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Transactions of the American Mathematical Society

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Existence of quasicrystals and universal stable sampling and interpolation in LCA groups


Authors: Elona Agora, Jorge Antezana, Carlos Cabrelli and Basarab Matei
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 42C15, 94A20; Secondary 42C30, 43A25
DOI: https://doi.org/10.1090/tran/7723
Published electronically: May 20, 2019
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Abstract: We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and project scheme that produces it. On the other hand, if an LCA group $ G$ admits a simple quasicrystal, we prove that recent results of Meyer and Matei for the case of the Euclidean space $ \mathbb{R}^n$ can be extended to $ G$. More precisely, we prove that simple quasicrystals are universal sets of stable sampling and universal sets of stable interpolation in generalized Paley-Wiener spaces.


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Additional Information

Elona Agora
Affiliation: Instituto Argentino de Matemática “Alberto P. Calderón” (IAM-CONICET), 1083 Buenos Aires, Argentina
Email: elona.agora@gmail.com

Jorge Antezana
Affiliation: Departamento de Matemática, Universidad Nacional de La Plata and Instituto Argentino de Matemática “Alberto P. Calderón” (IAM-CONICET), Buenos Aires, Argentina
Email: antezana@mate.unlp.edu.ar

Carlos Cabrelli
Affiliation: Departamento de Matemática, Universidad de Buenos Aires and Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina
Email: cabrelli@dm.uba.ar

Basarab Matei
Affiliation: Institut Galilée and Université Paris 13, Paris, France
Email: matei@lipn.univ-paris13.fr

DOI: https://doi.org/10.1090/tran/7723
Keywords: Quasicrystals, universal sampling and interpolation, Landau-Beurling's densities, Poisson measures, locally compact abelian groups
Received by editor(s): September 17, 2017
Received by editor(s) in revised form: September 3, 2018
Published electronically: May 20, 2019
Additional Notes: The first author was supported in part by Grants MTM2013-40985-P, MTM2016-75196-P, PIP 112201501003553CO, UBACyT 20020130100422BA
The second author was supported in part by Grants CONICET-PIP 152, UNLP-11X585, MTM2016-75196-P
The third author was supported in part by Grants PICT 2014-1480 (ANPCyT), CONICET PIP 11220110101018, UBACyT 20020130100403BA, UBACyT 20020130100422BA
Article copyright: © Copyright 2019 American Mathematical Society