The Wiener property for a class of discrete hypergroups

1989 ◽  
Vol 202 (2) ◽  
pp. 271-274 ◽  
Author(s):  
Oliver Gebuhrer ◽  
Ajay Kumar
Keyword(s):  
Discrete Hypergroups

scholarly journals CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS

2018 ◽  
Vol 106 (03) ◽  
pp. 361-426
Author(s):  
MICHAEL VOIT
Keyword(s):  
Double Coset ◽  
Association Schemes ◽  
Stochastic Matrices ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Infinite Case ◽  
Compact Case ◽  
Commutative Hypergroup ◽  
Double Coset Hypergroup ◽  
Discrete Hypergroups

Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$ . We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.

scholarly journals Generalized hypergroups and orthogonal polynomials

2007 ◽  
Vol 82 (3) ◽  
pp. 369-394 ◽  
Author(s):  
Rupert Lasser ◽  
Josef Obermaier ◽  
Holger Rauhut
Keyword(s):  
Fourier Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
The Relationship ◽  
Discrete Hypergroups

AbstractThe concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

scholarly journals Weighted discrete hypergroups

2016 ◽  
Vol 65 (2) ◽  
pp. 423-451 ◽  
Author(s):  
Mahmood Alaghmandan ◽  
Ebrahim Samei
Keyword(s):  
Discrete Hypergroups

Compact Almost Discrete Hypergroups

1996 ◽  
Vol 48 (1) ◽  
pp. 210-224 ◽  
Author(s):  
Michael Voit
Keyword(s):  
Projective System ◽  
Successive Substitution ◽  
The One ◽  
One Point Compactification ◽  
Countably Infinite ◽  
Compact Hypergroup ◽  
Discrete Hypergroups ◽  
Totally Disconnected

AbstractA compact hypergroup is called almost discrete if it is homeomorphic to the one-point-compactification of a countably infinite discrete set. If the group Up of all p-adic units acts multiplicatively on the p-adic integers, then the associated compact orbit hypergroup has this property. In this paper we start with an exact projective sequence of finite hypergroups and use successive substitution to construct a new surjective projective system of finite hypergroups whose limit is almost discrete. We prove that all compact almost discrete hypergroups appear in this way—up to isomorphism and up to a technical restriction. We also determine the duals of these hypergroups, and we present some examples coming from partitions of compact totally disconnected groups.

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