Shift-invariant system on the Heisenberg Group

Abstract

In this paper, a shift-invariant system of the form $$\{L_{a\nu }g_s:\nu \in \varGamma ,s\in {{\mathbb {Z}}}\}$$ for $$a>0$$ is studied on the Heisenberg group $${{\mathbb {H}}^n}$$ , where $$L_x$$ denotes the left translation operator on $${{\mathbb {H}}^n}$$ and $$\varGamma $$ is a lattice in $${{\mathbb {H}}^n}$$ . The characterizations for the shift-invariant system to be a Bessel sequence and a frame sequence are given in terms of certain operators arising from the fiber map corresponding to this system. For a shift invariant system to be a frame sequence (Riesz sequence) the characterization is given in terms of dual Gramian (Gramian). Moreover, the problem of characterizing a pair of shift-invariant systems to be dual frames is also discussed.