Subsets of \(\mathbb{R}\) which support hypergroups with polynomial characters
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 and classify those families of orthogonal polynomials in one or several variables which satisfy a certain type of product formula.
scientific article; zbMATH DE number 903563
scientific article