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I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. Please suggest some books or links on the web.

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Fourier analysis: an introduction, by Stein and Shakarchi. – Giuseppe Negro Jun 7 '11 at 9:22
@dossonance : I couldn't see anything on complex valued functions from the contents page of the book. – Rajesh Dachiraju Jun 7 '11 at 9:50
Rajesh: It is not quite clear to me what you're really asking for. There is absolutely no difficulty nor any essential difference between real-valued and complex-valued functions. @dissonance's suggestion is a very good one. Alternatively, you might want to have a look at Katznelson's book on Harmonic Analysis. – t.b. Jun 7 '11 at 11:19
The difference is: rather than using sin and cos, you can instead use complex exponentials as the orthonormal basis. Virtually all math texts do it this way. Only very elementary math texts or engineering texts use sin and cos. – GEdgar Jun 7 '11 at 15:16

Chapter 3 of Grafakos's book Classical Fourier Analysis threats these questions.

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