Fourier transformable measures with weak Meyer set support and their lift to the cut-and-project scheme

Abstract

AbstractIn 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${\mathcal L} \cap (G \times W)$and the Fourier transformable measures onGsupported inside${\LARGE \curlywedge }(W)$. We provide a closed formula relating the Fourier transform of the original measure and the Fourier transform of the projection. We show that this formula can be used to re-derive some known results about Fourier analysis of measures with weak Meyer set support.