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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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    The difference is: rather than using sin and cos, you can instead use com$p$lex 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.2011-06-07

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