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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. That's one interesting practical (for some definition of "practical") application. – Tim S. May 4 '14 at 18:11

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

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