Abstract
Let \(H\) be a locally compact group and \(K\) be an LCA group also let \(\tau :H\rightarrow Aut(K)\) be a continuous homomorphism and \(G_\tau =H\ltimes _\tau K\) be the semidirect product of \(H\) and \(K\) with respect to \(\tau \). In this article we define the Zak transform \(\mathcal{Z }_L\) on \(L^2(G_\tau )\) with respect to a \(\tau \)-invariant uniform lattice \(L\) of \(K\) and we also show that the Zak transform satisfies the Plancherel formula. As an application we analyze how these technique apply for the semidirect product group \(\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2\) and also the Weyl-Heisenberg groups.
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Acknowledgments
The authors would like to gratefully acknowledge financial support from the Numerical Harmonic Analysis Group (NuHAG) at the Faculty of Mathematics, University of Vienna. Thanks are also due to the referees for several valuable comments.
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Arefijamaal, A.A., Ghaani Farashahi, A. Zak transform for semidirect product of locally compact groups. Anal.Math.Phys. 3, 263–276 (2013). https://doi.org/10.1007/s13324-013-0057-6
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DOI: https://doi.org/10.1007/s13324-013-0057-6