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Multifractal analysis of the divergence of Fourier series | Frédéric Bayart
; Yanick Heurteaux
; | Date: |
15 Mar 2011 | Abstract: | A 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 | Services: | Forum | Review | PDF | Favorites |
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