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

2010/04/21 by Kozma, Gady, Olevskii, Alexander
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1004.3631

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: (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 l². We also give examples of non-symmetry which may occur for measures with "small" support. A number of open questions are stated.

Related

• Singular distributions and symmetry of the spectrum
• Fourier transforms of spherical distributions on compact symmetric spaces
• A measure with small support and p-summable Fourier transform
• Higher-order Fourier dimension and frequency decompositions
• The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures