Hölder regularity of arithmetic Fourier series arising from modular forms

Abstract

Given a modular form which is not a cusp form $M_k(z)={\sum }_{n=0}^{\infty }{r}_n{e}^{2\pi inz}$ of weight $k\ge 4$, we define the series $M_{k,s}(x)={\sum }_{n=1}^{\infty }\frac{r_n}{n^s}\sin (2\pi nx),$ which converges for all $x\in \mathbb{R}$ when $s>k$. In this paper, we compute the Hölder regularity exponent of $M_{k,s}$ at irrational points. In our analysis we apply wavelets methods proposed by Jaffard in 1996 in the study of the Riemann series. We find that the Hölder regularity exponent at a point $x$ is related to the fine diophantine properties of $x$, in a very precise way.

19 pages, added references, improved results

Cited by in corpus (2)

• Differentiability of arithmetic Fourier series arising from Eisenstein series
• The Hölder exponent of some Fourier series