Harmonic Analysis: From Fourier to Wavelets
About this Title
María Cristina Pereyra, The University of New Mexico, Albuquerque, NM and Lesley A. Ward, University of South Australia, Mawson Lakes Campus, Adelaide, Australia
Publication: The Student Mathematical Library
Publication Year 2012: Volume 63
ISBNs: 978-0-8218-7566-7 (print); 978-0-8218-8786-8 (online)
MathSciNet review: MR2933659
MSC: Primary 42-02; Secondary 42Axx, 42B25, 42C40
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.
Undergraduate and beginning graduate students interested in harmonic analysis.
Table of Contents
- Chapter 1. Fourier series: Some motivation
- Chapter 2. Interlude: Analysis concepts
- Chapter 3. Pointwise convergence of Fourier series
- Chapter 4. Summability methods
- Chapter 5. Mean-square convergence of Fourier series
- Chapter 6. A tour of discrete Fourier and Haar analysis
- Chapter 7. The Fourier transform in paradise
- Chapter 8. Beyond paradise
- Chapter 9. From Fourier to wavelets, emphasizing Haar
- Chapter 10. Zooming properties of wavelets
- Chapter 11. Calculating with wavelets
- Chapter 12. The Hilbert transform
- Appendix. Useful tools