The H\"older exponent of some Fourier series

TL;DR

This paper investigates the local regularity of fractional integrals of Fourier series, especially those from modular forms, revealing that cusp forms typically produce pure fractals rather than multifractals, supported by examples and visualizations.

Contribution

It provides a detailed analysis of the H"older regularity of fractional integrals of Fourier series, highlighting the fractal nature of cusp forms and including explicit examples and visualizations.

Findings

Cusp forms generally produce pure fractals.

The study offers explicit examples and computer plots.

Different definitions of H"older exponent are considered.

Abstract

In this paper we study the local regularity of fractional integrals of Fourier series using several definitions of the H\"older exponent. We especially consider series coming from fractional integrals of modular forms. Our results show that in general cusp forms give rise to pure fractals (as opposed to multifractals). We include explicit examples and computer plots.