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I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.

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Not a book but couldn't resist recommending:youtube.com/playlist?list=PL1780FAF4A29FE679 –  Inquest Feb 12 '12 at 17:36
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An excellent book is "Fourier analysis" by T.W. Körner. –  Andres Caicedo Feb 12 '12 at 17:37
    
@AndresCaicedo: Is that the same as Exercises in Fourier Analysis? –  Ross Feb 12 '12 at 17:52
    
No, the book on Exercises is a companion to the main text. –  Andres Caicedo Feb 12 '12 at 18:29
    
"Fourier Analysis" by Stein and Shakarchi is a lovely book. It may look like it is aimed at a lower level (it is supposed to be an introductory text to analysis) but the material covered there is incredibly broad and wonderfully treated. –  Chris Janjigian Feb 12 '12 at 18:43

3 Answers 3

A good place to start is Tolstov's little book on Fourier Analysis. It is published by Dover and it's inexpensive. There are nice problems.

Here is the Amazon page for it. The Stein books are wonderful but they do demand a serious level of skill with analysis. This book is a good place for you to start.

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Dym & McKean's book Fourier Series and Integrals has a better collection of applications than most (I suspect?) books do. Physics, number theory, probability, isoperimetric problems, ..... lots of stuff.

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