3,456 research outputs found
Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
Based on the theory of Dunkl operators, this paper presents a general concept
of multivariable Hermite polynomials and Hermite functions which are associated
with finite reflection groups on \b R^N. The definition and properties of
these generalized Hermite systems extend naturally those of their classical
counterparts; partial derivatives and the usual exponential kernel are here
replaced by Dunkl operators and the generalized exponential kernel K of the
Dunkl transform. In case of the symmetric group , our setting includes the
polynomial eigenfunctions of certain Calogero-Sutherland type operators. The
second part of this paper is devoted to the heat equation associated with
Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem
is governed by a positive one-parameter semigroup; this is assured by a maximum
principle for the generalized Laplacian. The explicit solution to the Cauchy
problem involves again the kernel K, which is, on the way, proven to be
nonnegative for real arguments.Comment: 24 pages, AMS-LaTe
Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC
In this paper, we derive explicit product formulas and positive convolution
structures for three continuous classes of Heckman-Opdam hypergeometric
functions of type . For specific discrete series of multiplicities these
hypergeometric functions occur as the spherical functions of non-compact
Grassmann manifolds over one of the (skew) fields We write the product formula of these spherical
functions in an explicit form which allows analytic continuation with respect
to the parameters. In each of the three cases, we obtain a series of hypergroup
algebras which include the commutative convolution algebras of -biinvariant
functions on
A positive radial product formula for the Dunkl kernel
It is an open conjecture that generalized Bessel functions associated with
root systems have a positive product formula for non-negative multiplicity
parameters of the associated Dunkl operators. In this paper, a partial result
towards this conjecture is proven, namely a positive radial product formula for
the non-symmetric counterpart of the generalized Bessel function, the Dunkl
kernel. Radial hereby means that one of the factors in the product formula is
replaced by its mean over a sphere. The key to this product formula is a
positivity result for the Dunkl-type spherical mean operator. It can also be
interpreted in the sense that the Dunkl-type generalized translation of radial
functions is positivity-preserving. As an application, we construct Dunkl-type
homogeneous Markov processes associated with radial probability distributions.Comment: 25 page
Olshanski spherical functions for infinite dimensional motion groups of fixed rank
Consider the Gelfand pairs associated
with motion groups over the fields
with and fixed as well as the inductive limit ,the
Olshanski spherical pair . We classify all Olshanski
spherical functions of as functions on the cone
of positive semidefinite -matrices and show that they appear as
(locally) uniform limits of spherical functions of as .
The latter are given by Bessel functions on . Moreover, we determine all
positive definite Olshanski spherical functions and discuss related positive
integral representations for matrix Bessel functions. We also extend the
results to the pairs which
are related to the Cartan motion groups of non-compact Grassmannians. Here
Dunkl-Bessel functions of type B (for finite ) and of type A (for
) appear as spherical functions
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