Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
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
v
viii Table of Contents
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
i
x Table of Contents
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