Wavelets: Theory and Applications

Contents

Preface

Notation

Introduction

1  The Continuous Wavelet Transform

1.1  Definition and Elementary Properties

1.2  Affine Operators

1.3  Filter Properties of the Wavelet Transform

1.3.1  Phase-Space Representations and Localization Operators

1.3.2  Wavelet Transform versus Windowed Fourier Transform

1.4  Approximation Properties

1.4.1  Asymptotic Behaviour in the Frequency Parameter

1.4.2  Remarks About the Order of Wavelets

1.5  Decay Behaviour

1.6  Group-Theoretical Foundations and Generalizations

1.6.1  The Orthogonality Relation for Locally Compact Groups

1.6.2  The Left Transform

1.6.2.1  The Wavelet Transform in L²(ℝ)

1.6.2.2  The Windowed Fourier Transform

1.6.2.3  The Wavelet Transform inn L²(ℝ)²

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

Exercises

2  The DiscreteWavelet Transform

2.1  Wavelet Frames

2.1.1  Introduction and Definition

2.1.2  The Frame Operator

2.2  Multiscale Analysis

2.2.1  One-Dimensional Multiscale Analysis

2.2.2  Multidimensional Multiscale Analysis

2.3  Fast Wavelet Transform

2.4  One-Dimensional Orthogonal Wavelets

2.4.1  Spline Wavelets

2.4.2  Solving Scaling Equations

2.4.3  Orthogonal Wavelets with Compact Support

2.4.4  Properties of the Daubechies Wavelets

2.4.5  Biorthogonal Wavelets

2.4.6  Wavelets Adapted to Operators

2.4.6.1  Wavelet–Vaguelette Decompositions

2.4.6.2  Wavelet–Wavelet Decompositions

2.4.7  Notes

2.4.7.1  Wavelets and Derivatives

2.4.7.2  Wavelets on the Interval ...