2$\pi$-peridodic and continuous function with non summable Fourier coeffficients

Question

Can someone tell me an example of 2$\pi$-periodic and continuous function, f, with Fourier coefficients $\hat{f}(n)\;\forall{n\in{\mathbb{Z}}}$ such that $\sum|\hat{f}(n)|>\infty$?

Thanks.

Answer 1

Convergence of Fourier Series is a delicate business but everything is known. The theorem you are looking for is the following:

(Du Bois-Reymond 1873) There is a continuous function $f: [-\pi, \pi] \rightarrow \mathbb{C}$ such that $\limsup_{N} \sum_{|n| \le N} \widehat{f}(n) \rightarrow \infty$.

A standard proof of this result can be found all over the internet but I think the most elementary construction is given here.

Just as a reference, if you know that your function is in $C^1$ then the Fourier series converges uniformly. Furthermore, if you know that your function is in $L^p$ for $p > 1$ then your Fourier series converges pointwise for everything in your compact interval except possibly on a set of measure $0$. This is the celebrated Carleson-Hunt Theorem.