Dunkl operators are differential-difference operators on R which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we introduce two systems of biorthogonal polynomials with respect to Dunkl’s Gaussian distributions in a quite canonical way. These systems, called Appell systems, ad...

Abstract. We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times t of a stochastic differential equation with a singular drift (see [11] for the original proof and [4] for a proof under an additional restriction), the first hitting time of the Weyl ...

In this paper, we introduce a new differential-difference operator Tξ (ξ ∈ R ) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this o...

In this paper, we construct a linear positive operators q-parametric Szász Mirakjan operators generated by the q-Dunkl generalization of the exponential function. We obtain Korovkin’s type approximation theorem for these operators and compute convergence of these operators by using the modulus of continuity. Furthermore, the rate of convergence of the operators for functions belonging to the Li...

This paper consists in a first study of the Hardy space H in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H. These results are proved here in the one-dimensional case and in the product case.

For a locally compact Hausdorff semigroup S, the L representation algebra R(S) was extensively studied by Dunkl and Ramirez. The FourierStieltjes algebra F (S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.

In this paper we prove inversion formulas for the Dunkl intertwining operator Vk and for its dual tVk and we deduce the expression of the representing distributions of the inverse operators V −1 k and V −1 k , and we give some applications.

This paper is intented to establish the analogue of Titchmarsh?s theorem for Dunkl generalized transform on real line.

This paper is concerned with energy properties of the wave equation associated to the Dunkl-Cherednik Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case. Mathematics Subject Index 2000: Primary 35L05 ; Secondary 22E30, 33C67, 35L65, 58J45