A generalized sampling theorem for locally compact abelian groups

Abstract

We present a sampling theorem for locally compact abelian groups. The sampling sets are finite unions of cosets of a closed subgroup. This generalizes the well-known case of nonequidistant but periodic sampling on the real line. For nonbandlimited functions an L 1 {L_1} -type estimate for the aliasing error is given. We discuss the application of the theorem to a class of sampling sets in R s {{\mathbf {R}}^s} , give a general algorithm for a computer implementation, present a detailed description of the implementation for the s-dimensional torus group, and point out connections to lattice rules for numerical integration.