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.
an example of a continuous function whose Fourier series diverges at a dense set of points
3 Answers
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).
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
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0@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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0@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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0@J.M. : What is concept on which the result is based on ? – 2010-12-19
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0@Rajesh: Ask George; I only cleaned up his answer a bit. – 2010-12-19
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2@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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0@George. ...although the series may still diverge on a dense subset. – 2010-12-19
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0@TCL: Agree, $f$ continuous $\Rightarrow f\in L^2$. – 2010-12-19
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0@TCL : please give an example of such a function. – 2010-12-19
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0@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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0@Rajesh: See Y. Katznelson,"An introduction to Harmonic Analysis" 2nd ed. p. 52. – 2010-12-19
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.