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

Let $H$ and $K$ be locally compact groups and also $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ 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 $H\ltimes_{\hat{\tau}}\hat{K}$, 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.