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I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the point. Further, I am not terribly interested in applications and would rather be exposed to how Fourier Analysis fits into the broader framework of analysis.

For background, I used Baby Rudin for a one-year course in advanced calculus, I am currently taking a course from Kolmogorov and Fomin's Introductory Real Analysis and I have taken complex analysis (using Conway's text, Functions of One Complex Variable) as well as topology (using Munkres as well as Engelking) at the graduate level, but I have not yet been introduced to the Lebesgue integral.

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"I am not terribly interested in applications" - Dang, and I was about to recommend Brown and Churchill too. –  J. M. Sep 11 '10 at 23:55
If you really want to learn Fourier analysis (and there are in fact plenty of aspects to the subject) I would recommend you to start by Lebesgue integration. –  AD. Sep 12 '10 at 5:08

6 Answers 6

You mentioned graduate level. So you really should first learn Lebesgue integration. (Stein and Shakarchi volume 3 is not bad, as are many of the usual suspects -- Big Rudin and Royden's book on measure theory, just to name a couple.)

Then I would recommend any/all of the following:

  • Stein and Weiss, Introduction to Fourier Analysis on Euclidean Spaces (after that you may also be interested in Stein's Singular Integrals and Differentiability Properties of Functions and Harmonic Analysis)
  • Grafakos, Classical and Modern Fourier Analysis (which has been republished in the GTM series as two separate books; you should start with the Classical Fourier Analysis volume).
  • Sogge, Fourier Integrals in Classical Analysis

For one aspect of how Fourier analysis fits into the broader framework of analysis, I also recommend studying some distribution theory, and theory of partial/pseudo/para-differential operators. Some interesting texts in that regard include:

  • Friendlander and Joschi, Introduction to the theory of distributions
  • Hörmander, Analysis of Linear Partial Differential Operators, volumes 1-4 (the first volume includes a quick "review" of the parts of Fourier analysis used; I put the word in quotes because, well, it is Hörmander...)
  • Alinhac and Gérard, Pseudo-differential Operators and the Nash-Moser Theorem (and if you read French, you should consider looking at the French original)
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Aren't the books you suggest a bit hard (except for Grafakos) when one is not even up to Lebesgue integration? +1 for Alinhac and Gérard though. –  Jonas Teuwen Mar 7 '11 at 10:20
@Jonas: That's why I suggest to first learn Lebesgue integration! And I consider the list above to be about just right for a graduate level introduction to Fourier analysis. If the question were posed for an undergraduate level, I'd agree with you that my list is perhaps not appropriate. Also, if you are willing to take some facts on Lebesgue integration for granted, neither Sogge nor Stein and Weiss are very difficult. The last three are meant to be "further study" anyway, so I took a bit of leeway in suggesting even harder stuff. –  Willie Wong Mar 7 '11 at 12:57
Right. Fair enough. –  Jonas Teuwen Mar 7 '11 at 13:58

I can't help but recommend G. Folland's Tata notes on PDE, which are light, but not conversational/sloppy. It becomes immediately clear how Fourier transforms help.

Rudin's "Functional Analysis" treats Fourier transforms carefully, but gives the impression that he doesn't care about them very much. Not inspirational.

Hormander's volume I of his expanded PDE books is (unlike the later volumes) readable by everyone, and very useful.

The case of Fourier series in one variable is treated in a fashion meant to be down-to-earth, but also forward-looking, in my notes functions on circles, which includes discussion of Sobolev spaces and distributions on circles.

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I just wanted to thank you, @paul garrett, for those wonderful notes! They contain many little insights/perspectives which are difficult to find in the standard treatments (e.g. defining a Frechet space as a countable limit of Banach spaces). –  wildildildlife Feb 12 '12 at 10:24
Thank you, @wildildildlife! Encouragement is always helpful! Perhaps it's not surprising that in "Functional Analysis for Number Theorists" (e.g.) the criteria for "clarification" may be different than in "Fun Analaysis for PDE people", etc. Also "Functional Analysis the Second Time Through", and such. Thanks again for your kind words, and I would encourage everyone to give feedback to on-line authors whose work has been found useful! :) –  paul garrett Feb 13 '12 at 14:09

Here are the ones which i would recommend:

The second one is very good and you can comprehend it if you are familiar with the analysis which you have mentioned. I would also like to advice you to read "A radical approach to Lebesgue theory of Integration" by D.M.Bressoud, link https://www.maa.org/EbusPPRO/DynamicSearch/ProductDetailsAdvancedSearch/tabid/176/ProductId/1498/Default.aspx this is a beautiful book to learn Measure theory

Update You might be interested to see this link: http://www.cargalmathbooks.com/#FourierAnalysis

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I love Korner's book, but I'm not sure that I'd describe it as less conversational and more to the point. :) –  Byron Schmuland Sep 11 '10 at 18:58
I'll add another recommendation for Korner, but also second Bryon's comment that it almost certainly could not be described as less conversational. –  cch Mar 7 '11 at 5:23

I'd like to suggest Fourier Series and Integrals by Dym and McKean. It's old, but still an excellent book. Chapters 3 and 4 show how Fourier analysis fits in with some other parts of mathematics.

From the Preface:

The level of preparation expected is a thorough knowledge of advanced calculus. To this must be added a willingness to believe in (or to study up on) the Lebesgue integral.

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Fantastic book. –  5space Apr 23 at 5:38

Oops I misread the question, but I actually like Shakarchi and Stein volume 1! In my opinion the second volume is a lot better than the first one however.

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Hm, I would have suggested Stein+Shakarchi as well, but alright.

You say you want a book that's to the point, and you've already gone through Baby Rudin... Did you like Baby Rudin? If so, you might consider Big Rudin.

The relevant chapters are Chapters 4, 5, 9, which discuss Hilbert spaces (hence Fourier series), Banach spaces, and the Fourier transform, respectively. Admittedly, Chapters 1-3 focus on the Lebesgue integral and $L^p$ spaces, but I'm not sure how much of that you'll actually need. (Well, alright, you might need the most basic properties of $L^p$, but that shouldn't be a problem.) My only warning would be that Big Rudin is, of course, somewhat terse and rather abstract.

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