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These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable(More)
1. Introduction and results. In recent years, the theory of Dunkl operators has found a wide area of applications in mathematics and mathematical physics. Besides their use in the study of multivariable orthogonality structures associated with root systems (see, for example, [D1], [D2], [He], [vD], and [R]), these operators are closely related to certain(More)
We present a construction of a wavelet-type orthonormal basis for the space of radial L 2-functions in R 3 via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial(More)
In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras F = R, C or H which interpolate the convolution algebras of radial bounded Borel measures on a matrix space M p,q (F) with p q. Radiality in this context means invariance under the action of the unitary group U(More)
This paper studies the asymptotic behavior of the integral kernel of the Dunkl transform, the so-called Dunkl kernel, when one of its arguments is fixed and the other tends to infinity either within a Weyl chamber of the associated reflection group, or within a suitable complex domain. The obtained results are based on the asymptotic analysis of an(More)
In this paper, we derive explicit product formulas and positive con-volution structures for three continuous classes of Heckman-Opdam hy-pergeometric functions of type BC. For specific discrete series of multi-plicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the (skew) fields F = R,(More)
Dunkl operators are differential-difference operators on R N 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 poly-nomials with respect to Dunkl's Gaussian distributions in a quite canonical way.(More)
We study convolution algebras associated with Heckman–Opdam polynomials. For root systems of type BC we derive three continuous classes of positive convolution algebras (hypergroups) by interpolating the double coset convolution structures of compact Grassmannians U/K with fixed rank over the real, complex or quaternionic numbers. These convolution algebras(More)