A new approach to the Fourier analysis on semi-direct products of groups

Abstract

Let $H$ and $K$ be locally compact groups and also $\tau :H\to Aut(K)$ 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 $\tau$. This paper presents a novel approach to the Fourier analysis of $G_{\tau }$, when $K$ is abelian. We define the $\tau$-dual group $G_{\hat{\tau }}$ of $G_{\tau }$ as the semi-direct product $H{⋉}_{\hat{\tau }}\hat{K}$, where $\hat{\tau }:H\to Aut(\hat{K})$ defined via (\ref{A}). We prove a Ponterjagin duality Theorem and also we study $\tau$-Fourier transforms on $G_{\tau }$. As a concrete application we show that how these techniques apply for the affine group and also we compute the $\tau$-dual group of Euclidean groups and the Weyl-Heisenberg groups.

arXiv commentThis paper has been withdrawn by the author due to a crucial sign error in equations and references