On the Fourier Transformability of Strongly Almost Periodic Measures

TL;DR

This paper characterizes when strongly almost periodic measures can be Fourier transformed, providing conditions based on their Fourier Bohr series and exploring implications for measures from cut and project schemes.

Contribution

It offers a new integrability criterion for Fourier transformability of strongly almost periodic measures and characterizes when such measures are Fourier transforms of other measures.

Findings

Fourier transformability characterized by Fourier Bohr series integrability

Necessary and sufficient conditions for measures to be Fourier transforms

Application to measures from cut and project formalism

Abstract

In this paper we characterize the Fourier transformability of a strongly almost periodic measure in terms of an integrability condition for its Fourier Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be a Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\RR^d$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.