Laws of Large Numbers, Spectral Translates and Sampling Over LCA Groups

Abstract

Kluvánek extended the Whittaker-Kotel’nikov-Shannon theorem to the abstract harmonic analysissetting over a LCA group G. In this context, the classical condition for f ∈ L2(R) to be band limited isreplaced by fb having its support essentially contained in a transversal set of a compact quotient group.This condition was later shown to be necessary in general. Moreover, the classical interpolation formulais also equivalent to a Plancherel like isometric formula involving the L2(G) norm of f and the norm ofthe sequence of its samples over a subgroup H. Here, recalling some Laws of Large Numbers, we willprove an equivalent result for the support of the spectral measure µX of a Gaussian stationary randomprocess X, indexed over a LCA group G. The conditions are formulated in terms of an almost sureisometric formula involving the sample variances of X, and its samples over a subgroup H respectively