Wavelets: Theory and Applications

Preface ix

Notation xi

Introduction xv

1 The Continuous Wavelet Transform 1

1.1. Definition and Elementary Properties 1

1.2 Affine Operators 10

1.3 Filter Properties of the Wavelet Transform 12

1.4 Approximation Properties 22

1.5 Decay Behaviour 32

1.6 Group-Theoretical Foundations and Generalizations 36

1.7 Extension of the One-Dimensional Wavelet Transform to Sobolev Spaces 59

Exercises 69

2 The Discrete Wavelet Transform 73

2.1 Wavelet Frames 73

2.2 Multiscale Analysis 97

2.3 Fast Wavelet Transform 121

2.4 One-Dimensional Orthogonal Wavelets 131

2.5 Two-Dimensional Orthogonal Wavelets 203

Exercises 226

3 Applications of the Wavelet Transform 231

3.1 Wavelet Analysis of One-Dimensional Signals 231

3.2 Quality Control of Texture 235

3.3 Data Compression in Digital Image Processing 239

3.4 Regularization of Inverse Problems 251

3.5 Wavelet – Galerkin Methods for Two-Point boundary Value Problems 259

3.6 Schwarz Iterations Based on Wavelet Decompositions 278

3.7 An Outlook on Two-Dimensional Boundary Value Problems 300

Exercises 306

Appendix The Fourier Transform 309

References 313

Index 321