Fourier transform on compact Hausdorff groups

Abstract

This article deals with the generalization of the abstract Fourier analysis on the compact Hausdorff group. In this paper, the generalized Fourier transform \textbf{\textit{$F$}} is defined as \textit{$F\left(\psi \right)\left(\alpha \right)=\int \psi \left(h\right)M_{\alpha } \left(h^{-1} \right) \; d\mu \left(h\right)$}\textbf{\textit{ }} for all $\psi \in L^{2} \left(G\right)\bigcap L^{1} \left(G\right)$, where $M_{\alpha } $ is a continuous unitary representation $M_{\alpha } \; :\; G\to UC\left(C^{n\left(\alpha \right)} \right)$ of the group $G$ in $C^{n\left(\alpha \right)} $, and its properties are studied. Also, we define the symplectic Fourier transform and the generalized Wigner function $W_{A} \left(\psi ,\, \varphi \right)$ and establish the Moyal equality for the Wigner function.

We show that the homomorphism $\pi \; :\; G\to U\left(L^{2} \left(G/K,H_{1} \right)\right)$ induced by $\Lambda \; :\; G\times \left(G/K\right)\to U\left(H_{1} \right)$ by $\left(\pi \left(\psi \right)\right)\left(g,h\right)=\left(\Lambda \left(h^{-1} ,g\right)\right)^{-1} \left(\psi \left(h^{-1} g\right)\right)$, $g\in G/K,$ $h\in G$, $\psi \in L^{2} \left(G/K,H_{1} \right)$ is a unitary representation of the group $G$, assuming the mapping $h\mapsto \left(\pi \left(\psi \right)\right)\left(g,h\right)$ is continuous as morphism $G\to U\left(L^{2} \left(G/K,H_{1} \right)\right)$.

We study the unitary representation $\tilde{\pi }\; :\; G\to H$ induced by the unitary representation $V\; :\; K\to U\left(H_{1} \right)$ given by $\tilde{\pi }_{g} \left(\psi \right)\left(t\right)=\psi \left(g^{-1} t\right)$ for all $t\in G/K$.