Localization operators for generalized Weyl–Heisenberg group

Abstract

In this paper, we consider the generalized Weyl–Heisenberg group $${\mathbb {H}}(G_\tau )$$ associated with the semi direct product group $$G_\tau =H \times _\tau K$$ in which H is a locally compact group, K is a locally compact abelian group and $$\tau :H \rightarrow \hbox {Aut}(K)$$ is a continuous homomorphism. We give a square integrable representation on $${\mathbb {H}}(G_\tau )$$ and then as a result, we obtain admissible wavelets on this group. Moreover, we define the localization operator on $${\mathbb {H}}(G_\tau )$$ and we show that it is a bounded and compact operator. Furthermore, we investigate the boundedness and compactness of localization operators with two admissible wavelets on $$L^p({\hat{K}})$$ , ( $${\hat{K}}$$ the dual group of K), for the generalized Weyl–Heisenberg group. Finally, some example are given.