Almost periodic distributions and crystalline measures

TL;DR

This paper explores the properties of almost periodic distributions and crystalline measures, establishing connections with Fourier transforms and constructing a novel crystalline measure that is neither almost periodic nor a Fourier quasicrystal.

Contribution

It introduces a new crystalline measure on the real line that is distinct from known classes like almost periodic distributions and Fourier quasicrystals.

Findings

Constructed a crystalline measure not classified as almost periodic or a Fourier quasicrystal.

Analyzed the relationship between Fourier transforms and coefficients of almost periodic distributions.

Established properties of temperate almost periodic distributions in Euclidean space.

Abstract

Based on the properties of distributions and measures with discrete support, we investigate temperate almost periodic distributions on the Euclidean space and connection with their Fourier transforms. We also study relations between the Fourier transform of almost periodic distributions and their Fourier coefficients. The main result of the article is the construction of a crystalline measure on the real line, which is neither almost periodic distribution, nor a Fourier quasicrystal.