Denote the 1-torus with $\mathbb{T}$ and if $f\in L^1(\mathbb{T})$ denote the Fourier transform of $f$ by $\hat{f}$.
I know that $$\exists f\in C(\mathbb{T}), t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{int}\notin L^\infty(\mathbb{T}).$$ The reason is that from the theory of Fourier series this must be the case, otherwise there would be convergence in the uniform norm for every Fourier series of every continuous function and this is not the case (see e.g the book An introduction to harmonic analysis by Katznelson). However I don't know any explicit function that does this job... Can anyone provide an explicit example of this fact?