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 $\{T_{(k,l)}^t\varphi :k,l\in {\mathbb{Z}}^n\}$ is a frame sequence iff $0<A\le \left[\varphi ,\varphi \right](\xi ,{\xi }^{{}^{\prime }})\le B<\infty ,$ for a.e $(\xi ,{\xi }^{{}^{\prime }})\in {\Omega }_{\varphi },$ where ${\Omega }_{\varphi }=\{(\xi ,{\xi }^{{}^{\prime }})\in {\mathbb{T}}^n\times {\mathbb{T}}^n:\left[\varphi ,\varphi \right](\xi ,{\xi }^{{}^{\prime }})\ne 0\}$. We also prove a similar result for the system $\{T_{(k,l)}^t\varphi :k,l\in {\mathbb{Z}}^n\}$ to be a Riesz sequence. For a given function belonging to the principal twisted shift-invariant space $V^t(\varphi )$, we find a necessary and sufficient condition for the existence of a canonical biorthogonal function. Further, we obtain a characterization for the system $\{T_{(k,l)}^t\varphi :k,l\in \mathbb{Z}\}$ to be a Schauder basis for $V^t(\varphi )$ in terms of a Muckenhoupt ${\mathcal{A}}_2$ weight function.