Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme

TL;DR

This paper establishes a bijection between Fourier transformable measures supported on a cut-and-project scheme and those supported on Meyer sets, providing a new way to analyze their Fourier transforms and re-derive known results.

Contribution

It introduces a bijection between measures supported on a cut-and-project scheme and Meyer sets, linking their Fourier transforms and enabling new analytical approaches.

Findings

Established a bijection between measures supported on cut-and-project schemes and Meyer sets.

Derived relations between Fourier transforms of these measures.

Re-derivation of known Fourier analysis results for Meyer set supported measures.

Abstract

In this paper, we prove that given a cut-and-project scheme $(G, H, \mathcal{L})$ and a compact window $W \subseteq H$, the natural projection gives a bijection between the Fourier transformable measures on $G \times H$ supported inside the strip $\cL \cap (G \times W)$ and the Fourier transformable measures supported inside $\oplam(W)$, and relate their Fourier transforms. We use this formula to relate the Fourier transforms of the measures, and explain how one can use this relation to re-derive some known results about Fourier analysis of measures with Meyer set support.