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.
Highlights
Background and Plan of the ArticleSampling concerns the problem of reconstructing a function f from its restriction f | to a subdomain
One can show that the existence of an upper bound for stable sampling is equivalent to the existence of an upper bound for stable interpolation, see [38, p. 129]
Recall that S K = { f ∈ L1(G) ∩ L2(G) : pf |K c = 0} is a subspace of PW K, which is dense in PW K if and only if K has almost no inner boundary
Summary
Sampling concerns the problem of reconstructing a function f from its restriction f | to a subdomain. Interpolation concerns the question of how to extend a function defined on the subdomain. Both problems are classical in harmonic analysis. We are interested in irregular sampling domains arising from model sets in locally compact abelian (LCA) groups
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