Harmonic Analysis: From Fourier to Wavelets

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American Mathematical Soc., 2012 - Mathematics - 410 pages
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In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently. This book is published in cooperation with IAS/Park City Mathematics Institute.
 

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Contents

Some motivation
1
Analysis concepts
21
Pointwise convergence of Fourier series
55
Summability methods
77
Meansquare convergence of Fourier series
107
A tour of discrete Fourier and Haar analysis
127
The Fourier transform in paradise
161
Beyond paradise
189
From Fourier to wavelets emphasizing Haar
221
Zooming properties of wavelets
261
Calculating with wavelets
303
The Hilbert transform
329
Appendix Useful tools
371
Bibliography
391
Name index
401
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