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Pure point measures with sparse support and sparse Fourier-Bohr support

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Baake, M, Strungaru, N and Terauds, V ORCID: 0000-0002-6891-5333 2020 , 'Pure point measures with sparse support and sparse Fourier-Bohr support' , Transactions of the London Mathematical Society, vol. 7, no. 1 , pp. 1-32 , doi: 10.1112/tlm3.12020.

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Abstract

Fourier‐transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier–Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

Item Type: Article
Authors/Creators:Baake, M and Strungaru, N and Terauds, V
Keywords: aperiodicity, pure point measures, Fourier transform
Journal or Publication Title: Transactions of the London Mathematical Society
Publisher: John Wiley & Sons Ltd
ISSN: 2052-4986
DOI / ID Number: 10.1112/tlm3.12020
Copyright Information:

Copyright 2020 The Authors. Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) http://creativecommons.org/licenses/by-nc-nd/4.0/

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