Sine functions on hypergroups

Abstract

In a recent paper we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm-Liouville hypergroups, etc. A non-commutative hypergroup is also considered.

Moment Functions on Affine Groups

“…Finally, substitution into (9) and (8) gives (5). Conversely, if m has the desired form then (5) implies that m is exponential. Functions ϕ with this property are called K-radial functions on V .…”

“…Clearly, K is a compact subgroup of G. Finally, G is topologically isomorphic to the affine group Aff (K) = K C. For more about this group see e.g. [9, p. 201] or [5]. The normalized Haar measure on K is given by for each continuous function ϕ : K → C. It is easy to check that K is not a normal subgroup, hence the hypergroup structure on the double coset space G//K is not induced by a group operation.…”

“…Obviously, m ≡ 1 is an exponential on any hypergroup, and 1-sine functions are exactly the additive functions. The description of sine functions on some types of hypergroups can be found in [5]. Let G be a locally compact group with identity e and K a compact subgroup with the normalized Haar measure ω.…”

“…In the case g 0 ≡ 1 the first order moment functions are exactly the additive functions, which, for general K, can be called K-additive functions. In the generalized moment function sequence (g n ) 1≤n≤N with N ≥ 1 the function g 1 is a g 0 -sine function, according to the terminology introduced in [7] (see also [5]), which, in the general case can be called K-sine functions associated with the K-spherical function g 0 . Generalized moment functions and generalized moment function sequences have been described on different types of commutative hypergroups (see e.g.…”

Exponential monomials on hypergroup joins

“…For more about spectral analysis and spectral synthesis see the monograph [12]. Characterization of these functions classes and related functional equations on different types of hypergroups have been studied in several papers(see e.g [2,8]). In this paper we describe generalized exponential classes on hypergroup joins.…”