Zak Transform Associated with the Weyl Transform and the System of Twisted Translates on $\mathbb {R}^{2n}$

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 $\{{T_{(k,l)}^t}{\phi }: k,l\in \mathbb {Z}^{n}\}$ is a frame sequence iff $0<A\le \left[ {\phi },{\phi }\right] (\xi ,\xi ^{'})\le B<\infty ,$ for a.e $(\xi ,\xi ^{'})\in \Omega _{{\phi }},$ where $\Omega _{{\phi }}=\{({\xi },{\xi ^{'}})\in {\mathbb {T}^{n}}\times {\mathbb {T}^{n}}: \left[ {\phi },{\phi }\right] (\xi ,\xi ^{'})\ne 0\}$. We also prove a similar result for the system $\{{T_{(k,l)}^t}{\phi }: k,l\in \mathbb {Z}^{n}\}$ 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 $\{{T_{(k,l)}^t}{\phi }: k,l\in \mathbb {Z}\}$ to be a Schauder basis for $V^{t}({\phi })$ in terms of a Muckenhoupt $\mathcal {A}_{2}$ weight function.