A dynamical approach to sampling and interpolation in unimodular groups

Abstract

We introduce a notion of covolume for point sets in locally compact groups that simultaneously generalizes the covolume of a lattice and the reciprocal of the Beurling density for amenable, unimodular groups. This notion of covolume arises naturally from transverse measure theory applied to the hull dynamical system associated to a point set. Using groupoid techniques, we prove necessary conditions for sampling and interpolation in reproducing kernel Hilbert spaces on unimodular groups in terms of this new notion of covolume. These conditions generalize previously known density theorems for compactly generated groups of polynomial growth, while also covering important new examples, in particular model sets arising from cut-and-project schemes.

v2: 29 pages. The paper has been restructured and shortened. Sections 3 and 4 in the first version now constitute section 2, while sections 2 and 5 in the first version now constitute section 3. Some of the results on groupoids of point sets that appearedn in section 3 of the first version have been moved to an appendix