# Good intro to fourier analysis?

• desti

#### desti

Hi,

I am starting in a Ph.D. program in math next fall and the prerequisites for the first-year graduate course sequences included basics of Fourier analysis. The only thing I know about it is that you calculate a projection of a function on a certain infinite dimensional subspace, so I do know how to derive many of the really basic formulas.

I would like to find a good book that assumes you know linear algebra, complex analysis and basic real analysis (measure theoretic). What would you recommend?

## Answers and Replies

You might want to look at "Fourier Analysis and Its Applications" by Gerald B. Folland. If this is not at a high enough level, then have a look at the Fourier analysis chapter in the Real Analysis grad text by the same author.

In my Communication Systems degree we used this book:-

Applied Fourier Analysis

Author: Hwei Hsu
Format: Paperback
Publication Date: October 1984
Publisher: Harcourt College Pub
Dimensions: 10.75"H x 8.25"W x 0.5"D; 1.05 lbs.
ISBN-10: 0156016095
ISBN-13: 9780156016094

Dym and McKean starts with a basic intro to the Lesbegue measure and integration so if you skip the few sections of the first chapter, the rest of the book is what you're looking for.

Stein would also be a good choice; however, his book on Fourier series only treats riemann integrable functions. But since the Fourier book is the first in a series, he develops the theory more and more in his complex and measure theory books.

Dym and McKean starts with a basic intro to the Lesbegue measure and integration so if you skip the few sections of the first chapter, the rest of the book is what you're looking for.

Stein would also be a good choice; however, his book on Fourier series only treats riemann integrable functions. But since the Fourier book is the first in a series, he develops the theory more and more in his complex and measure theory books.

I like Stein's book as well, but in addition to the restriction to Riemann integrable functions, which still allows a pretty good treatment of Fourier series, he also limits his treatment of Fourier transforms in Volume 1 to those of Schwartz (smooth and rapidly decaying) functions. Volume 3 covers the Lebesgue integral and then covers the Fourier transform on L^1 and L^2.

There was supposed to be a Volume 4 which would cover distribution theory (linear functionals on the space of Schwartz functions), which allows you to define Fourier transforms for a much broader range of function-like objects, but this volume appears never to have materialized.

[Note to the original poster to avoid confusion: we are referring above to the set of "Princeton Lectures in Analysis" undergraduate books by Stein and Shakarchi (volume 2 covers complex analysis, with a bit of Fourier content there as well), NOT the much more advanced graduate-level series by Stein and Weiss.]

There are other good books that cover the same material, though. I've only looked at excerpts of the following two, but from what I have seen, they look really nice and don't require the machinery of the Lebesgue intergral and measure theory (i.e., for the most part you can interpret the integrals as either Riemann or Lebesgue):

Kammler, "A First Course in Fourier Analysis" - this "feels" to me like a more rigorous version of the treatment you would see in an engineering-oriented book, particularly in terms of the applications covered. I seem to recall that Folland's Fourier analysis book is at about this level as well.

Howell, "Principles of Fourier Analysis" - more mathematically oriented, on about the same level as Stein and Shakarchi's Volume 1, but with a much more comprehensive treatment (and double the page count).

A really nice, efficient book that presumes you know the basic Lebesgue theory is Katznelson's "Introduction to Harmonic Analysis."

Finally, no discussion of Fourier analysis books should go without mentioning Körner's "Fourier Analysis," which is a bit unorthodox and not really a place to learn Fourier analysis in a systematic way, but rather a fascinating collection of mini-essays, mostly covering a wide variety of applications of Fourier analysis (both in the sense of "applied math" and in the sense of applications to other branches of pure math). This is well worth checking out.