Harmonic Analysis: From Fourier to Wavelets is 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.
The book consists roughly of four parts of uneven length: chapters 1–5 deal with Fourier series, chapter 6 with finite Fourier analysis, chapters 7–8 with the classical Fourier transform on the line, and chapters 9–12 survey a number of transforms leading to wavelets and conclude with several points of view on the Hilbert transform.
There are twenty-four projects included at the end of the chapters throughout the book. Each project is carefully chosen so that students at different levels can find it approachable and challenging at the same time. References for getting started are incorporated in the project description, with brief directions as to where the main focus is. At the end of each section there are exercises of varying difficulty.
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 is Assistant Professor of Mathematics at St. John’s University in Queens, New York.