Table of Contents
Preface to first edition v
Preface to second edition ix
List of Figures xvii
1 Orthogonal Series 1
1.1 General theory 1
1.2 Examples 5
1.2.1 Trigonometric system 6
1.2.2 Haar system 10
1.2.3 The Shannon system 12
1.3 Problems 15
2 A Primer on Tempered Distributions 19
2.1 Intuitive introduction 20
2.2 Test functions 22
2.3 Tempered distributions 25
2.3.1 Simple properties based on duality 27
2.3.2 Further properties 29
2.4 Fourier transforms 30
2.5 Periodic distributions 32
2.6 Analytic representations 33
2.7 Sobolev spaces 35
2.8 Problems 35
3 An Introduction to Orthogonal Wavelet Theory 37
3.1 Multiresolution analysis 38
3.2 Mother wavelet 44
3.3 Reproducing kernels and a moment condition 53
3.4 Regularity of wavelets as a moment condition 55
3.4.1 More on example 3 59
3.5 Mallat's decomposition and reconstruction algorithm 64
3.6 Filters 65
3.7 Problems 70
4 Convergence and Summability of Fourier Series 73
4.1 Pointwise convergence 73
4.2 Summability 79
4.3 Gibbs phenomenon 81
4.4 Periodic distributions 84
4.5 Problems 87
5 Wavelets and Tempered Distributions 91
5.1 Multiresolution analysis of tempered distributions 92
5.2 Wavelets based on distributions 95
5.2.1 Distribution solutions of dilation equations 95
5.2.2 A partial distributional multiresolution analysis 99
5.3 Distributions with point support 100
5.4 Approximation with impulse trains 104
5.5 Problems 107
6 Orthogonal Polynomials 109
6.1 General theory 109
6.2 Classical orthogonal polynomials 114
6.2.1 Legendre polynomials 115
6.2.2 Jacobi polynomials 119
6.2.3 Laguerre polynomials 120
6.2.4 Hermite polynomials 121
6.3 Problems 126
7 Other Orthogonal Systems 129
7.1 Self adjoint eigenvalue problems on finite intervals 130
7.2 Hilbert-Schmidt integral operators 132
7.3 An anomaly: the prolate spheroidal functions 134
7.4 A lucky accident? 135
7.5 Rademacher functions 140
7.6 Walsh function 142
7.7 Periodic wavelets 143
7.7.1 Periodizing wavelets 144
7.7.2 Periodic wavelets from scratch 146
7.8 Local sine or cosine basis 150
7.9 Biorthogonal wavelets 154
7.10 Problems 159
8 Pointwise Convergence of Wavelet Expansions 161
8.1 Reproducing kernel delta sequences 162
8.2 Positive and quasi-positive delta sequences 163
8.3 Local convergence of distribution expansions 169
8.4 Convergence almost everywhere 172
8.5 Rate of convergence of the delta sequence 173
8.6 Other partial sums of the wavelet expansion 177
8.7 Gibbs phenomenon 178
8.8 Positive scaling functions 181
8.8.1 A general construction 181
8.8.2 Back to wavelets 182
8.9 Problems 186
9 A Shannon Sampling Theorem in Wavelet Subspaces 187
9.1 A Riesz basis of Vm 188
9.2 The sampling sequence in Vm 189
9.3 Examples of sampling theorems 191
9.4 The sampling sequence in Tm 195
9.5 Shifted sampling 197
9.6 Gibbs phenomenon for sampling series 199
9.6.1 The Shannon case revisited 201
9.6.2 Back to wavelets 201
9.7 Irregular sampling in wavelet subspaces 212
9.8 Problems 214
10 Extensions of Wavelet Sampling Theorems 217
10.1 Oversampling with scaling functions 218
10.2 Hybrid sampling series 223
10.3 Positive hybrid sampling 225
10.4 The convergence of the positive hybrid series 228
10.5 Cardinal scaling functions 232
10.6 Interpolating multiwavelets 240
10.7 Orthogonal finite element multiwavelets 242
10.7.1 Sobolev type norm 244
10.7.2 The mother multiwavelets 245
10.8 Problems 252
11 Translation and Dilation Invariance in Orthogonal Systems 255
11.1 Trigonometric system 255
11.2 Orthogonal polynomials 256
11.3 An example where everything works 257
11.4 An example where nothing works 258
11.5 Weak translation invariance 259
11.6 Dilations and other operations 265
11.7 Problems 267
12 Analytic Representations Via Orthogonal Series 269
12.1 Trigonometric series 270
12.2 Hermite series 274
12.3 Legendre polynomial series 280
12.4 Analytic and harmonic wavelets 282
12.5 Analytic solutions to dilation equations 286
12.6 Analytic representation of distributions by wavelets 287
12.7 Wavelets analytic in the entire complex plane 291
12.8 Problems 293
13 Orthogonal Series in Statistics 295
13.1 Fourier series density estimators 296
13.2 Hermite series density estimators 299
13.3 The histogram as a wavelet estimator 301
13.4 Smooth wavelet estimators of density 305
13.5 Local convergence 309
13.6 Positive density estimators based on characteristic functions 310
13.7 Positive estimators based on positive wavelets 312
13.7.1 Numerical experiment 316
13.8 Density estimation with noisy data 318
13.9 Other estimation with wavelets 322
13.9.1 Spectral density estimation 322
13.9.2 Regression estimators 324
13.10 Threshold Methods 324
13.11 Problems 326
14 Orthogonal Systems and Stochastic Processes 329
14.1 K-L expansions 329
14.2 Stationary processes and wavelets 332
14.3 A series with uncorrected coefficients 335
14.4 Wavelets based on band limited processes 341
14.5 Nonstationary processes 345
14.6 Problems 349
Bibliography 351
Index 363
