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Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.

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Kolmogorov improved his result to a Fourier series diverging everywhere. Original papers, in French:

Kolmogorov, A. N.: Une série de Fourier-Lebesgue divergente presque partout, Fund. Math., 4, 324-328 (1923).

Kolmogorov, A. N.: Une série de Fourier-Lebesgue divergente partout, Comptes Rendus, 183, 1327-1328 (1926).

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Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.

For an explicit example, you can consider a Riesz product of the form:

$$ \prod_{k=1}^\infty \left( 1+ i \frac{\cos 10^k x}{k}\right)$$

which is divergent. For more examples, see here and here.


Edit: (response to comment). Yes, you are right, du Bois Reymond did indeed construct the examples of Fourier series diverging at a dense set of points. However the result of Kolmogorov is stronger in that it gives almost everywhere divergence.

The papers of du Bois Reymond are:

Ueber die Fourierschen Reihen

available for free download here also another one here.

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    @George S. : It is given in this link http://www-history.mcs.st-and.ac.uk/history/Biographies/Du_Bois-Reymond.html that Du Bois-Reymond gave such function.2010-12-19
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    @J.M. : I am not able to read them as they are not in english. Please suggest where to search for english versions. Also is the function given by Du Bois Reymond continuous everywhere ?2010-12-19
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    @J.M. : What is concept on which the result is based on ?2010-12-19
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    @Rajesh: Ask George; I only cleaned up his answer a bit.2010-12-19
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    @George. Kolmogorov's example is for an $L^1$ function, not continuous. By a famous theorem of Carleson, for a continuous function, the Fourier series converges almost everywhere.2010-12-19
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    @George. ...although the series may still diverge on a dense subset.2010-12-19
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    @TCL: Agree, $f$ continuous $\Rightarrow f\in L^2$.2010-12-19
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    @TCL : please give an example of such a function.2010-12-19
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    @Rajesh. I thought the above mentioned du Bois Reymond's example is one. Existence of such examples is also garunteed by Baire Category theorem.2010-12-19
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    @Rajesh: See Y. Katznelson,"An introduction to Harmonic Analysis" 2nd ed. p. 52.2010-12-19
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As I mentioned in comments below, Kolmogorov's example is for a discontinuous function in $L^1$.

For a continuous function whose Fourier series diverges at all rational multiples of $2\pi$ (and hence on a dense set) see Katznelson's book: An Introduction to Harmonic Analysis Chapter 2, Remark after proof of Theorem 2.1. Note that the Fourier series of such a continuous function still converges almost everywhere by Carleson's theorem.