Hypergroups of compact type
type: Item
instance of: scholarly article
title: Hypergroups of compact type
zbMATH Open document ID: 1077.43005
author: Rupert Lasser
author: Ryszard Szwarc
author: Frank Filbir
published in: Journal of Computational and Applied Mathematics
publication date: 2005-04-18T00:00:00Z
review text: The aim of this paper is to study commutative hypergroups \(X\) for which the translation operator \(T_{x}-I\) is compact on the space \(L^{2}(X,m)\) (resp. \(L^{1}(X,m)\)) for each \(x\in X\), where \(m\) is the Haar measure on \(X\). These hypergroups are called hypergroups of compact type (resp. strong compact type). The main results are as follows: Hypergroups of compact type are necessarily discrete. Hypergroups of strong compact type have the property that the support of the Plancherel measure coincides with the set of all bounded characters; this kind of hypergroups are necessarily of compact type and symmetric, i.e. every bounded character is self-adjoint. It is also shown that for the class of hypergroups of strong compact type the ``modified '' Reiter condition \(\mathcal{P}_{1}\), introduced in [\textit{F. Filbir, R. Lasser, R. Szwarc}, Monatsh. Math. 143, 189-203 (2004; Zbl 1059.43005)] is satisfied for each nontrivial character. Finally, a full description of polynomial hypergroups of compact type (or equivalently in this particular case, of strong compact type) is presented and examples, such as the polynomial hypergroups generated by the little \(q\)-Legendre polynomials, are given.
reviewed by: Liliana Pavel
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2157669
zbMATH Keywords: hypergroups of compact type
zbMATH Keywords: strong compact type
zbMATH Keywords: polynomial hypergroups
MaRDI profile type: Publication
full work available at URL: j.cam.2004.03.028
OpenAlex ID: W2140419747
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DOI: 10.1016/J.CAM.2004.03.028
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description: scientific article; zbMATH DE number 2157669