{
  "id": "1103.3027",
  "version": "v1",
  "published": "2011-03-15T20:37:49.000Z",
  "updated": "2011-03-15T20:37:49.000Z",
  "title": "Multifractal analysis of the divergence of Fourier series",
  "authors": [
    "Frédéric Bayart",
    "Yanick Heurteaux"
  ],
  "journal": "Annales Scientifiques de l'\\'Ecole Normale Sup\\'erieure 45 (2012) 927-946",
  "categories": [
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-15T20:37:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier series",
      "multifractal analysis",
      "exceptional sets",
      "quasi-all functions",
      "divergence index equal"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3027B"
    }
  }
}