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

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 ${\mathcal L} \cap (G \times W)$ and the Fourier tr...

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Bibliographic Details
Published inCanadian mathematical bulletin Vol. 66; no. 3; pp. 1044 - 1060
Main Author Strungaru, Nicolae
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.09.2023
Cambridge University Press
Subjects
Banach spaces
Fourier analysis
Fourier transforms
43A60
Meyer sets
43A25
Cut-and-project schemes
Fourier transform of measures
52C23
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Summary: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 ${\mathcal L} \cap (G \times W)$ and the Fourier transformable measures on G supported 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.
ISSN:0008-4395
1496-4287
DOI:10.4153/S0008439523000164