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On symmetry of discrete polynomial hypergroups

Authors: Marc-Olivier Gebuhrer and Ryszard Szwarc
Journal: Proc. Amer. Math. Soc. 127 (1999), 1705-1709
MSC (1991): Primary 43A62, 42C05
Published electronically: February 11, 1999
MathSciNet review: 1476129
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Abstract: Let $H$ be a discrete polynomial hypergoup on $\mathbb{N}$ with Plancherel measure $\mu.$ If the hypergroup $H$ is symmetric, the set of characters $\widehat{H}$ can be identified with a compact subset of the real line which contains the support of $\mu. $ We show that the lower and upper bounds of $\widehat{H}$ and $\supp \mu$ coincide. In particular, the trivial character belongs to the support of the Plancherel measure.

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Marc-Olivier Gebuhrer
Affiliation: Institut de Recherche Mathématique Avancée Université Louis Pasteur et C.N.R.S. 7, rue René Descartes 67084 Strasbourg Cedex France

Ryszard Szwarc
Affiliation: Institute of Mathematics Wrocław University pl. Grunwaldzki 2/4 50–384 Wroc- ław, Poland

Keywords: Hypergroup, symmetry, orthogonal polynomials
Received by editor(s): January 21, 1997
Received by editor(s) in revised form: September 9, 1997
Published electronically: February 11, 1999
Additional Notes: This work has been partially supported by KBN (Poland) under grant 2 P03A 030 09. The work has been done while the second author was visiting IRMA, University Louis Pasteur in February 1994.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society