Generalized hypergroups and orthogonal polynomials

Abstract

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.2000 Mathematics subject classification: primary 43A62,43A99,46J10, 05E35, 33C80.

Moment functions on polynomial hypergroups

“…An extensively studied class of Hermitian hypergroups is represented by the polynomial hypergroups in one variable (see [6]). …”

Point derivations on the L1-algebra of polynomial hypergroups

“…To have a good reference and for the sake of completeness we recall briefly the basic facts on polynomial hypergroups. For more details and proofs we refer to [16] and [17].…”

“…We shall suppose throughout this paper that the coefficients g(m, n; k) are non-negative. There are many orthogonal polynomial systems which have this property (see [5,16,17]). We can define convolution multiplication on N 0 by the formula (3) δ m * δ n = n+m k=|n−m| g(m, n; k)δ k , where δ k is the point measure at k ∈ N 0 .…”

“…With this convolution, the involutionñ = n and the discrete topology the set of natural numbers N 0 is a commutative hypergroup. Such a hypergroup is called the polynomial hypergroup induced by (R n ) n∈N 0 (see [16]). The Haar measure on the polynomial hypergroup N 0 is the counting measure with weights h(n) = g(n, n; 0) −1 of the points n ∈ N 0 .…”

“…The Hermitean dual spaceN 0 of N 0 (i.e. the Hermitean structure space of l 1 (h)) can be identified with the set (4) {x ∈ R : |R n (x)| ≤ 1 for all n ∈ N 0 } via the mapping x → α x , α x (n) := R n (x) (see [16]). Hence we consider N 0 as a compact subset of R which contains 1 ∈ R (since R n (1) = 1).…”