Continuous Gabor transform for semi-direct product of locally compact groups

Abstract

Let $H$ be a locally compact group, $K$ be an LCA group, $\tau :H\to Aut(K)$ be a continuous homomorphism and $G_{\tau }=H{⋉}_{\tau }K$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $\tau$. In this article we introduce the $\tau \times \hat{\tau }$-time frequency group $G_{\tau \times \hat{\tau }}$. We define the $\tau \times \hat{\tau }$-continuous Gabor transform of $f\in L^2(G_{\tau })$ with respect to a window function $u\in L^2(K)$ as a function defined on $G_{\tau \times \hat{\tau }}$. It is also shown that the $\tau \times \hat{\tau }$-continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of $L^2(G_{\tau })$ as a window function. Finally, we illustrate application of these methods in the case of some well-known semi-direct product groups.