Shift-invariant system on the Heisenberg Group

In this paper, a shift-invariant system of the form { Laνgs: ν∈ Γ, s∈ Z} for a> 0 is studied on the Heisenberg group Hn, where Lx denotes the left translation operator on Hn and Γ is a lattice in Hn. 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.