The existence of Zak transform in locally compact hypergroups

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 (HK)). 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.