Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

TL;DR

This paper investigates tempered distributions with Fourier transforms that are translation bounded measures, establishing their order, and introduces a generalized Eberlein decomposition compatible with previous versions, with applications to measures on Meyer sets.

Contribution

It proves the existence of a generalized Eberlein decomposition for this class of distributions and explores its properties and compatibility with earlier decompositions.

Findings

Distributions have order at most 2d.

Existence of a generalized Eberlein decomposition.

Analysis of absolutely continuous spectrum on Meyer sets.

Abstract

In this paper, we study the class of tempered distributions whose Fourier transform is a translation bounded measure and show that each such distribution in $\mathbb{R}^d$ has order at most $2d$. We show the existence of the generalized Eberlein decomposition within this class of distributions, and its compatibility with all previous Eberlein decompositions. The generalized Eberlein decomposition for Fourier transformable measures and properties of its components are discussed. Lastly, we take a closer look at the absolutely continuous spectrum of measures supported on Meyer sets.