Wavelets and Other Orthogonal Systems With Applications

Gilbert G. Walter

(g) CRC Press

Boca Raton Ann Arbor London Tokyo

Table of Contents

Preface

1 Orthogonal Series l 1.1 General theory 1 1.2 Examples 5

1.2.1 Trigonometrie System 5 1.2.2 Haar System 9 1.2.3 Shannon System 10

1.3 Problems 13

2 A Primer on Tempered Distributions 15

2.1 Tempered distributions 16 2.2 Fourier transforms 21 2.3 Periodic distributions 23 2.4 Analytic representations 24 2.5 Sobolev Spaces 26 2.6 Problems 26

3 An Introduction to Orthogonal Wavelet Theory 28

3.1 Multiresolution analysis 29 3.2 Mother wavelet 33 3.3 Reproducing kerneis and a moment condition 38 3.4 Regularity of wavelets as a moment condition 39 3.5 Mallat's decomposition and reconstruction algorithm 43 3.6 Filters 45 3.7 Problems 48

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4 Convergence and Summability of Fourier Series 50

4.1 Pointwise convergence 50 4.2 Summability 55 4.3 Gibbs' phenomenon 56 4.4 Periodic distributions 58 4.5 Problems 60

5 Wavelets and Tempered Distributions 63

5.1 Multiresolution analysis of tempered distributions 64 5.2 Wavelets based on distributions 67

5.2.1 Distribution Solutions of dilation equations 67 5.2.2 A partial distributional multiresolution

analysis 70 5.3 Distributions with point support 71 5.4 Problems 75

6 Orthogonal Polynomials 76 6.1 General theory 76 6.2 Classical orthogonal polynomials 81

6.2.1 Legendre polynomials 81 6.2.2 Jacobi polynomials 85 6.2.3 Laguerre polynomials 85 6.2.4 Hermite polynomials 86

6.3 Problems 91

7 Other Orthogonal Systems 93 7.1 Seif adjoint eigenvalue problems on a finite interval 93 7.2 Hilbert-Schmidt integral Operators 96 7.3 An anomaly—the prolate spheroidal functions 97 7.4 A lucky accident? 99 7.5 Rademacher functions 103 7.6 Walsh functions 104 7.7 Periodic wavelets 105 7.8 Local sine or cosine bases 107 7.9 Biorthogonal wavelets 111 7.10 Problems 114

Table of Contents ix

8

8.1 8.2 8.3 8.-1 8.5 8.6 8.7

Pointwise Convergence of Wavelet Expansions Quasi-positive delta sequences Local convergence of distribution expansions Convergence almost everywhere Rate of convergence of the delta sequence Other partial sums of the wavelet expansion Gibbs ' phenomenon Problems

116 117 120 124 124 128 130 132

9

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

A Shannon Sampling Theorem in Vm A Riesz basis of Vm

The sampling sequence in Vm

Examples of sampling theorems The sampling sequence in Tm

Shifted sampling Oversampling with scaling functions Cardinal scaling functions Problems

134 135 136 138 140 143 145 149 156

10

10.1 10.2 10.3 10.4 10.5 10.6 10.7

Translation and Dilation Invari-ance in Orthogonal Systems Trigonometrie System Orthogonal polynomials An example where everything works An example where nothing works Weak translation invariance Dilations and other Operations Problems

158 158 159 160 161 161 167 168

11

11.1 11.2 11.3 11.4 11.5 11.6 11.7

Analytic Representations Via Orthogonal Series Trigonometrie series Hermite series Legendre polynomial series Analytic and harmonic wavelets Analytic Solutions to dilation equations Analytic representation of distributions by wavelets Problems

170 170 174 179 180 183 184 188

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12 Orthogonal Series in Statistics 190 12.1 Fourier series density estimators 191 12.2 Hermite series density estimators 194 12.3 The histogram as a wavelet estimator 195 12.4 Smooth wavelet estimators of density 199 12.5 Local convergence 203 12.6 Positive density estimators 204 12.7 Other estimation with wavelets 205

12.7.1 Mixture problems 206 12.7.2 Spectral density estimation 211 12.7.3 Regression estimators 212

12.8 Problems 213

13 Orthogonal Systems and Stochastic Processes 215

13.1 K-L expansions 215 13.2 Stationary processes and wavelets 218 13.3 A series with uncorrelated coefficients 220 13.4 Wavelets based on band limited processes 226 13.5 Nonstationary processes 229 13.6 Problems 231

Bibliography 233

Index 241