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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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.
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.
"Fourier analysis" by T.W. Körner and its accompanied exercise book are the best.
please don't propose books that are impossible to find on internet, there are enough books and courses that are free to read on internet,
and the scientifics that you are should understand it is a huge social progress that everyone can FREELY access to the scientific knowledge.
look at any of the first links for :
google / fourier transform pdf
google / discrete fourier transform pdf
google / distribution fourier transform pdf
google / laplace transform pdf
google / filter convolution pdf
google / linear differential equation pdf
but be careful, it can take ~ 5 to 10 years to fully study the subject, it is so vast