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

摘要

Let H and K be locally compact groups and tau : H -> Aut(K) be a continuous homomorphism. Further let G(tau) = H proportional to(tau) K be the semi-direct product of H and K with respect to the continuous homomorphism . This paper presents a unified approach for the partial Fourier analysis on G(tau) - H proportional to(tau) K, when K is Abelian. The tau-dual group (partial dual group) G((tau) over cap) of G(tau) is defined as the semi-direct product group H ((tau) over cap)w (K) over cap where (tau) over cap : H -> Aut((K) over cap) is given via (tau) over caph(omega) : = omega omicron tau(h)-1 for all h is an element of H and omega is an element of(K) over cap 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.