Subsets of \(\mathbb{R}\) which support hypergroups with polynomial characters
type: Item
instance of: scholarly article
title: Subsets of \(\mathbb{R}\) which support hypergroups with polynomial characters
zbMATH Open document ID: 0857.43007
DOI: 10.1016/0377-0427(95)00101-8
author: William C. Connett
author: Alan L. Schwartz
published in: Journal of Computational and Applied Mathematics
publication date: 1997-03-03T00:00:00Z
review text: This interesting and well-written paper deals with so-called hypergroups (in the sense of Dunkl, Jewett, and Spector). Let \(H\) be a locally compact Hausdorff space, let \(M(H)\) denote the space of all bounded regular Borel measures on \(H\) with the total variation norm. Roughly speaking \(H\) is a hypergroup if \(M(H)\) is a Banach algebra with multiplication * (called convolution) and involution, and some natural axioms are satisfied. In particular, a convolution of probability measures is a probability measure and the mapping \((\mu,\nu) \mapsto \mu*\nu\) is positive-continuous from \(M(H) \times M(H)\) into \(M(H)\). These ideas and methods of the theory of group representations can be extended to the case of hypergroups while it is convenient to treat representations and characters of \(H\) as representations and characters of the corresponding hypergroup algebra \(M(H)\). Let \(\widehat H\) denote the set of characters of \(H\). Suppose that \(H\) is an infinite subset of \(\mathbb{R}\) and that \(\widehat H\) contains a family of polynomials \({\mathcal P} = \{p_n\}\), where \(n=0,1,2, \dots\) and \(p_n\) has exact degree \(n\). In this case the authors show that if either \(H\) is compact or the polynomial characters are orthogonal with respect to a measure supported on \(H\), then those polynomials are essentially the Jacobi polynomials (up to a linear change of variables). Related results are obtained for polynomial product formulas. The following duality theorem is also proved: The only discrete polynomial hypergroups on \(\{0,1,2, \dots\}\) which have dual hypergroups in any sense of this word are generated by Jacobi polynomials. -- Two examples of families of polynomials which have product formulas not of hypergroup type are also described. The results presented in this paper are essential for the authors' results in the more complicated two-variable case (as well as in higher dimensions). This is part of the authors' well-known programme to identify…
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 903563
zbMATH Keywords: hypergroup
zbMATH Keywords: Hausdorff space
zbMATH Keywords: Banach algebra
zbMATH Keywords: probability measure
zbMATH Keywords: representations
zbMATH Keywords: characters
zbMATH Keywords: Jacobi polynomials
reviewed by: Grigori L. Litvinov
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00101-8
OpenAlex ID: W1964907177
cites work: Q3314162
cites work: Q4714458
cites work: Q4840459
cites work: Product formulas, hypergroups, and the Jacobi polynomials
cites work: Q4833865
cites work: The Measure Algebra of a Locally Compact Hypergroup
cites work: A CHARACTERIZATION THEOREM FOR POSITIVE DEFINITE SEQUENCES ON THE KRAWTCHOUK POLYNOMIALS1
cites work: Linearization of the Product of Jacobi Polynomials. II
cites work: Banach algebras for Jacobi series and positivity of a kernel
cites work: Q4040884
cites work: Spaces with an abstract convolution of measures
cites work: The Product Formula and Convolution Structure for the Generalized Chebyshev Polynomials
cites work: Topologies on Spaces of Subsets
cites work: Mesures Invariantes Sur Les Hypergroupes
cites work: Orthogonal Polynomials and a Discrete Boundary Value Problem I
cites work: Orthogonal Polynomials and a Discrete Boundary Value Problem II
cites work: Q4840488
description: scientific article; zbMATH DE number 903563