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

Abstract

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