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I am starting in a Ph.D. program in math next fall and the prerequisites for the first-year graduate course sequences included basics of Fourier analysis. The only thing I know about it is that you calculate a projection of a function on a certain infinite dimensional subspace, so I do know how to derive many of the really basic formulas.

I would like to find a good book that assumes you know linear algebra, complex analysis and basic real analysis (measure theoretic). What would you recommend?

Thanks in advance.