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

We introduce the Zak transform on \(L^{2}(\mathbb {R}^{2n})\) $L^2({\mathbb{R}}^{2n})$ associated with the Weyl transform. By making use of this transform, we define a bracket map and prove that the system of twisted translates \(\{{T_{(k,l)}^t}{\phi }: k,l\in \mathbb {Z}^{n}\}\) $\{T_{(k,l)}^t\varphi :k,l\in {\mathbb{Z}}^n\}$ is a frame sequence iff \(0<A\le \left[ {\phi },{\phi }\right] (\xi ,\xi ^{'})\le B<\infty ,\) $0<A\le \left[\varphi ,,,\varphi \right](\xi ,{\xi }^{{}^{\prime }})\le B<\infty ,$ for a.e \((\xi ,\xi ^{'})\in \Omega _{{\phi }},\) $(\xi ,{\xi }^{{}^{\prime }})\in {\mathrm{\Omega }}_{\varphi },$ where \(\Omega _{{\phi }}=\{({\xi },{\xi ^{'}})\in {\mathbb {T}^{n}}\times {\mathbb {T}^{n}}: \left[ {\phi },{\phi }\right] (\xi ,\xi ^{'})\ne 0\}\) ${\mathrm{\Omega }}_{\varphi }=\{(\xi ,{\xi }^{{}^{\prime }})\in {\mathbb{T}}^n\times {\mathbb{T}}^n:\left[\varphi ,,,\varphi \right](\xi ,{\xi }^{{}^{\prime }})\ne 0\}$ . We also prove a similar result for the system \(\{{T_{(k,l)}^t}{\phi }: k,l\in \mathbb {Z}^{n}\}\) $\{T_{(k,l)}^t\varphi :k,l\in {\mathbb{Z}}^n\}$ to be a Riesz sequence. For a given function belonging to the principal twisted shift-invariant space \(V^{t}({\phi })\) $V^t(\varphi )$ , we find a necessary and sufficient condition for the existence of a canonical biorthogonal function. Further, we obtain a characterization for the system \(\{{T_{(k,l)}^t}{\phi }: k,l\in \mathbb {Z}\}\) $\{T_{(k,l)}^t\varphi :k,l\in \mathbb{Z}\}$ to be a Schauder basis for \(V^{t}({\phi })\) $V^t(\varphi )$ in terms of a Muckenhoupt \(\mathcal {A}_{2}\) ${\mathcal{A}}_2$ weight function.