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.
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Acknowledgements
We thank Prof. C. Heil for his helpful comments on Lemma 6.1. We thank the referee for meticulously reading the manuscript and giving us valuable comments and suggestions which helped us to improve the presentation of the earlier version of the manuscript to the current version.
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The second named author is a recipient of senior research fellowship from University Grants Comission, India.
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Ramakrishnan, R., Velsamy, R. Zak Transform Associated with the Weyl Transform and the System of Twisted Translates on \(\mathbb {R}^{2n}\). Results Math 79, 65 (2024). https://doi.org/10.1007/s00025-023-02088-x
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DOI: https://doi.org/10.1007/s00025-023-02088-x