Moment Functions on Affine Groups

Abstract

Moments of probability measures on a hypergroup can be obtained from so-called (generalized) moment functions of a given order. The aim of this paper is to characterize generalized moment functions on a non-commutative affine group. We consider a locally compact group G and its compact subgroup K. First we recall the notion of the double coset space G//K of a locally compact group G and introduce a hypergroup structure on it. We present the connection between K-spherical functions on G and exponentials on the double coset hypergroup G//K. The definition of the generalized moment functions and their connection to the spherical functions is discussed. We study an important class of double coset hypergroups: specyfing K as a compact subgroup of the group of inverible linear transformations on a finitely dimensional linear space V we consider the affine group Aff K. Using the fact that in the finitely dimensional case (Aff K, K) is a Gelfand pair we give a description of exponentials on the double coset hypergroup Aff K//K in terms of Kspherical functions. Moreover, we give a general description of generalized moment functions on Aff K and specific examples for K = SO(n), and on the so-called ax + b-group.Mathematics Subject Classification. 20N20, 43A62, 39B99.

Moment Functions on Groups

“…More on moment functions can be found e.g. in [2,[4][5][6][10][11][12] and references therein. As already mentioned groups are special hypergroups and in this paper we are going to focus on a generalized moment problem on Abelian groups.…”

Moment functions and exponential monomials on commutative hypergroups

“…We say that the functions (ϕ k ) k∈{0,1,...,N } form a generalized moment function sequence of order N. For the sake of simplicity, in this paper we shall omit the adjective "generalized" and we refer to moment functions and moment function sequences. We note that in [2], a more general concept of moment function sequences was introduced.…”

Moment functions on groups

“…More on moment functions can be found e.g. in [2,4,5,6,10,11,12] and references therein. As already mentioned groups are special hypergroups and in this paper we are going to focus on a generalized moment problem on Abelian groups.…”

“…In addition, this paper of J. Aczél has provided motivation for the present research. Namely, it is shown there that if pG, `q is a grupoid and R is a commutative ring, then functions ϕ n : G Ñ R satisfying (1) for each n in N are of the form (2) ϕ n ptq " n!…”