Singular distributions, dimension of support, and symmetry of fourier transform

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Abstract

We study the "Fourier symmetry" of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (i) A one-side extension of Frostman's theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support; (ii) A construction of compacts of "critical" size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l2. We also give examples of non-symmetry which may occur for measures with "small" support. A number of open questions are stated.

Original languageEnglish
Pages (from-to)1205-1226
Number of pages22
JournalAnnales de l'Institut Fourier
Volume63
Issue number4
DOIs
StatePublished - 2013

Keywords

  • Fourier symmetry
  • Frostman's theorem
  • Hausorff dimension

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