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A First Course in Wavelets with Fourier Analysis, 2nd Edition

ISBN: 978-1-118-21115-1
336 pages
September 2011
A First Course in Wavelets with Fourier Analysis, 2nd Edition (1118211154) cover image
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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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.

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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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  • Offers self-contained exposition of Fourier analysis and wavelets
  • Contains expanded applications to signal processing and additional exercises throughout the book
  • Provides complete proofs of the presented theory in addition to solutions to selected exercises in the back of the book
  • Features improved sections devoted to continuous wavelets and two-dimensional wavelets
  • Presents updated Matlab code in the appendix and online via a related website

 

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"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." (Mathematical Reviews, 2011)

"The discussions of applications avoid the deep jargon of signal processing … accessible to a wider audience." (Book News, December 2009)

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