# an example of a continuous function whose Fourier series diverges at a dense set of points

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.

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.

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

• @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. – TCL Dec 19 '10 at 16:13