# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.04778