On sampling and interpolation by model sets

TL;DR

This paper refines the understanding of stable sampling and interpolation for model sets in locally compact abelian groups, using Fourier analysis and duality, and provides elementary proofs away from critical density.

Contribution

It improves existing results on sampling and interpolation for model sets, introducing a refined duality and elementary proofs based on the Poisson Summation Formula.

Findings

Refined duality between sampling and interpolation for model sets.

Elementary proof of stable sampling and interpolation away from critical density.

Application of Fourier analysis to unbounded Radon measures.

Abstract

We refine a result of Matei and Meyer on stable sampling and stable interpolation for simple model sets. Our setting is model sets in locally compact abelian groups and Fourier analysis of unbounded complex Radon measures as developed by Argabright and de Lamadrid. This leads to a refined version of the underlying model set duality between sampling and interpolation. For rather general model sets, our methods also yield an elementary proof of stable sampling and stable interpolation sufficiently far away from the critical density, which is based on the Poisson Summation Formula.