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Zak transform for semidirect product of locally compact groups

Zak transform for semidirect product of locally compact groups 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Zak transform for semidirect product of locally compact groups

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References (17)

Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer Basel
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-013-0057-6
Publisher site
See Article on Publisher Site

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.

Journal

Analysis and Mathematical Physics – Springer Journals

Published: Apr 10, 2013

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