Universal sampling and interpolation in locally compact abelian groups

Abstract

Let G be a locally compact abelian group whose dual group Gˆ is compactly generated. If G admits a quasicrystal, then it is known that this quasicrystal is a universal sampling and universal interpolation set. However, there exist simple groups without quasicrystals. In this work, we fill this gap. We prove that G admits universal sampling and interpolation sets, provided that G is also compactly generated. If G is not assumed to be compactly generated, we prove the existence of universal sampling or interpolating sets in a slightly weaker sense.