**Student Mathematical Library**

Volume: 63;
2012;
410 pp;
Softcover

MSC: Primary 42;

Print ISBN: 978-0-8218-7566-7

Product Code: STML/63

List Price: $58.00

Individual Price: $46.40

**Electronic ISBN: 978-0-8218-8786-8
Product Code: STML/63.E**

List Price: $58.00

Individual Price: $46.40

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#### Supplemental Materials

# Harmonic Analysis: From Fourier to Wavelets

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*María Cristina Pereyra; Lesley A. Ward*

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

# Table of Contents

## Harmonic Analysis: From Fourier to Wavelets

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- List of figures xi12 free
- List of tables xiii14 free
- IAS/Park City Mathematics Institute xv16 free
- Preface xvii18 free
- Fourier series: Some motivation 126 free
- Interlude: Analysis concepts 2146
- Pointwise convergence of Fourier series 5580
- Summability methods 77102
- Mean-square convergence of Fourier series 107132
- A tour of discrete Fourier and Haar analysis 127152
- The Fourier transform in paradise 161186
- Beyond paradise 189214
- From Fourier to wavelets, emphasizing Haar 221246
- Zooming properties of wavelets 261286
- Calculating with wavelets 303328
- The Hilbert transform 329354
- Useful tools 371396
- Alexander’s dragon 389414
- Bibliography 391416
- Name index 401426
- Subject index 403428
- Back Cover Back Cover1437

#### Readership

Undergraduate and beginning graduate students interested in harmonic analysis.

#### Reviews

This is a gentle introduction to Fourier analysis and wavelet theory that requires little background but still manages to explain some of the applications of Fourier and wavelet methods and touch on several current research topics. ... The authors have taken care to be accessible to undergraduate mathematicians. ... Compared to standard texts, this book is characterised by more personal and historical information, including footnotes. ... It comes with many projects for interested students, and lists a number of open-ended problems that suggest further developments and should engage interested students. ... In summary, this is a well-written and lively introduction to harmonic analysis that is accessible and stimulating for undergraduates and instructive and amusing for the more sophisticated reader. It could also be argued that the material herein should be part of the knowledge of most undergraduates in mathematics, given that the modern world relies more and more on data compression. It is therefore timely as well. It has certainly earned my enthusiastic recommendation.

-- Michael Cowling, Gazette of the Australian Mathematical Society

A wonderful introduction to harmonic analysis and applications. The book is intended for advanced undergraduate and beginning graduate students and it is right on target. Pereyra and Ward present in a captivating style a substantial amount of classical Fourier analysis as well as techniques and ideas leading to current research. ... It is a great achievement to be able to present material at this level with only a minimal prerequisite of advanced calculus and linear algebra and a set of Useful Tools included in the appendix. I recommend this excellent book with enthusiasm and I encourage every student majoring in math to take a look.

-- Florin Catrina, MAA Reviews

[T]he panorama of harmonic analysis presented in the book includes very recent achievements like the connection of the dyadic shift operator with the Hilbert transform. This gives to an interested reader a good chance to see concrete examples of contemporary research problems in harmonic analysis. I highly recommend this book as a good source for undergraduate and graduate courses as well as for individual studies.

-- Krzysztof Stempak, Zentralblatt MATH