Applications of the γ*-Relation to Polygroups

The γ*-relation was introduced by Freni. In this article, we use the γ*-relation in a given polygroup. In this way, the γ*-relation is the smallest equivalence relation on a polygroup P such that P/γ* is an abelian group. Results are obtained on extension polygroups, derived hypergroups, γ-parts, and semi-direct hyperproduct.     

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,α_*), where K=[0,+∞[×R and α_* a convolution product on K coming from the product formula satisfied by the Laguerre functions (m∈N, α?0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μα_*k=μα_*?α_*μ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.     

Normed Convergence Property for Hypergroups Admitting An Invariant Measure

We prove that a hypergroup admitting a countable basis and an invariant Haar measure has normed convergence property if and only if it is compact.AMS Subject Classification (2001): 43A62, 60B99     

Isomorphism on Hypergroups

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Stochastic processes indexed by hypergroups II

Further results on weakly stationary processes indexed by hypergroups are presented. The concept of translation operators is developed; processes on orbit spaces and double coset spaces are constructed. It is shown that every weakly stationary process indexed by a hypergroupK with centerC contains a maximalK//C-weakly stationary component. New examples forK-weakly stationary processes are continuous estimates of the mean of a weakly stationary process, isotropic random fields, andK-oscillations.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.     

A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.