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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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3 Answers 3

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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.

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  • $\begingroup$ Why the function in Katznelson's book doesn't satisfy Fourier's Theorem? I know it is continuous but I suppose I can't derive it term by term to check if the derivate is piecewise continuous. $\endgroup$ Jul 13, 2020 at 22:20
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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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  • $\begingroup$ @George S. : It is given in this link www-history.mcs.st-and.ac.uk/history/Biographies/… that Du Bois-Reymond gave such function. $\endgroup$
    – Rajesh D
    Dec 19, 2010 at 11:26
  • $\begingroup$ @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 ? $\endgroup$
    – Rajesh D
    Dec 19, 2010 at 12:37
  • $\begingroup$ @J.M. : What is concept on which the result is based on ? $\endgroup$
    – Rajesh D
    Dec 19, 2010 at 12:38
  • $\begingroup$ @Rajesh: Ask George; I only cleaned up his answer a bit. $\endgroup$ Dec 19, 2010 at 12:42
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    $\begingroup$ @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. $\endgroup$
    – TCL
    Dec 19, 2010 at 16:13
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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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