Multifractal analysis of the divergence of Fourier series: the extreme cases

TL;DR

This paper investigates the Hausdorff dimension of subsets of the circle where Fourier series of generic functions in various function spaces exhibit extreme divergence behavior, using concepts of genericity like Baire category and prevalence.

Contribution

It provides a multifractal analysis of divergence sets for Fourier series of generic functions across different function spaces, highlighting the size and structure of these sets.

Findings

Hausdorff dimension of divergence sets characterized

Divergence behavior analyzed for functions in $L^1$, $L^p$, and continuous spaces

Results relate to genericity notions like Baire category and prevalence

Abstract

We study the size, in terms of the Hausdorff dimension, of the subsets of $\mathbb T$ such that the Fourier series of a generic function in $L^1(\TT)$, $L^p(\TT)$ or in $\mathcal C(\mathbb T)$ may behave badly. Genericity is related to the Baire category theorem or to the notion of prevalence.