Zak transform associated with the Weyl transform and the system of twisted translates on R^{2n}

Abstract

We introduce the Zak transform on $L^{2}(\mathbb{R}^{2n})$ associated with the Weyl transform. By making use of this transform, we define a bracket map and prove that the system of twisted translates $\{T^{t}_{(k,l)}\phi : k,l\in \mathbb{Z}^{n}\}$ is a frame sequence iff $0<A\leq \left[\phi,\phi\right](\xi,\xi^{'})\leq B<\infty,$ for a.e $(\xi,\xi^{'})\in \Omega_{\phi},$ where $\Omega_{\phi}=\{(\xi,\xi^{'})\in \mathbb{T}^{n}\times\mathbb{T}^{n} : \left[\phi,\phi\right](\xi,\xi^{'})\neq 0\}$. We also prove a similar result for the system $\{T^{t}_{(k,l)}\phi : k,l\in \mathbb{Z}^{n}\}$ to be a Riesz sequence. For a given function belonging to the principal twisted shift-invariant space $V^{t}(\phi)$, we find a necessary and sufficient condition for the existence of a canonical biorthogonal function. Further, we obtain a characterization for the system $\{T^{t}_{(k,l)}\phi : k,l\in\mathbb{Z}\}$ to be a Schauder basis for $V^{t}(\phi)$ in terms of a Muckenhoupt $\mathcal{A}_{2}$ weight function.