Fourier transform on compact Hausdorff groups

Mykola Yaremenko

Filomat, Tome 37 (2023) no. 22, p. 7329

Résumé

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.

Détail

DOI : 10.2298/FIL2322329Y

Classification : 42A16, 35S30, 42A38, 42A16, 42A38
Keywords: Fourier transform, Wigner function, Compact group, Peter-Weyl theorem