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I am a bit confused. I have heard today someone saying that the Fourier series of any continues periodic function $f$, say with period 1 for concreteness, converges pointwise to $f$. Wikipedia here explicitly says otherwise, but claims that the proof is not constructive and somewhat advanced. So what is it?

I have to apologize for this question, since I could just pick any book about Fourier analysis and check it up myself, but since the answer is a simple "yes" or "no" I hope it is not to much of an effort to answer this for someone how knows the answer.

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On Page 83 of Stein-Shakarchi's Fourier Analysis, they give an example of a continuous period function $f$ whose Fourier series diverges at a point. –  ADF Nov 1 '11 at 13:46
    
In general it's not true. You need some additional hypotheses that in most of concrete examples are satisfied, for instance that it has bounded variation. –  Valerio Capraro Nov 1 '11 at 13:48
    
It's only true of "nice" continuous functions, eg. with piecewise continuous derivatives, which are the ones that one spontaneously (naively?) thinks of as continuous functions, and often finds in applications. –  leonbloy Nov 1 '11 at 14:22

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up vote 8 down vote accepted

See Theorem 2.2 in this pdf which shows (the proof is not really complete) that there is a continuous periodic function with divergent fourier series in some point. The first example of such a function was given by DuBois-Reymond in 1873, in the meantime this has even been extended (see the note on page 13):

For every null set $N \subset ]-\pi,\pi]$ there is a continuous periodic function for which the Fourier series is divergent in each point of $N$.

Shown in Y. Katznelson, An introduction to harmonic analysis. Wiley 1968.

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Thanks! This is very nice. –  KotelKanim Nov 1 '11 at 14:24

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