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.