Volume 1, Issue 3

In this paper we prove that there exists a unique topological isomorphism V k from (the space of C ∞ -functions on ) onto itself which intertwines the Cherednik operators T j , j = 1, 2, . . . , d , and the partial derivatives , j = 1, 2, . . . , d , called the trigonometric Dunkl intertwining operator (this name has been proposed by G. J. Heckman). To define and study the operator V k we have introduced first the trigonometric Dunkl dual intertwining operator t V k . The operators V k and t V k are the analogue in the Dunkl theory of the Dunkl intertwining operator and its dual (see [Dunkl, Can. J. Math. 43: 1213–1227, 1991, Trimèche, Integrals Transforms Special Funct. 12: 349–374, 2001]).

We define and study a [0, 1] m -valued process depending on three positive real parameters p, q, β that specializes for β = 1, 2 to the eigenvalues process of the real and complex matrix Jacobi processes on the one hand and that has the distribution of the β -Jacobi ensemble as stationary distribution on the other hand. We first prove that this process, called β -Jacobi process, is the unique strong solution of the stochastic differential equation defining it provided that β > 0, p ∧ q > m – 1 + 1/ β . When specialized to β = 1, 2, our results actually improve well known results on eigenvalues of matrix Jacobi processes. While proving the strong uniqueness, the generator of the β -Jacobi process is mapped into the radial part of the Dunkl–Cherednik Laplacian associated with the non reduced root system of type BC . The transformed process is then valued in the principal Weyl alcove and this allows to define the Brownian motion in the Weyl alcove corresponding to all multiplicities equal one. Second, we determine, using stochastic calculus and a comparison theorem, the range of β, p, q for which the m components of the β -Jacobi process first collide, the smallest one reaches 0 and the largest one reaches 1. This is equivalent to the first hitting time of the boundary of the principal Weyl alcove by the transformed process. Finally, we write down its semi group density.

In this paper, we analyze the class of the symmetric companion of a Laguerre–Hahn linear functional. We conclude with some illustrative examples.

In this paper we introduce and study two new sequences of positive linear operators acting on the space of all Lebesgue integrable functions defined, respectively, on the N -dimensional hypercube and on the N -dimensional simplex ( N ≥ 1). These operators represent a natural generalization to the multidimensional setting of the ones introduced in [Altomare and Leonessa, Mediterr. J. Math. 3: 363–382, 2006] and, in a particular case, they turn into the multidimensional Kantorovich operators on these frameworks. We study the approximation properties of such operators with respect both to the sup-norm and to the L p -norm and we give some estimates of their rate of convergence by means of certain moduli of smoothness.

We study the existence of solutions for a class of quasilinear elliptic equations involving the anisotropic -Laplace operator, on a bounded domain with smooth boundary. Since our differential operator involves partial derivatives with different variable exponents, we work on the anisotropic variable exponent Sobolev spaces. Using the Ekeland's variational principle and the mountain-pass theorem of Ambrosetti and Rabinowitz, we establish two existence results.