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Zak transform for semidirect product of locally compact groups

Research paper by Ali Akbar Arefijamaal, Arash Ghaani Farashahi

Indexed on: 10 Apr '13Published on: 10 Apr '13Published in: Analysis and Mathematical Physics



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.