Multifractal analysis of the divergence of Fourier series

Bayart, Frédéric, and Heurteaux, Yanick. "Multifractal analysis of the divergence of Fourier series." Annales scientifiques de l'École Normale Supérieure 45.6 (2012): 927-946. <http://eudml.org/doc/272112>.

@article{Bayart2012,
abstract = {A famous theorem of Carleson says that, given any function $f\in L^p(\mathbb \{T\})$, $p\in (1,+\infty )$, 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 $L^p(\mathbb \{T\})$ 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 surprising.},
author = {Bayart, Frédéric, Heurteaux, Yanick},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Fourier series; multifractal analysis; divergence; Baire category theorem},
language = {eng},
number = {6},
pages = {927-946},
publisher = {Société mathématique de France},
title = {Multifractal analysis of the divergence of Fourier series},
url = {http://eudml.org/doc/272112},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Bayart, Frédéric
AU - Heurteaux, Yanick
TI - Multifractal analysis of the divergence of Fourier series
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 6
SP - 927
EP - 946
AB - A famous theorem of Carleson says that, given any function $f\in L^p(\mathbb {T})$, $p\in (1,+\infty )$, 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 $L^p(\mathbb {T})$ 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 surprising.
LA - eng
KW - Fourier series; multifractal analysis; divergence; Baire category theorem
UR - http://eudml.org/doc/272112
ER -