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I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond.

I am going to dedicate quite some time on the subject, so I expect something with a lot of exercises (calculus, demonstrations) and solutions, from the basics up to the most complex topics.

Could be nice to also have some exercises (with solutions) with practical applications in Matlab or Python/Numpy.

Any pointers? Tutorials, books, chapters, websites?

thanks! :)

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In case you're not already aware, FFTs are how we do the calculations needed to prove the largest known prime numbers today. mersenne.org/various/math.php That's one interesting practical (for some definition of "practical") application. – Tim S. May 4 '14 at 18:11
    
it depends on your level, and what you really need, and how much time do you have. – user1952009 May 19 at 18:09

I don't know if it is the best place to start for you -- after all, these things can depend on background and "mathematical maturity" -- but if you are going to be studying Fourier analysis in depth, then it would be a great pity to miss out on Thomas Körner's beautifully written and insightful Fourier Analysis (CUP, 1988).

"Ah, but there are no exercises ...."

Oh yes there are, but you have to look along the shelves to find Körner's accompanying book Exercises in Fourier Analysis (CUP, 1993) which gives chapter-by-chapter exercises for the original book, over three hundred pages of them!

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book by Körner definitely is a gem, but arguably not the book "to start learning..." – Artem May 4 '14 at 16:26

If you are interested in "pure mathematical theory" of Fourier analysis, start with the advice by @PeterSmith. If you are looking for a somewhat more applied point of view, here is a nice source:

A First Course in Fourier Analysis

It can be supplemented with the Stanford course of Fourier Analysis:

web page of the course

with video lectures, nice lecture notes and assignments.

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Is the First Course in Fourier Analysis by Kammler? The link is no longer valid. Also the link to the Stanford course is broken now too. – William May 19 at 14:15
    
@William Yes, Kammler, but the link works fine for me. I will fix the second link. – Artem May 19 at 18:05
    
@Artem : the answer depends on his level, and what he really needs, and how much time does he have. – user1952009 May 19 at 18:10
    
For some reason the rads.stackexchange.com links never work for me, I don't know why. – William May 19 at 18:10
    
and a link to amazon is just ridiculus... come on my friend ! we are on internet, the place where everything is downloadable for free ! – user1952009 May 19 at 18:11

The Fourier Transform And Its Applications by Ronald N. Bracewell.

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