An uncertainty principle for the windowed Bochner-Fourier transform with the complex-valued window function

The generalization of Hardy uncertainty principle for the windowed Bochner-Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Psi:G -> X, where X is a com-pletely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. Weestablish that if phi is an element of C-C (G) is anon-trivial window function and Psi is an element of L- 2(G)satisfies & Vert;V phi(Psi)(g,chi)& Vert; HB <= c(2)(g)exp(-beta & Vert;chi & Vert;(2)),beta>0, then Psi = 0 almost every where.