# Wavelets and Multiwavelets

 Wavelets and Multiwavelets Fritz Keinert Studies in advanced mathematics, vol. 42 Chapman & Hall/CRC Press, Boca Raton, FL ISBN 1-58488-304-9 QA403.3 K45 2003 515'.2433—dc22 CRC Press web site for this book

## Back Cover

(with description of contents)

## Errata

Known errors are listed in file errata.pdf. If you find any others, please report to the author.

## Software

A toolbox of Matlab subroutines is made available with the book. Documentation (in both .pdf and .ps format) is included in the archive.

 mw.tar Unix tar format, 960 KB mw.tar.Z compressed Unix tar format, 604 KB mw.zip PC zip archive of Matlab subroutines, 460 KB

Preface

Contents

Part I - Scalar Wavelets

1 Basic Theory

1.1 Refinable Functions

1.2 Orthogonal MRAs and Wavelets

1.3 Wavelet Decomposition

1.4 Biorthogonal MRAs and Wavelets

1.5 Moments

1.6 Approximation Order

1.7 Symmetry

1.8 Point Values and Normalization

2 Practical Computation

2.1 Discrete Wavelet Transform

2.2 Pre- and Postprocessing

2.3 Handling Boundaries

2.3.1 Data Extension Approach

2.3.2 Matrix Completion Approach

2.3.3 Boundary Function Approach

2.4 Putting It All Together

2.5 Modulation Formulation

2.6 Polyphase Formulation

2.7 Lifting

2.8 Calculating Integrals

2.8.1 Integrals with Other Refinable Functions

2.8.2 Integrals with Polynomials

2.8.3 Integrals with General Functions

3 Creating Wavelets

3.1 Completion Problem

3.1.1 Finding Wavelet Functions

3.1.2 Finding Dual Scaling Functions

3.2 Projection Factors

3.3 Techniques for Modifying Wavelets

3.4 Techniques for Building Wavelets

3.5 Bezout Equation

3.6 Daubechies Wavelets

3.6.1 Bezout Approach

3.6.2 Projection Factor Approach

3.7 Coiflets

3.7.1 Bezout Approach

3.7.2 Projection Factor Approach

3.7.3 Generalized Coiflets

3.8 Cohen Wavelets

3.9 Other Constructions

4 Applications

4.1 Signal Processing

4.1.1 Detection of Frequencies and Discontinuities

4.1.2 Signal Compression

4.1.3 Denoising

4.2 Numerical Analysis

4.2.1 Fast Matrix--Vector Multiplication

4.2.2 Fast Operator Evaluation

4.2.3 Differential and Integral Equations

5 Existence and Regularity

5.1 Distribution Theory

5.2 L^1-Theory

5.3 L^2-Theory

5.3.1 Transition Operator

5.3.2 Sobolev Space Estimates

5.4 Pointwise Theory

5.5 Smoothness and Approximation Order

5.6 Stability

Part II - Multiwavelets

6 Basic Theory

6.1 Refinable Function Vectors

6.2 MRAs and Multiwavelets

6.2.1 Orthogonal MRAs and Multiwavelets

6.2.2 Biorthogonal MRAs and Multiwavelets

6.3 Moments

6.4 Approximation Order

6.5 Point Values and Normalization

7 Practical Computation

7.1 Discrete Multiwavelet Transform

7.2 Pre- and Postprocessing

7.2.1 Interpolating Prefilters

7.2.3 Hardin--Roach Prefilters

7.2.4 Other Prefilters

7.3 Balanced Multiwavelets

7.4 Handling Boundaries

7.4.1 Data Extension Approach

7.4.2 Matrix Completion Approach

7.4.3 Boundary Function Approach

7.5 Putting It All Together

7.6 Modulation Formulation

7.7 Polyphase Formulation

7.8 Calculating Integrals

7.8.1 Integrals with Other Refinable Functions

7.8.2 Integrals with Polynomials

7.8.3 Integrals with General Functions

7.9 Applications

7.9.1 Signal Processing

7.9.2 Numerical Analysis

8 Two-Scale Similarity Transforms

8.1 Regular TSTs

8.2 Singular TSTs

8.3 Multiwavelet TSTs

8.4 TSTs and Approximation Order

8.5 Symmetry

9 Factorizations of Polyphase Matrices

9.1 Projection Factors

9.1.1 Orthogonal Case

9.1.2 Biorthogonal Case

9.2 Lifting Steps

9.3 Raising Approximation Order by Lifting

10 Creating Multiwavelets

10.1 Orthogonal Completion

10.1.1 Using Projection Factors

10.1.2 Householder-Type Approach

10.2 Biorthogonal Completion

10.3 Other Approaches

10.4 Techniques for Modifying Multiwavelets

10.5 Techniques for Building Multiwavelets

11 Existence and Regularity

11.1 Distribution Theory

11.2 L^1-Theory

11.3 L^2-Theory

11.3.1 Transition Operator

11.3.2 Sobolev Space Estimates

11.4 Pointwise Theory

11.5 Smoothness and Approximation Order

11.6 Stability

Appendix A: Standard Wavelets

A.1 Scalar Orthogonal Wavelets

A.2 Scalar Biorthogonal Wavelets

A.3 Orthogonal Multiwavelets

A.4 Biorthogonal Multiwavelets

Appendix B: Mathematical Background

B.1 Notational Conventions

B.2 Derivatives

B.3 Functions and Sequences

B.4 Fourier Transform

B.5 Laurent Polynomials

B.6 Trigonometric Polynomials

B.7 Linear Algebra

Appendix C: Computer Resources

C.1 Wavelet Internet Resources

C.2 Wavelet Software

C.3 Multiwavelet Software

References

Index