A Unified Group Theoretical Method for the Partial Fourier Analysis on Semi-Direct Product of Locally Compact Groups

Abstract

Let H and K be locally compact groups and ${\tau : H \to Aut(K)}$ be a continuous homomorphism. Further let ${G_\tau = H \ltimes_\tau K}$ be the semi-direct product of H and K with respect to the continuous homomorphism ${\tau}$. This paper presents a unified approach for the partial Fourier analysis on ${G_\tau = H \ltimes_\tau K}$, when K is Abelian. The ${\tau}$-dual group (partial dual group) ${G_{\widehat{\tau}}}$ of ${G_\tau}$ is defined as the semi-direct product group ${H \ltimes_{\widehat{\tau}}\widehat{K}}$, where ${\widehat{\tau}: H \to Aut(\widehat{K})}$ is given via ${\widehat{\tau}_h(\omega) : = \omega \circ \tau_{h^{-1}}}$ for all ${h \in H}$ and ${\omega \in \widehat{K}}$. We will prove a Pontrjagin duality theorem and we introduce a unitary partial Fourier transform on ${G_\tau}$. As examples, we shall study these techniques for some well-known semi-direct product groups.