A First Course in Wavelets with Fourier Analysis

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John Wiley & Sons, Sep 8, 2009 - Mathematics - 315 pages
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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.

 

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Contents

Inner Product Spaces
1
Exercises
34
Fourier Series
38
The Fourier Transform
92
Discrete Fourier Analysis
132
Haar Wavelet Analysis
160
Multiresolution Analysis
190
Exercises
228
The Daubechies Wavelets
234
Other Wavelet Topics
250
Technical Matters
273
Solutions to Selected Exercises
287
MATLAB Routines
305
Index
313
Copyright

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About the author (2009)

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.

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