Moment functions on groups

Abstract

The main purpose of this work is to prove characterization theorems for generalized moment functions on groups. According one of the main results these are exponential polynomials that can be described with the aid of complete (exponential) Bell polynomials. These characterizations will be immediate consequences of our main result about the characterization of generalized moment functions of higher rank.

Moment functions and exponential monomials on commutative hypergroups

“…The above notions suggest that generalized moment functions may play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative hypergroups. In our former paper [2], we described generalized moment functions on commutative groups using Bell polynomials, even in the higher rank case. In fact, the notion of exponential monomials is not easy to handle, compared to that of generalized moment functions: the functional equations characterizing generalized moment functions are more convenient than those for exponential monomials.…”

“…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.…”

Spectral synthesis via moment functions on hypergroups

“…Let X be a commutative hypergroup, r a positive integer, and for each multi-index α in N r , let f α : X Ñ C be continuous function, such that f α ‰ 0 for |α| " 0. We say that pf α q αPN r is a generalized moment sequence of rank r, if (1.1) f α px ˚yq " ÿ βďα ˆα β ˙fβ pxqf α´β pyq holds whenever x, y are in X (see [4]). It is obvious that the variety of f α is finite dimensional: in fact, every translate of f α is a linear combination of the finitely many functions f β with β ď α, by equation (1.1).…”

“…The main result in [4] is that moment sequences of rank r can be described by using Bell polynomials if the underlying hypergroup is an Abelian group. The point is that in this case the situation concerning (1.1) can be reduced to the case where the generating exponential is the identically 1 function and the problem reduces to a problem about polynomials of additive functions.…”