A comprehensive, self-contained treatment of Fourier analysis and
wavelets—now in a new edition
Through expansive coverage and easy-to-follow explanations, A First Course in Wavelets
with Fourier Analysis, Second Edition provides a self-contained mathematical treatment of
Fourier analysis and wavelets, while uniquely presenting signal analysis applications and
problems. Essential and fundamental ideas are presented in an effort to make the book
accessible to a broad audience, and, in addition, their applications to signal processing
are kept at an elementary level.
The book begins with an introduction to vector spaces, inner product spaces,
and other preliminary topics in analysis. Subsequent chapters feature:
- The development of a Fourier series, Fourier transform, and discrete Fourier
analysis
- Improved sections devoted to continuous wavelets and two-dimensional wavelets
- The analysis of Haar, Shannon, and linear spline wavelets
- The general theory of multi-resolution analysis
- Updated MATLAB® code and expanded applications to signal processing
- The construction, smoothness, and computation of Daubechies' wavelets
- Advanced topics such as wavelets in higher dimensions, decomposition and
reconstruction, and wavelet transform
Applications to signal processing are provided throughout the book, most involving the
filtering and compression of signals from audio or video. Some of these applications are
presented first in the context of Fourier analysis and are later explored in the chapters
on wavelets. New exercises introduce additional applications, and complete proofs
accompany the discussion of each presented theory. Extensive appendices outline more
advanced proofs and partial solutions to exercises as well as updated MATLAB® routines
that supplement the presented examples.
A First Course in Wavelets with Fourier Analysis, Second Edition is an excellent book
for courses in mathematics and engineering at the upper-undergraduate and graduate levels.
It is also a valuable resource for mathematicians, signal processing engineers, and
scientists who wish to learn about wavelet theory and Fourier analysis on an elementary
level.
ALBERT BOGGESS, PhD, is Professor of Mathematics at Texas A&M
University. Dr. Boggess has over twenty-five years of academic experience and has authored
numerous publications in his areas of research interest, which include overdetermined
systems of partial differential equations, several complex variables, and harmonic
analysis.
FRANCIS J. NARCOWICH, PhD, is Professor of Mathematics and Director of the
Center for Approximation Theory at Texas A&M University. Dr. Narcowich serves as an
Associate Editor of both the SIAM Journal on Numerical Analysis and Mathematics
of Computation, and he has written more than eighty papers on a variety of topics in
pure and applied mathematics. He currently focuses his research on applied harmonic
analysis and approximation theory.
Table of Contents
Preface and Overview.
0 Inner Product Spaces.
0.1 Motivation.
0.2 Definition of Inner Product.
0.3 The Spaces L2 and l2.
0.4 Schwarz and Triangle Inequalities.
0.5 Orthogonality.
0.6 Linear Operators and Their Adjoints.
0.7 Least Squares and Linear Predictive Coding.
Exercises.
1 Fourier Series.
1.1 Introduction.
1.2 Computation of Fourier Series.
1.3 Convergence Theorems for Fourier Series.
Exercises.
2 The Fourier Transform.
2.1 Informal Development of the Fourier Transform.
2.2 Properties of the Fourier Transform.
2.3 Linear Filters.
2.4 The Sampling Theorem.
2.5 The Uncertainty Principle.
Exercises.
3 Discrete Fourier Analysis.
3.1 The Discrete Fourier Transform.
3.2 Discrete Signals.
3.3 Discrete Signals & Matlab.
Exercises.
4 Haar Wavelet Analysis.
4.1 Why Wavelets?
4.2 Haar Wavelets.
4.3 Haar Decomposition and Reconstruction Algorithms.
4.4 Summary.
Exercises.
5 Multiresolution Analysis.
5.1 The Multiresolution Framework.
5.2 Implementing Decomposition and Reconstruction.
5.3 Fourier Transform Criteria.
Exercises.
6 The Daubechies Wavelets.
6.1 Daubechies’ Construction.
6.2 Classification, Moments, and Smoothness.
6.3 Computational Issues.
6.4 The Scaling Function at Dyadic Points.
Exercises.
7 Other Wavelet Topics.
7.1 Computational Complexity.
7.2 Wavelets in Higher Dimensions.
7.3 Relating Decomposition and Reconstruction.
7.4 Wavelet Transform.
Appendix A: Technical Matters.
Appendix B: Solutions to Selected Exercises.
Appendix C: MATLAB® Routines.
Bibliography.
Index.
336 pages, Hardcover