Twice Fourier transformable measures and diffraction theory

TL;DR

This paper reviews and unifies mathematical diffraction theory using Wiener amalgams, simplifying previous approaches and applying to translation bounded measures and weighted model sets.

Contribution

It introduces a unified framework for diffraction theory using Wiener amalgams, simplifying existing methods and extending to weighted model sets.

Findings

Unified approach to diffraction theory with Wiener amalgams

Simplification of previous Fourier and distribution methods

Application to weighted Meyer model sets

Abstract

Mathematical diffraction theory has been developed since about 1995. Hof's initial approach relied on tempered distributions in euclidean space. Nowadays often the Fourier theory by Argabright and Gil de Lamadrid is used, which applies to appropriate measures on locally compact abelian groups. We review diffraction theory using Wiener amalgams as test function spaces. For translation bounded measures, this unifies and simplifies the former two approaches. We treat weighted versions of Meyer's model sets as examples.