On the Fourier Analysis of Measures with Meyer Set Support

TL;DR

This paper establishes the generalized Eberlein decomposition for Fourier transformable measures supported on Meyer sets, showing each component's support and periodicity properties, and characterizing Fourier transformability via positive definite measures.

Contribution

It proves the existence of the Eberlein decomposition for measures with Meyer set support and characterizes Fourier transformability in terms of positive definite measures with Meyer support.

Findings

Each component of the decomposition is Fourier transformable with Meyer support.

Fourier transform of such measures is norm almost periodic.

Fourier transformability characterized by positive definite measures with Meyer support.

Abstract

In this paper we show the existence of the generalized Eberlein decomposition for Fourier transformable measures with Meyer set support. We prove that each of the three components is also Fourier transformable and has Meyer set support. We obtain that each of the pure point, absolutely continuous and singular continuous components of the Fourier transform is a strong almost periodic measure, and hence is either trivial or has relatively dense support. We next prove that the Fourier transform of a measure with Meyer set support is norm almost periodic, and hence so is each of the pure point, absolutely continuous and singular continuous components. We show that a measure with Meyer set support is Fourier transformable if and only if it is a linear combination of positive definite measures, which can be chosen with Meyer set support, solving a particular case of an open problem. We complete the paper by discussing some applications to the diffraction of weighted Dirac combs with Meyer set support.