Generalized Hypergroups and Orthogonal Polynomials
نویسندگان
چکیده
We study in this paper a generalization of the notion of a discrete hypergroup with particular emphasis on the relation with systems of orthogonal polynomials. The concept of a locally compact hypergroup was introduced by Dunkl [8], Jewett [12] and Spector [25]. It generalizes convolution algebras of measures associated to groups as well as linearization formulae of classical families of orthogonal polynomials, and many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commutative hypergroups; see Heyer [11], Litvinov [17], Ross [22], and references cited therein. Orthogonal polynomials have been studied in terms of hypergroups by Lasser [15] and Voit [31], see also the works of Connett and Schwartz [6] and Schwartz [23] where a similar spirit is observed. The special case of a discrete hypergroup, particularly in the commutative case, goes back earlier. In fact the ground-breaking paper of Frobenius
منابع مشابه
Generalized hypergroups and orthogonal polynomials
The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed which are more special then generalized hypergroups introduced by Obata and Wildberger and which are more general then discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In case of commutativity we define the dual objects and prove some basic theorems o...
متن کاملDiscrete commutative hypergroups
The concept of a locally compact hypergroup was introduced by Dunkl [6], Jewett [14] and Spector [26]. Hypergroups generalize convolution algebras of measures associated to groups as well as linearization formulae of classical families of special functions, e.g. orthogonal polynomials. Many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commuta...
متن کاملAlmost periodic sequences with respect to orthogonal polynomials
Let (Rn(x))n∈N0 be an orthogonal sequence inducing a polynomial hypergroup on N0. The basic facts on polynomial hypergroups and their characters can be found in the monograph [1] or in the papers [6, 7]. A recent review is [8]. The Banach space of almost periodic functions on hypergroups is introduced and studied by the author in [5]. Weakly almost periodic functions are the topic of [12]. It i...
متن کاملProduct Formulas and Associated Hypergroups for Orthogonal Polynomials on the Simplex and on a Parabolic Biangle
Explicit product formulas are obtained for families of multivariate polynomials which are orthogonal on simplices and on a parabolic biangle in R. These product formulas are shown to give rise to measure algebras which are hypergroups. The article also includes an elementary proof that the Michael topology for the space of compact subsets of a topological space (which is used in the definition ...
متن کاملHypergroups on the Set of All Integers
Stating a two-parameter class of examples, a — positive — answer is given to the question “Are there any nontrivial hypergroups on Z?” . 1. Background and Introduction In 1989 Zeuner [14] proved that no nontrivial hypergroups (i. e. not the group) exist on the set of real numbers R. Five years later Rösler [5] showed that by weakening the axioms by considering signed hypergroups some of these s...
متن کامل