Let Pr n Com(G) denotes the probability that a randomly ordered ntuples of elements in a finite group G be a mutually commuting n-tuples. We aim to generalize the above concept to a compact topological group which generally not only finite but also even uncountable. The results are mostly new or improvements of known results in finite case given in [1], [3] and [5].
This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L2(G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Riesz bases generated by shifts of a countable set of generators from L2(ℝn) to L2(G).
We investigate shift invariant subspaces of L 2 (G), where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.
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.
Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space [Formula: see text]. The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.