Uniqueness theorem for Fourier transformable measures on LCA groups

TL;DR

This paper proves a uniqueness theorem for Fourier transformable measures on LCA groups, showing that under certain conditions, two measures with similar support and mass behavior must be identical, with exceptions for thick measures.

Contribution

It establishes a new uniqueness criterion for Fourier transformable measures on LCA groups, extending to almost periodic measures and highlighting limitations for thick measures.

Findings

Uniqueness holds for discrete 'not very thick' Fourier transformable measures.

Counterexamples exist for 'thick' measures where uniqueness fails.

Constructed an almost periodic measure with diminishing masses at support points.

Abstract

We show that if points of supports of two discrete "not very thick" Fourier transformable measures on LCA groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete "thick" measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.