Left Translates of a Square Integrable Function on the Heisenberg group

The aim of this paper is to study some properties of left translates of a square integrable function on the Heisenberg group. First, a necessary and sufficient condition for the existence of the canonical dual to a function $\phi \in L^2({\mathbb{R}}^{2n})$ is obtained in the case of twisted shift-invariant spaces. Further, characterizations of ${\mathrm{\ell }}^2$-linear independence and the Hilbertian property of the twisted translates of a function $\phi \in L^2({\mathbb{R}}^{2n})$ are obtained. Later these results are shown in the case of the Heisenberg group.