Harmonic Analysis: From Fourier to Wavelets (Student Mathematical Library) (Student Mathematical Library - IAS/Park City Mathematical Subseries)

✍ Scribed by María Cristina Pereyra, Lesley A. Ward

• Publisher: American Mathematical Society
• Year: 2012
• Tongue: English
• Leaves: 437
• Series: Student Mathematical Library - IAS/Park City Mathematical Subseries
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Title page
Contents
List of figures
List of tables
IAS/Park City Mathematics Institute
Preface
Fourier series: Some motivation
Interlude: Analysis concepts
Pointwise convergence of Fourier series
Summability methods
Mean-square convergence of Fourier series
A tour of discrete Fourier and Haar analysis
The Fourier transform in paradise
Beyond paradise
From Fourier to wavelets, emphasizing Haar
Zooming properties of wavelets
Calculating with wavelets
The Hilbert transform
Useful tools
Alexander’s dragon
Bibliography
Name index
Subject index
Back Cover

✦ Subjects

Harmonic Analysis, Fourier Analysis