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28 March 2024
 
  » arxiv » 1103.3027

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Multifractal analysis of the divergence of Fourier series
Frédéric Bayart ; Yanick Heurteaux ;
Date 15 Mar 2011
AbstractA famous theorem of Carleson says that, given any function $fin L^p(TT)$, $pin(1,+infty)$, its Fourier series $(S_nf(x))$ converges for almost every $xin mathbb T$. 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 $eta$ such that $S_nf(x)=O(n^eta)$ and we are interested in the size of the exceptional sets $E_eta$, namely the sets of $xinmathbb T$ with divergence index equal to $eta$. 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 $etain[0,1/p]$, $E_eta$ has Hausdorff dimension equal to $1-eta p$. 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.
Source arXiv, 1103.3027
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