Left Translates of a Square Integrable Function on the Heisenberg group

TL;DR

This paper investigates properties of left translates of square integrable functions on the Heisenberg group, providing conditions for duality, linear independence, and Hilbertian properties in twisted shift-invariant spaces.

Contribution

It extends the analysis of twisted shift-invariant spaces to the Heisenberg group, establishing new criteria for duality and independence of translates.

Findings

Necessary and sufficient condition for the existence of a canonical dual in twisted shift-invariant spaces.

Characterizations of $ ext{l}^2$-linear independence of twisted translates.

Results on the Hilbertian property of twisted translates on the Heisenberg group.

Abstract

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 $\varphi\in L^{2}(\mathbb{R}^{2n})$ is obtained in the case of twisted shift-invariant spaces. Further, characterizations of $\ell^{2}$-linear independence and the Hilbertian property of the twisted translates of a function $\varphi\in L^{2}(\mathbb{R}^{2n})$ are obtained. Later these results are shown in the case of the Heisenberg group.