Bandlimited Wavelets on the Heisenberg Group

Abstract

Let $\mathbb{H}$ be the three-dimensional Heisenberg group. We introduce a structure on the Heisenberg group which consists of the biregular representation of $\mathbb{H\times H}$ restricted to some discrete subset of $\mathbb{H\times H}$ and a free group of automorphisms $H$ singly generated and acting semi-simply on $\mathbb{H}$. Using well-known theorems borrowed from Gabor theory, we are able to construct simple and computable bandlimited discrete wavelets on the Heisenberg group. Moreover, we provide necessary and sufficient conditions for the existence of these wavelets.