On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups

Abstract

Let H be a locally compact group and K be a locally compact abelian group. Also let G = H × τ K denote the semidirect product group of H and K, respectively. Then the unitary representation (U,is called the quasi regular representation. The properties of this representation in the case K = (R n , +), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) δ H ≡ 1, and (ii) the stabilizers H ω are compact for a.e. ω ∈ K; both are necessary for square integrability of U . Furthermore, we shall consider some sufficient conditions for the square integrability of U . Also, for the square integrability of subrepresentations of U , we will introduce a concrete form of the Duflo-Moore operator.

Generalized quasi-regular representation and its applications for shearlet transforms

“…[8], Subsection 2.6). The class of locally compact semidirect product groups, as a large class of non-Abelian groups, has significant roles in theories connecting mathematical physics and mathematical theory of coherent states analysis [2,5,14]. To introduce 3-fold semidirect product block shearlet group, we need to review some of the standard facts on semidirect product groups and summarize some results on them.…”

Abstract Relative Fourier Transforms Over Canonical Homogeneous Spaces of Semi-Direct Product Groups With Abelian Normal Factor

“…The abstract notion of relative-convolution operators over homogeneous spaces of locally compact groups introduced in [16] and studied comprehensively in [10,18]. The class of locally compact semi-direct product groups as a large class of non-Abelian groups, has significant roles in theories connecting mathematical physics, mathematical theory of coherent states analysis [1,8,9,12,21] and covariant transforms, see [2,3,4,5,18,19,20] and standard references therein.…”

Finite equal norm Parseval wavelet frames over prime fields

“…Traditionally, the classical Gabor transforms and wavelet transforms have been used to perform time-frequency (resp. time-scale) analysis of a given function/signal in a Hilbert space, see [1][2][3]. In the last decades, generalized methods of Gabor transforms and wavelet transforms have been developed [14].…”