Linear independence of time-frequency shifts under a generalized Schrödinger representation

Abstract.

Let $\rho_{\mathbb{R}}$ be the classical Schrödinger representation of the Heisenberg group and let $\Lambda$ be a finite subset of $\mathbb{R} \times \mathbb{R}$. The question of when the set of functions $\{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\}$ is linearly independent for all $f \in L^2(\mathbb{R}), f \neq 0$, arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset $\Lambda$ of $G \times \widehat{G}$ and $\rho_G$ the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set $\{\rho_G (x, w, 1)f : (x, w) \in \Lambda\}$ to be linearly independent for all $f \in L^2(G), f \neq 0$.