Sergey Favorov: Measures and temperate distributions with discrete support and spectrum

Abstract

Consider a measure on Euclidean space with discrete support such that its distributional Fourier transform also a measure with discrete support (so called crystalline measure). We investigate conditions for support of the measure to be a finite union of translated full-rank lattices and give a complete description of such measures. The proof is based on Cohen's Idempotent Theorem and a new local analog of Wiener's Theorem on trigonometric series. Also we show generalizations of these results to temperate distributions with discrete support. Moreover, as application of above results we find a new sufficient condition for a discrete set in Euclidean space to be a coherent set of frequencies.