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.