Sampling theorems for the Heisenberg groups

Abstract

In the first part of the paper a general notion of sampling expansions for locally compact groups is introduced, and its close relationship to the discretisation problem for generalised wavelet transforms is established. In the second part, attention is focussed on the simply connected nilpotent Heisenberg group $\HH$. We derive criteria for the existence of discretisations and sampling expansions associated to lattices in $\HH$. Analogies and differences to the sampling theorem over the reals are discussed, in particular a notion of bandwidth on $\HH$ will figure prominently. The main tools for the characterisation are the Plancherel formula of $\HH$ and the theory of Weyl-Heisenberg frames. In the last section we compute an explicit example