function, then orthonormality requires that (i) the integer translates of Q be a.e.

INTRODUCTI ON TO WAVELETS

363

disjoint, and (ii) IQI = I. These imply that/Q(x) := LyEZ:" 1Q(x + y) = I a.e., since (i) implies that/Q(x) .:=: 1 a.e. and (ii) implies that I = IQI = J; ., "/Q(x)dx. Hence 0 1 0 = fto.IJ"(l - fQ(x)) dx, which proves thatfQ(x) = I a.e. Now the scaling relation is written (6.7.18)

IQ(x) =

L a y IQ(Ax- y). )' f-

=~J

Conversel y, if we are given a set Q for which LyEZ" 1Q(x + y) = 1 a.e. and which satisfies a scaling relation (6.7.18), then we obtain an MRA wavelet, since 1Q E L 1 (JRd) implies that is continuou s at l; = 0 with d>(O) = l. Exercise 6.7.18. lf

(6.7.19) where s1 E S. The series is clearly convergen t, since we have the estimate lA -J xj ::S: ca-ijxl for any dilation matrix, where 0 < a < I and C > 0. Furthermo re Q satisfies a scaling equation, since for any x E Q 00

x = A- 1 (sJ) +A- 1 LA-i(s1 + 1 ) = A- 1(sJ) +A- 1y j=l

withy E Q. This means that x E Q if and only if for some i, we have Ax- ri E Q for some i = 1, ... , q. Thus IQ(x) = Lf=I 1Q(Ax- r,). The scaling relation (6.7.18) is satisfied with ay = I if y = r, for some i and ay = 0 otherwise. In order to prove the orthonorm ality, we appeal to Theorem 6. 7 .16. This requires that we study the properties of the scaling filter 1 mo(l;) = q

(6.7.20)

L q

e-2Jul;

k,.

1=1

If we can find a set K to satisfy the conditions of Theorem 6. 7 .16, then we can assert that I Q is the scaling function of an MRA wavelet. The details will depend on the choice of the representa tives k 1 , ••• , kq. Example 6.7 .19.

If

A=(~ ~) wechoose k1

=

(0, 0), k2

=

(1, 0), k3 = (0, 1), k 4 (l, I)asaseto frepresen tatives.

364

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

Then

mo(O = =

mo(~I, ~2)

.!_ 0

+ e-2rril;, + e-2rril;2 +

e-2rrt(l; 1 +1;2))

4

= .!_ (1

4

+

e-2rril; 1 )(1

+

e-2rrt1; 2 ).

-1, 1]

2 , thenm 0 (~) =I= 0 onK, sothattheconditionsofTheorem6.7.2 IfwechooseK = [ are satisfied. The set Q is simply the set of binary expansions of pairs of real numbers in [0, 1], hence Q = [0, 1 ] 2 , so that we obtain th~ two-dimensional Haar wavelet, discussed previously.

Example 6.7.20. With the same choice of A, let k 1 (0, 1), k4 = (1, 2).

-

(0, 0), k 2

-

(1, 1), k 3

-

In this case the scaling filter is

mo(~)

1, 1]

=

~(1 +

e-2rrt(1;,+1;2)

+ e-2rril;2 + e-2rri(/; +1;2J). 1

2 , we again infer that the m 0 ( ~) =I= 0 on K. In this case the set Q is Letting K = [the rhombus obtained as the convex hull of the four points k 1 , k 2 , k 3 , k 4 .

REFER ENCES

R. Banuelos and C. N. Moore. 1999. Probabilistic Behavior of Harmonic Functions. Basel: Birkhauser. W. Beckner. 1975. Inequalities in Fourier analysis on JR:.n. Proceedings of the National Academy of Sciences 72: 638-641. M. Benedicks. 1985. On Fourier transforms of functions supported on sets of finite Lebesgue measure. Journal of Mathematical Analysis and Applications 106: 180-183. D. Bernoulli. 1753. Reflexions et eclairements sur les nouvelles vibrations des cordes. Mem de l'Academie des Sciences de Berlin 9: 173ff. S. N. Bernstein. 1912. On the best approximatio n of continuous functions by polynomials of a given degree. Mem. Acad. Roy. Belgique 4: 1-104. P. Billingsley. 1999. Convergence of Probability Measures. New York: Wiley. R. P. Boas. 1972. Summation formulas and band-limited signals. Tohoku Journal of Mathematics 24: 121-125. S. Bochner. 1931. Ein konvergenzsa tz ftir mehrvariablig e Fouriersche Integrate. Math Zeit 34: 440-447. S. Bochner. 1936. Summation of multiple Fourier series by spherical means. Transactions of the American Mathematical Society 40: 175-207. L. Brandolini and L. Colzani. 1999. Localization and convergence of eigenfunction expansions. Journal of Fourier Analysis and Applications 5: 431-447. A. Calderon and A. Zygmund. 1952. On the existence of certain singular integrals. Acta Mathematica 8: 85-139. A. Calderon and A. Zygmund. 1956. On singular integrals. American Journal of Mathematics 18: 289-309. L. Carleson. 1966. On convergence and growth of partial sums of Fourier series. Acta Mathematica 116: 135-157. I. Daubechies. 1992. Ten Lectures on Wavelets. Vol. 61. CBS-NSF Regional Conference in Applied Mathematics, SIAM. B. Davis. 1974. On the weak type (1, 1) inequality for conjugate functions. Proceedings of the AMS 44: 307-311. M. Davis and S. Y. Chang. 1987. Lectures on Bochner-Ries z Means. LMS Lecture Notes No. 114. Cambridge: Cambridge University Press.

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REFERENCES

P. G. Dirichlet. 1829. Sur Ia convergence des series trigonometriq ues qui servent a representer une fonction arbitraire entre des limites donnees. J. Reine und Angewand Math. 4: 157-169. H. Dym and H. P. McKean. 1972. Fourier Series and Integrals. San Diego: Academic Press. C. Fefferman. 1970. Inequalities for strongly singular convolution operators. Acta Mathematica 124: 9-36. C. Fefferman. 1973. Pointwise convergence of Fourier series. Annals ofMathematics 98: 55 1-571. G. B. Folland. 1995. Introduction to Partial D~fferential Equations. Princeton, NJ: Princeton University Press. G. B. Folland and A. Sitaram. 1997. The uncertainty principle: A mathematical survey. Journal of Fourier Analysis and Applications 3: 207-238. G. B. Folland. 1999. Real Analysis. New York: John Wiley. J. B. Fourier. 1822. Theoric Analvtique de La Chaleur. Paris: Dover reprint, 1955, New York. L. Garding. 1997. Some points of analysis and their history. Providence, RI: American Mathematical Society. F. Grunbaum. 1990. Trying to beat Heisenberg. In C. Sadosky (ed.), Analysis and Partial D~erential Equations. New York: Marcel Dekker, pp 657-665. G. H. Hardy. 1949. Divcrf?ent Series. Oxford: Clarendon Press. E. Hernandez and G. Weiss. 1996. A First Course on Wavelets. Boca Raton. FL: CRC Press. R. Hunt. 1968. On the convergence of Fourier series. In Proceedings of the Conference on Orthogonal Expansion and their Continuous Analogues. Edwardsville, IL: Southern Illinois University Press, 234-255. S. lgari. 1996. Real Analysis-Wit h an Introduction to Wavelet Theory. Vol. 177. Providence, RI: AMS Translations of Mathematical Monographs. C. Jordan. 1881. Sur Ia serie de Fourier. Comptes Rendus de l'Academie des Sciences, Paris 92: 228-230. M. Kac. 1939. Power series with large gaps. American Journal of Mathematics 61: 473-476. J.P. Kahane andY. Katznelson. 1966. Surles ensembles de divergence des series trigonometriq ues. Studia Mathematica 26: 305-306. J.P. Kahane. 1968. Some Random Series of Functions. Lexington, MA: D.C. Heath. J.P. Kahane. 1995. Le phenomene de Pinsky et Ia geometric des surfaces. Comptes Rendus de l'Academie des Sciences, Paris 321: 1027-1029. J.P. Kahane and P. G. Lcmarie Rieusset. 1995. Fourier Series and Wavelets. Vol. 3. In Studies in the Development of Modern Mathematics. New York: Gordon Breach. Y. Katznelson. 1976. An Introduction to Harmonic Analysis. New York: Dover Reprint. D. G. Kendall. 1948. On the number of lattice points inside a random oval. Quarterly Journal of Mathematics, Oxford Series 19: 1-26. A. Kolmogorov. 1935. Sur Ies fonctions harmoniques conjugees et les series de Fourier. Fundamenta Mathematica 7: 23-28. A. Kolmogorov. 1926. Une serie de Fourier-Lebes gue divergente partout. Comptes Rendus de l'Academie des Sciences de Paris 183: 1327-1328. T. Komer. 1989. Fourier Analysis. Cambridge: Cambridge University Press. P. Levy. 1948. Processus stochastiques et mouvement brownien. Paris: Gauthiers-Vil lars. J. C. Maxwell. 1860. Illustrations of the dynamical theory of gases. Philosophical Magazine and Journal of Science 19: 19-32. Y. Meyer. Ondlettes. Hermann; translated as Wavelets and Operators. Cambridge, UK: Cambridge University Press, 1990. M. Pinsky. 1994. Pointwise Fourier inversion and related eigenfunction expansions. Communications in Pure and Applied Mathematics 47: 653-681. M. Pinsky and M. Taylor. 1997. Pointwise Fourier inversion-a wave equation approach. Journal of Fourier Analysis and Applications 3: 647-703.

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M. Pinsky, N. Stanton, and P. Trapa. 1993. Fourier seric~ of radial functions in several variables. Journal of Function al Analysis I 16: I I 1-132. M. Riesz. 1927. Sur le~ fonctions conjugee s. Mathema tische Zeitschri ft 27: 218-244. W. Rudin. 1987. Real and Complex Analysis. New York: McGraw- HilL R. Salem and A. Zygmund . 1948. On lacunary trignome tric series II. Proceedi ngs lJ{the National Academy of Sciences 33: 54-62. K. T Smith. 1956. A generaliz ation of an inequality of Hardy and Littlewoo d. Canadian Journal of Mathema tics 8: 157-170. C. D. Sogge. 1993. Fourier lnteRr.. 1971. Introduct ion to Fourier Analysis on Euclidean Spaces. Princeton , NJ: Princeton Universit y Press. R. Strichartz . 1994. Con~truction of orthonorm al wavelets. In J. J. Benedetto and J. W. Frazer (eds.), Wavelets, Mathema tics and Applicati ons. Boca Raton, FL: CRC Press. R. Strichartz . 2000. Gibbs' phenome non and arclength . Journal of Fourier Analysis and Applicati ons 6: 533-536. D. W. Stroock. 1993. Probabili ty Theory. Cambridg e, UK: Cambridg e Universit y Press. M. E. Taylor. 1996. Partial Different ial Equation s, 3-volume work. New York: Springer Verlag. M. E. Taylor. 1999. Schriidin ger Equation and Gauss Sums. Preprint by Universit y of North Carolina, Chapel HilL E. C. Titchmar sh. 1986. Introduct ion to the Theory of Fourier Integrals, 3rd ed. New York: Chelsea. P. Tomas. 1975. A restriction theorem for the Fourier transform . Bulletin of the American Mathema tical Society 81: 477-478. G. Watson. 1944. A Treatise on the Theory of Bessel Functions . Cambridg e, UK: Cambridg e Universit y Press. N. Wiener. 1933. The Fourier Integral and Certain ofIts Applicati ons. Cambridg e. UK: Cambridg e Universit y Pre>.s. P. Wojtaszc zyk. 1997. A Mathema tical Introduct ion to Wavelets. Cambridg e, UK: Cambridg e Universit y Press, LMS Student Texts No. 37. A. Zygmund . 1959. Trigonom etric Series. 2nd ed .. Cambridg e, UK: Cambridg e Universit y Press. A. Zygmund . 1974. On Fourier coefficien ts and transform s of functions of two variables. Studia Mathema tica 50: I 89-20 I .

NOTATIONS

A \ B is the difference of the two sets, defined as {x

E

A : x ¢: B}

AD.B is the symmetric difference of the two sets, defined as (A\ B) U (B \A)

IA is the indicator function of the set A, defined by 1A (n)

=

1 if n E A and 1A (n)

= 0 if

nfj:A

"iff" means "if and only if" lR = {x : -oo < x < oo}, the real number system

JR+ = {x E lR: 0:::: x < oo}, the nonnegative real numbers ]Rn = {(x 1 , ... , xn) : x; E lR for 1 < i < n}, then-dimensional Euclidean space Z = {0, ±1, ±2, ... }, the mtegers z+ = {0, 1, 2, ... }, the nonnegative integers LnEZ an is the symmetric infinite sum, defined as limN .... oo L~=-N an 1r = 1Rj2:rr Z, the circle, identified with ( -:rr, :rr] C('lr) = the space of complex-valued continuous functions on 'lr, identified with continuous 2:rr -periodic functions on lR IP('lr) is the space of complex-valued measurable functions on 1r with 11/llp .((2:rr)-1 f1r 1/(0)IP de) l/p < oo, where 1 < p < oo L 00 ('lr) is the space of complex-valued measurable functions on 1I' with esssup8 ETif(8)1 < oo

~~(lR) is the space of complex-valued measurable functions on lR with oo for each M > 0

11/lloo

:=

J!:!M lf(x)l dx

<

Var(j) is the total variation of the complex-valued measurable function/ on 'lr, defined as the supremum of I:, If (xi+ 1) - f (x;) I taken over all finite partitions of 1I' fh is the translate off E L 1 ('lr), defined by fh(O) =f(O- h)

(/ * g)(O)

= (lj2:rr) f1rf(¢)g(8- ¢) d¢, the convolution off, g

E

L 1 ("lr)

369

370

INTRODUCTION TO FOL'RIER ANALYSIS AND WAVELETS

2~

(f, g) =

f..wf(B)g((}) d(), the inner product off, g E L 1 ('lf). I - r2

Pr(B) =

I

+

r2 - 2r cos ()

, the Poisson kernel, defined for 0.:::; r < I,() E 1I'

2r sin () . p msson . k erne 1, d e fi ne d 10r c 0 .:::; r < I , () , the conJugate 1 + r2 - 2r cos () w(j; h)= sup11 E!L!II:<:h If(()+ y ) - j(B)I is the modulus of continuity off E C('JI') Qr(()) =

np(f; h)= suplyl.:<:h ( 2 ~ fr: If(() +y) -j(O)I"d())

f

11

P,

E

'11' JJ.

the U' modulus of continuity of

E LP('lf)

1 f(n) = - - JTj(())e-m 11 d() is the nth Fourier coefficient off E L 1 ('lf).

27T

sin(N + ~ )(} . 0 is the Dirichlet kernel, defined for N

DN(()) =

SJO

Ln =

2~

E

z+

and 0

#- ()

E

'li'

JT IDn(¢)1 d¢ is the n 1h Lebesgue constant

Si(x) = ~ SNJ(()) =

f

2

f;{

sin t /t dt is the Sine Integral function, defined for x

L~=-Nf(n)e"'

0

E

JR+

is the Nth symmetric partial sum of the Fourier series of

ELI ('lf)

aNf(B) = 1/(N +I) L~=-N(l Fourier series off E L 1 ('lf)

lni/(N + l))f(n)em 8 is the Nth Fejer mean of the

rn(()) = 2a2 ,._ 1 ( ( ) ) - a,_ 1 (B), the de la Vallee Poussin mean of order n

PrJ(())= LnEZ ri"IJ(n)emfl is the Abel mean of the Fourier series off E L 1 (1I') [x] is the integer part of x E IR (x) = x - [x] is the fractional part of x

E

IR

card(A) is the number of elements in the set A lA I is the Lebesgue measure of the set A PN is the space of trigonometric polynomials of degree N, functions of the formj(B) = '"'N Lk=-N

c k eik(l

Aa is the set of Holder continuous functions, with w(f; h) .:::; Cha for some C > 0,

0 < a .:::; 1 for all h > 0 A* is the Zygmund class, {f E C('lf) : lf(B +h) 0, () E 'lf}

+ j(()

-h) - 2j(B)I < Ch, Vh >

f 0:r 12 (sin Nujsin u) 4 [f(x + 2u) + j(x- 2u)] du, the Jackson mean of 4 at level N, where hN = fo:r 12 C:7:Uu) du.

JNf(x) = (lj2hN)

order 4

j
E

z+

c E A a} (da)n =an- an-I· the first difference of the sequence {a,.}

lm(t) = 2 ~ j_~:r e-unlletco' 0 d(), the m 1h modified Bessel function, defined form E and t E IR

z+

NOTATIONS

371

f(t) = O(g(t)), t ---+ ex> means that there exists M > 0 such that lf(t)l ~ Mg(t) for t > M f(t) = o(g(t)), t---+ ex> means that JimHxJ(t)j g(t) = 0 f(~) = (Ff)(~) is the Fourier transform off E L 1 (JR.n), defined by j'Jt,,f(x)e- 2rcit;.x dx (j

* g)(x)

= J'Jt,J(y)g(x - y) dy is the convolution off. g E L 1 (.!R")

a 11 ) is a multiindex, where a, E z+ · · · + a 11 is the norm of the multiindex

a = (a 1 ,

... ,

lal = a

+

1

Daf(x) =

aXIa,

acxf ...

aX u,

. th e m1xe . d partta . I d envattve . . o ff

IS

11

11/llk,m = sup~E'Jt".lal< m( I + lxl)k ID"'f(x) I S = {f : llfllk.m < CX!, Vk E ;:z:+, m E ;:z:+} e-lxl 2 /4r H,(x) = is the heat kernel of .!R" (4rrt)"/ 2 1 -cos Tx ., kr(x) = - - -- - is the FeJer kernel of lR 2 rrTx Py(x) =

"Y

rr(r

+ y2)

is the Poisson kernel of IR

Buc(IR 11 ) is the space of complex-val ued bounded and uniformly continuous functions on

.!R" Co(lR") = {f E Buc(.lR"): liml:xJ(x ) = 0} C 00 (.!R") is the set of continuous functions on IR" with compact support Bk is the Banach space of complex-val ued functions/ on lR with J'Jtf(x)j(l

+

lxlk) dx <

(X)

LP(.lR") is the space of complex-val ued measurable functions on .!R" with 11 11/llp = (J'R., lf(x)l" dx) " < ex>

Leb(f) is the Lebesgue set off E L/,x:(.!R"), defined as those x limr-+0 r-n J;y-xl 9 if(y)- f(x)l dy = 0

E

.!R" for which

P(x, y) is then-dimens ional Poisson kernel, defined by P(x, y) = J'Jt, e 2rc•x·.;e- 2rcyl.;l d~ DM(x) is the Dirichlet kernel of .!R, defined by DM(X) = si:~x forM > 0, 0 =I= X E lR SM,Nf(x) is the two-sided Fourier partial sum off

E

L 1 (lR), defined by J~Mj(~)e 2 rcix.; d~

SNf(x) = S-N.Nf(x) is the symmetric Fourier partial sum off E L 1 (JR) 00 Fe(~) = J 0 f(x) cos(rr~x/2) dx is the Fourier cosine transform off E L 1 (JR+) 00 F's(~) = j 0 f(x) sin(rr~x/2) dx is the Fourier sine transform off E L 1 (JR+) Do(f) = J'Jt x 2 if(x)i 2 dxj J:H. lf(x)l 2 dx is the dispersion about zero off E L 2 (.!R) 2 Hk(x) = (1,)k eUx-r 12 > lr=o is the k 1h Hermite polynomial

Rj[(x) = J'Jt" i~;/l~le 27Tit; '](~)d~ is thefh Riesz transform off E L 2 (.!R") SR.f(x)A is the spherical partial sum of the Fourier integral off E L 1 (.!R"), defined by J;.;l::::Rf(l;)e2rc,.;-x d~

372

INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS

](x) is the periodization off E L 1 (JR.d), defined by ](x) = LnEzdf(x- n) KM,a is the Bochner-Riesz kernel, defined as the Fourier transform of (1 1.; 12 / M 2 Y' liO,M] (I.; I)· 1/ra,b(x) is the rescaled function defined by lal- 112 1/f((x- b)/a). Wl/tf(a, b) is the wavelet transform off E L 2 (1R.), defined as the convolution off with the rescaled function 1/ra.o Fn is the dyadic partition of JR., consisting of the intervals ((k- l)/2n, k/2n1kEZ Vn is the n 1h space in a multiresolution analysis P nf is the orthogonal projection off onto the space Vn (t) is the scaling function for a multiresolution analysis \II (t) is the orthonormal wavelet associated with the scaling function mo(.;) is the scaling filter associated with the scaling function m 1 (~) is the wavelet filter associated to the wavelet \II

INDEX

A

Abel means, 45, 64, 261, 266 Abel sums, 49 Abel's lemma, 167 Abelian theorem, 54 Absolutely continuous functions, 21 Absolutely convergent trigonometric series, 3,43 Almost-every where convergence, 20, 28, 49, 153,346 Analytic function, 22, 172 Approximate identitites, 45, 97 Averaging, 118 B

Banach space, 75 Banuelos, R., 202 Beckner, W., 174, 175 Benedicks, M .• 240 Bernoulli, D., 1 Bernstein's inequality, 122 Bernstein's theorem, 43, 70, 129, 350 Berry-Esseen theorem, 276 Bessel functions, 6, 81, 140, 154, 164 Bessel's inequality, 36 Boas, R. P., 228 Bochner, S., 139 Bochner's method of subordination , 106 Bochner's theorem on positive-defin ite functions, 220, 270 Bochner-Ries z summability, 152

Bounded linear operator, 75 Bounded variation, 27, 60 Borel-Cantell i lemmas, 281 Brandolini, L., 149 Brownian motion, 299

c Calderon, A. P., 194, 196, 209 Calder6n-Zyg mund decompositio n, 209 Calder6n-Zyg mund theorem, 211 Cantor measure, 20 Cantor-Lebes gue theorem, 87 Cantor's uniqueness theorem, 86 Carleson, L., 73, 118 Central limit theorem, 260 Cesaro average, 119 Cohen's theorem, 338,262 Colzani, L., 149 Compactly supported wavelets, 326 Completeness of the Haar functions, 295 Completeness of the Hermite functions, 138 Conjugate function, 176 Conjugate Poisson kernel, 8, 179 Continuity theorem, 269 Continuum wavelet, 286 Contraction operator, 14 Convergence of wavelet expansions, 341 Convex function, 170 Convex sequence, 96 Convolution, 5, 14, 19, 92 Convolution sernigroups, 272

373

374

INDEX

D

H

Daubechies' formula, 331 de-Moivre Laplace theorem, I 00 Dilation matrix, 354 Dini condition, 27, 89 Dini's theorem, 115 Dirac measure, 20 Directed set, 46 Dirichlet, P. G., 27 Dirichlet kernel, 15, 25, 112, 140, 223 Dirichlet-Jord an theorem, 2 7, I 17 Distribution function, 181 Divergence of Fourier series, 73 Du Bois-Reymon d, 73, 117 Dyadic projections, 292

Haar function, 387, 306 Haar series, 291, 296 Haar wavelet, 291, 321 Hardy, G. H., 59 Hardy-Littlew ood maximal function, 197 Hardy's Tauberian theorem, 59 Harmonic functions, 212 Hausdorff- Young inequality, 174 Heat flow, 2 Heat kernel, 97, 203, 223 Herglotz theorem, 215 Hermite polynomials. 134 Hernandez-W eiss formula, 333 Higher-order approximatio n in C(1!'), 66 Hilbert transform on JR.. 184, 224 Hilbert transform on 'lr, 176, 224 Holder condition, 13, 40. 63, 116 Holder means, 53 Holder's inequality, 170 Homogeneou s Banach space, 47, 98 Hormander condition, 211 Hunt, R., 73, 118 Huygens' principle, 148

E

Equidistributi on mod 1, 57 Euler formula, 4 Exponential function, 6

F Factorial function, 6 Fatou's theorem, 50, 214 Fefferman, C., 73 Fejer means, 45, 54, 62, 98, 120 Fejer-Riesz lemma, 330 Finite measures, 19, 119, 256, 273 Fourier, J. B., 2 Fourier coefficient, 4, 13, 19 Fourier cosine transform, 124 Fourier reciprocity formula, 4, 14, 93, 97, 231 Fourier sine transform, 125 Fourier transforms, 89, 102, 256, 323

G Gap series, 262 Gaussian approximation , 80 Gaussian density function, 95, 257 Gaussian surnrnability, 104, 232 Gaussian wavelet, 288 Gauss-Weiers trass kernel, 97 Generalized h-transform, 125 Gibbs-Wilbra ham phenomenon, 3 1, 1 15 Grunbaum, A., 133

I

Independent random variables, 261 Infinitely differentiable function, 21 Integration of Fourier series, 29 Isoperimetric inequality, 38 J

Jackson's theorem, 65, 348 Jordan, C., 27

K Kac, M., 266 Kahane, J.P., 45, 73, 143 Katznelson, 1., 73, 180 Kendall's theorem, 241 Kolmogorov, A. N., 73, 118, 179 Kolmogorov's inequality, 192 L

Landau's asymptotic formula, 243 Laplace asymptotic method, 6, 9, 80

INDEX

Laplace's equation, 9, 106, 212 Large scale analysis, 345 Lattice ppints, 241 Laurent series, 5 Law of the iterated logarithm, 280 Lebesgue constants, 61, 75 Lebesgue differentiation theorem, 200 Lebesgue set, 124 Levy modulus of continuity, 302 Levy-Khintch ine theorem, 273 Lipschitz continuity, 62 Lusin, N., 73

Poisson summation formula on JR.d, 238 Positive-defin ite function, 257 Q

Quotient groups, 355 R

Marcinkiewic z, J., 169, 179 Marcinkiewic z interpolation theorem, 206 Maxwell's theorem, 109, 259 Mexican hat wavelet, 288 Meyer wavelets, 318 Modified Bessel function, 6, 81, 85 Modulus of continuity, 12, 20 Moore, C., 202 Multiple Fourier series, 230, 244 Multiple harmonic motion, I Multi-resoluti on analysis, 303

Radial functions, 92, 157, 202, 245 Random walk, 252 Rapidly decreasing function, 96 Rates of convergence in C(1f'), 61 Rates of convergence in L 2 (1f'), 39 Restriction theorem, 158 FUemann-Leb esguelemma, 18,94 FUemann localization, 3 1 FUemann means, 64 FUesz, M., 73 FUesz-Fischer theorem, 37 FUesz kernels, 195 FUesz potential kernel, 107 FUesz systems, 304, 356 FUesz-Thorin interpolation theorem, 169 Rotations, 194

N

s

M

Negative binomial series, 52 Newtonian potential kernel, 109 Non-tangentia l convergence, 202 Null-integrabi lity, 325

0 One-sided Fourier representation , 124 Orthogonality , 2, 137 Orthonormal wavelet, 319 p

Parseval's identity, 35 Periodic function, 13 Periodization, 223 Piecewise smooth function, 21, 145 Pinsky, M., 139, 145, 147 Plancherel's theorem, 128 Pointwise convergence criteria, 25, 115, 139, 233,298,347 Poisson kernel, 7, 15, 46, 90, 106, 189 Poisson summation formula on JR. 1 , 225

375

Salem, R., 266 Scaling equation, 310, 357 Scaling filter, 312 Scaling functions, 304, 319 Schrodinger equation and Gauss sums, 247 Schwartz class, 96 Schwartz distribution, 146 Self-adjoint operator, 15 Shannon sampling theorem, 228 Shannon scaling function, 306 Simple harmonic motion, 1 Simultaneous non-localizati on, 240 Sine integral, 24, 114, 185 Singular integrals, 183, 194, 210 Skew-adjoint operator, 16, 176 Smith, K. T., 202 Sobolev norm, 43, 290 Spherical Fourier inversion, 139 Spherical maximal function, 203 Spherical mean value, 140 Spherical partial sum, 139, 233 Stationary phase method, 162 Stein, E. M., 157, 159, 175,202,204 ,211 Stein's complex interpolation method, 159, 175

376

INDEX

Stirling's formula, 6 Strichartz, R., 33 Stronger summability method, 52 Structure constants, 310, 357 Sublinear operator, 197, 207 Summability matrix, 51 Summation by parts, 11, 79 Symmetric partial sum, 4, 112 Systems of wavelets, 352 Smoothness of wavelets, 334 Subgroups, 354

T Tail sum, 12 Tauberian theorem, 54 Taylor, M. E., 139, 147, 248 Tempered distribution, 108, 129 Three-lines theorem, 172 Tomas, P., 159 Translation error, 40

Uniqueness of Fourier coefficients, 17 Upper half plane, 216

v Vibrating string, 1 de la Vallee Poussin means, 68

w Wave equation, 171 Wavelets, 284 Wavelet transform, 285 Weak convergence of measures, 268, 327 Weierstrass M -test, 4 Weiss, G., 157 Whittaker, E. T., 228 Wiener's theorem, 57, 110 Wiener covering lemma, 198 y

u

Young's inequality for convolutions, 174

Uncertainty principle, 134 Uniform boundedness principle, 76, 84 Uniform continuity, 98 Uniform convergence, 4

z Zygmund,A.,20,44, 72,209,236,262,266 Zygmund's theorem, 44, 209