Preface
1. An Overview

1.1. Prom Fourier Analysis to Wavelet Analysis

1.2. the Integral Wavelet Transform and Time-Frequency Analysis

1.3. Inversion Formulas and Duals

1.4. Classification of Wavelets

1.5. Multi-Resolution Analysis, Splines, and Wavelets

1.6. Wavelet Decompositions and Reconstructions

2. Fourier Analysis

2.1. Fourier and Inverse Fourier Transforms

2.2. Continuous-Time Convolution and the Delta Function

2.3. Fourier Transform of Square-Integrable Functions

2.4. Fourier Series

2.5. Basic Convergence Theory and Poisson'S Summation Formula

3. Wavelet Transforms and Time-Frequency Analysis

3.1. the Gabor Transform

3.2. Short-Time Fourier Transforms and the Uncertainty Principle

3.3. the Integral Wavelet Transform

3.4. Dyadic Wavelets and Inversions

3.5. Frames

3.6. Wavelet Series

4. Cardinal Spline Analysis

4.1. Cardinal Spline Spaces

4.2. ß-Splines and Their Basic Properties

4.3. The Two-Scale Relation and An Interpolatory Graphical Display Algorithm

4.4. B-Net Representations and Computation of Cardinal Splines

4.5. Construction of Spline Approximation Formulas

4.6. Construction of Spline Interpolation Formulas

5. Scaling Functions and Wavelets

5.1. Multi-resolution Analysis

5.2. Scaling Functions with Finite Two-Scale Relations

5.3. Direct-Sum Decompositions of L2(R)

5.4. Wavelets and Their Duals

5.5. Linear-Phase Filtering

5.6. Compactly Supported Wavelets

6. Cardinal Spline-Wavelets

6.1. Interpolatory Spline-Wavelets

6.2. Compactly Supported Spline-Wavelets

6.3. Computation of Cardinal Spline-Wavelets

6.4. Euler-Frobenius Polynomials

6.5. Error Analysis in Spline-Wavelet Decomposition

6.6. Total Positivity, Complete Oscillation, Zero-Crossings

7. Orthogonal Wavelets and Wavelet Packets

7.1. Examples of Orthogonal Wavelets

7.2. Identification of Orthogonal Two-Scale Symbols

7.3. Construction of Compactly Supported Orthogonal Wavelets

7.4. Orthogonal Wavelet Packets

7.5. Orthogonal Decomposition of Wavelet Series

Notes

References

Subject Index

Appendix

Wavelet Analysis and its Applications, Volume 1: An Introduction to Wavelets provides an introductory treatise on wavelet analysis with an emphasis on spline-wavelets and time-frequency analysis.
This book is divided into seven chapters. Chapter 1 presents a brief overview of the subject, including classification of wavelets, integral wavelet transform for time-frequency analysis, multi-resolution analysis highlighting the important properties of splines, and wavelet algorithms for decomposition and reconstruction of functions. The preliminary material on Fourier analysis and signal theory is covered in Chapters 2 and 3. Chapter 4 covers the introductory study of cardinal splines, while Chapter 5 describes a general approach to the analysis and construction of scaling functions and wavelets. Spline-wavelets are deliberated in Chapter 6. The last chapter is devoted to an investigation of orthogonal wavelets and wavelet packets.
This volume serves as a textbook for an introductory one-semester course on "wavelet analysis¿ for upper-division undergraduate or beginning graduate mathematics and engineering students.