209 research outputs found
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
A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian
We consider compact Grassmann manifolds over the real, complex or
quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of
type . From an explicit integral representation of these polynomials we
deduce a sharp Mehler-Heine formula, that is an approximation of the
Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate
on the error term. This result is used to derive a central limit theorem for
random walks on the semi-lattice parametrizing the dual of , which are
constructed by successive decompositions of tensor powers of spherical
representations of . The limit is the distribution of a Laguerre ensemble in
random matrix theory. Most results of this paper are established for a larger
continuous set of multiplicity parameters beyond the group cases
A multivariate version of the disk convolution
We present an explicit product formula for the spherical functions of the
compact Gelfand pairs with ,
which can be considered as the elementary spherical functions of
one-dimensional -type for the Hermitian symmetric spaces with . Due to results of Heckman, they can be expressed in terms
of Heckman-Opdam Jacobi polynomials of type with specific half-integer
multiplicities. By analytic continuation with respect to the multiplicity
parameters we obtain positive product formulas for the extensions of these
spherical functions as well as associated compact and commutative hypergroup
structures parametrized by real . We also obtain explicit
product formulas for the involved continuous two-parameter family of
Heckman-Opdam Jacobi polynomials with regular, but not necessarily positive
multiplicities. The results of this paper extend well known results for the
disk convolutions for to higher rank
Limit theorems for radial random walks on pxq-matrices as p tends to infinity
The radial probability measures on are in a one-to-one correspondence
with probability measures on by taking images of measures w.r.t.
the Euclidean norm mapping. For fixed and each
dimension p, we consider i.i.d. -valued random variables
with radial laws corresponding to as above. We derive weak and strong
laws of large numbers as well as a large deviation principle for the Euclidean
length processes as k,p\to\infty in suitable ways.
In fact, we derive these results in a higher rank setting, where is
replaced by the space of matrices and by the cone
of positive semidefinite matrices. Proofs are based on the fact that
the form Markov chains on the cone whose transition
probabilities are given in terms Bessel functions of matrix argument
with an index depending on p. The limit theorems follow from new
asymptotic results for the as . Similar results are also
proven for certain Dunkl-type Bessel functions.Comment: 24 page
Biorthogonal polynomials associated with reflection groups and a formula of Macdonald
Dunkl operators are differential-difference operators on \b 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 polynomials with respect
to Dunkl's Gaussian distributions in a quite canonical way. These systems,
called Appell systems, admit many properties known from classical Hermite
polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian
distributions. In particular, these polynomials lead to a new proof of a
generalized formula of Macdonald due to Dunkl.
The ideas for this paper are taken from recent works on non-Gaussian white
noise analysis and from the umbral calculus.Comment: 14 pages, Latex2
A Limit Relation for Dunkl-Bessel Functions of Type A and B
We prove a limit relation for the Dunkl-Bessel function of type with
multiplicity parameters on the roots and on where tends to infinity and the arguments are suitably scaled. It
gives a good approximation in terms of the Dunkl-type Bessel function of type
with multiplicity . For certain values of an improved
estimate is obtained from a corresponding limit relation for Bessel functions
on matrix cones.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Integral representation and sharp asymptotic results for some Heckman-Opdam hypergeometric functions of type BC
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi
functions in one variable and include the spherical functions of non-compact
Grassmann manifolds over the real, complex or quaternionic numbers. There are
various limit transitions known for such hypergeometric functions. In the
present paper, we use an explicit form of the Harish-Chandra integral
representation as well as an interpolated variant, in order to obtain limit
results for three continuous classes of hypergeometric functions of type BC
which are distinguished by explicit, sharp and uniform error bounds. The first
limit realizes the approximation of the spherical functions of infinite
dimensional Grassmannians of fixed rank; here hypergeometric functions of type
A appear as limits. The second limit is a contraction limit towards Bessel
functions of Dunkl type
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