The existence of Zak transform in locally compact hypergroups

Subject Areas : Statistics

Seyyed Mohammad Tabatabaie ¹ , Soheila Jokar ²

1 - Department of Mathematics‎, ‎University of Qom‎, ‎Qom‎, ‎Iran.
2 - Department of Mathematics‎, ‎University of Qom‎, ‎Qom‎, ‎Iran

Received: 2018-05-06 Accepted : 2019-07-13 Published : 2020-04-01

Keywords: نگاشت بخش بورل, دامنه اساسی, دوگان ابرگروه, تبدیل فوریه,

Abstract :

Let K be a locally compact hypergroup. In this paper we initiate the concept of fundamental domain in locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fundamental domain. Finally, as an application we show that if K is a locally compact hypergroup and H is one of its commutative subhypergroup, then there exists an isometric transform Z from L^2 (K) to L^2 (H ̂,L^2 (H\K)). For this, we apply the dual of hypergroups and specially we use the Plancherel Theorem. This transform is a version of the Zak transform on locally compact hypergroups which can be considered as an extension of the usual notion of Zak transform in the case of locally compact groups.

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