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Does number theory have any role in the proof of convergence of Fourier series for certain functions? I vaguely remember reading in a book on signal processing, way back, that the proof (original proof by Dirichlet) of convergence of Fourier series for functions which satisfy Dirichlet's conditions is very complicated and involves some number theory as well. I'd like to know whether it is true and appreciate a reference in case it is. I am really sorry if it turned out to be a silly question based on vague memory. I've seen the Wikipedia page on convergence of Fourier series but couldn't find an answer.

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You may want to have a look at Elstrodt's article The life and Work of Gustav Lejeune Dirichlet (from Tschinkel's page). It contains a summary of his work on Fourier series in §8.A. A quick glance at the original text Über die Darstellung ganz willkürlicher Funktionen durch Sinus- und Cosinusreihen -- on the representation of completely arbitrary functions by sine and cosine series (Werke 1, p.133, see also p.117) doesn't seem to use too much number theory – t.b. Aug 5 '11 at 17:35
Is this the sort of thing you are looking for:… ? – Doug Chatham Aug 10 '11 at 14:07
@Doug : Thanks for the link. What i am looking for is the answer to this question. Proof of convergence of Fourier series is basically a real analysis problem. Why would we need results from number theory to prove it.(if it is true). – Rajesh Dachiraju Aug 10 '11 at 15:44

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