Multifractal analysis of the divergence of Fourier series

TL;DR

This paper studies the multifractal nature of divergence rates of Fourier series for functions in $L^p$ and continuous functions, revealing precise Hausdorff dimensions of divergence sets and surprising results in the continuous case.

Contribution

It introduces a multifractal analysis of divergence indices of Fourier series and characterizes the Hausdorff dimensions of the sets where divergence occurs at specific rates.

Findings

For quasi-all functions in $L^p$, divergence sets have Hausdorff dimension $1-eta p$.

In the continuous case, divergence behavior with logarithmic divergence is analyzed, yielding surprising results.

The study reveals a rich multifractal structure in the divergence of Fourier series.

Abstract

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without exceeding $O(n^{1/p})$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_nf(x)=O(n^\beta)$ and we are interested in the size of the exceptional sets $E_\beta$, namely the sets of $x\in\mathbb T$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p(\TT)$ have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in $L^p(\mathbb T)$, for all $\beta\in[0,1/p]$, $E_\beta$ has Hausdorff dimension equal to $1-\beta p$. We also investigate the same problem in $\mathcal C(\mathbb T)$, replacing polynomial divergence by logarithmic divergence. In this context, the results that we get on the size of the exceptional sets are rather surprizing.