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

Details Singular distributions, dimension of support, and symmetry of Fourier transform Singular distributions, dimension of support, and symmetry of Fourier transform

2010-04-21

arxiv.org/abs/1004.3631v2

Mathematics

Classical Analysis and ODEs

19 pages, to appear in Ann. Inst. Fourier

Scientific paper

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: (1) 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 the support; (2) A construction of compacts of "critical" size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l^2. We also give examples of non-symmetry which may occur for measures with "small" support. A number of open questions are stated.

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