This article deals with the generalization of the abstract Fourier analysis on the compact Hausdorff group. In this paper, the generalized Fourier transform F is defined as F ψ (α) = R ψ(h)Mα h−1 dμ (h) for all ψ ∈ L2 (G) T L1 (G), where Mα is a continuous unitary representation Mα : G → UC Cn(α) of the group G in Cn(α), and its properties are studied. Also, we define the symplectic Fourier transform and the generalizedWigner function WA ψ, φ and establish the Moyal equality for theWigner function. We show that the homomorphism π : G → U L2 (G/K,H1) induced by Λ : G × (G/K) → U(H1) by π ψ 1, h = Λ h−1, 1 −1 ψ h−11 , 1 ∈ G/K, h ∈ G, ψ ∈ L2 (G/K,H1) is a unitary representation of the group G, assuming the mapping h 7→ π ψ 1, h is continuous as morphism G → U L2 (G/K,H1) . We study the unitary representation ˜π : G → H induced by the unitary representation V : K → U(H1) given by ˜π1 ψ (t) = ψ 1−1t for all t ∈ G/K.