arXiv subjects

Michael Voit

Publications and source records attributed to Michael Voit.

50 records · Page 3Linked to original sources

arXiv2013

Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$ are constructed via product formulas for the spherical functions of the symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing. Our paper is related to recent positive product formulas of Rösler for three series of Heckman-Opdam hypergeometric functions of type BC as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$.

math.CA↗

arXiv2012

Limit transition between hypergeometric functions of type BC and type A

Let $F_{BC}(λ,k;t)$ be the Heckman-Opdam hypergeometric function of type BC with multiplicities $k=(k_1,k_2,k_3)$ and weighted half sum $ρ(k)$ of positive roots. We prove that $F_{BC}(λ+ρ(k),k;t)$ converges for $k_1+k_2\to\infty$ and $k_1/k_2\to \infty$ to a function of type A for $t\in\b R^n$ and $λ\in\b C^n$. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H$ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite dimensional Grassmann manifold in the sense of Olshanski.

math.CA↗

arXiv2012

Olshanski spherical functions for infinite dimensional motion groups of fixed rank

Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski spherical functions of $(G_\infty,K_\infty)$ as functions on the cone $Π_q$ of positive semidefinite $q\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to\infty$. The latter are given by Bessel functions on $Π_q$. 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 $(M_{p,q} \rtimes (U_p\times U_q),(U_p\times U_q))$ which are related to the Cartan motion groups of non-compact Grassmannians. Here Dunkl-Bessel functions of type B (for finite $p$) and of type A (for $p\to\infty$) appear as spherical functions.

math.CA↗

arXiv2012

Uniform oscillatory behavior of spherical functions of $GL_n/U_n$ at the identity and a central limit theorem

Let $\mathbb F=\mathbb R$ or $\mathbb C$ and $n\in\b N$. Let $(S_k)_{k\ge0}$ be a time-homogeneous random walk on $GL_n(\b F)$ associated with an $U_n(\b F)$-biinvariant measure $ν\in M^1(GL_n(\b F))$. We derive a central limit theorem for the ordered singular spectrum $σ_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix. The main ingredient for the proof will be a oscillatory result for the spherical functions $ϕ_{iρ+λ}$ of $(GL_n(\b F),U_n(\b F))$. More precisely, we present a necessarily unique mapping $m_{\bf 1}:G\to\b R^n$ such that for some constant $C$ and all $g\in G$, $λ\in\b R^n$, $$|ϕ_{iρ+λ}(g)- e^{iλ\cdot m_{\bf 1}(g)}|\le C\|λ\|^2.$$

math.CA↗

arXiv2012

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Let $p,q$ positive integers. The groups $U_p(\b C)$ and $U_p(\b C)\times U_q(\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\b C)\times \b R$ canonically as groups of automorphisms where $M_{p,q}(\b C)$ is the vector space of all complex $p\times q$-matrices. The associated orbit spaces may be identified with $Π_q\times \b R$ and $Ξ_q\times \b R$ respectively with the cone $Π_q$ of positive semidefinite matrices and the Weyl chamber $Ξ_q={x\in\b R^q: x_1\ge...\ge x_q\ge 0}$. In this paper we compute the associated convolutions on $Π_q\times \b R$ and $Ξ_q\times \b R$ explicitly depending on $p$. Moreover, we extend these convolutions by analytic continuation to series of convolution structures for arbitrary parameters $p\ge 2q-1$. This leads for $q\ge 2$ to continuous series of noncommutative hypergroups on $Π_q\times \b R$ and commutative hypergroups on $Ξ_q\times \b R$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on $Π_q$ and $Ξ_q$. In particular, we give a non-positive product formula for these Laguerre functions on $Ξ_q$. The paper extends the known case $q=1$ due to Koornwinder, Trimeche, and others as well as the group case with integers $p$ due to Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, it is closely related to product formulas for multivariate Bessel and other hypergeometric functions of Rösler.

math.CA↗

arXiv2012

Central Limit Theorems for Radial Random Walks on $p\times q$ Matrices for $p\to\infty$

Let $ν\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in\b N$, let $(X_n^p)_{n\ge1}$ be i.i.d. $\b R^p$-valued radial random variables with radial distribution $ν$. We derive two central limit theorems for $ \|X_1^p+...+X_n^p\|_2$ for $n,p\to\infty$ with normal limits. The first CLT for $n>>p$ follows from known estimates of convergence in the CLT on $\b R^p$, while the second CLT for $n<<p$ will be a consequence of asymptotic properties of Bessel convolutions. Both limit theorems are considered also for $U(p)$-invariant random walks on the space of $p\times q$ matrices instead of $\b R^p$ for $p\to\infty$ and fixed dimension $q$.

math.PR↗

arXiv2012

Central limit theorems for hyperbolic spaces and Jacobi processes on $[0,\infty[$

We present a unified approach to a couple of central limit theorems for radial random walks on hyperbolic spaces and time-homogeneous Markov chains on the positive half line whose transition probabilities are defined in terms of the Jacobi convolutions. The proofs of all results are based on limit results for the associated Jacobi functions. In particular, we consider the cases where the first parameter (i.e., the dimension of the hyperbolic space) tends to infinity as well as the cases $ϕ_{iρ-λ}^{(α,β)}(t)$ for small $λ$, and $ϕ_{iρ-nλ}^{(α,β)}(t/n)$ for $n\to\infty$. The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the limit results for Jacobi functions and of the CLTs are known, other improve known ones, and other are completely new.

math.PR↗

arXiv2008

A Limit Relation for Dunkl-Bessel Functions of Type A and B

We prove a limit relation for the Dunkl-Bessel function of type $B_N$ with multiplicity parameters $k_1$ on the roots $\pm e_i$ and $k_2$ on $\pm e_i\pm e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type $A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones.

math.CA↗

arXiv2007

Limit theorems for radial random walks on pxq-matrices as p tends to infinity

The radial probability measures on $R^p$ are in a one-to-one correspondence with probability measures on $[0,\infty[$ by taking images of measures w.r.t. the Euclidean norm mapping. For fixed $ν\in M^1([0,\infty[)$ and each dimension p, we consider i.i.d. $R^p$-valued random variables $X_1^p,X_2^p,...$ with radial laws corresponding to $ν$ as above. We derive weak and strong laws of large numbers as well as a large deviation principle for the Euclidean length processes $S_k^p:=\|X_1^p+...+X_k^p\|$ as k,p\to\infty in suitable ways. In fact, we derive these results in a higher rank setting, where $R^p$ is replaced by the space of $p\times q$ matrices and $[0,\infty[$ by the cone $Π_q$ of positive semidefinite matrices. Proofs are based on the fact that the $(S_k^p)_{k\ge 0}$ form Markov chains on the cone whose transition probabilities are given in terms Bessel functions $J_μ$ of matrix argument with an index $μ$ depending on p. The limit theorems follow from new asymptotic results for the $J_μ$ as $μ\to \infty$. Similar results are also proven for certain Dunkl-type Bessel functions.

math.CA↗

arXiv2006

Bessel convolutions on matrix cones: Algebraic properties and random walks

Bessel-type convolution algebras of bounded Borel measures on the matrix cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C, \mathbb H$ were introduced recently by Rösler. These convolutions depend on some continuous parameter, generate commutative hypergroup structures and have Bessel functions of matrix argument as characters. Here, we first study the rich algebraic structure of these hypergroups. In particular, the subhypergroups and automorphisms are classified, and we show that each quotient by a subhypergroup carries a hypergroup structure of the same type. The algebraic properties are partially related to properties of random walks on matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks on these hypergroups are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.

math.CA↗

arXiv2004

SU(d)--biinvariant random walks on SL(d,C) and their Euclidean counterparts

We establish a deformation isomorphism between the algebras of $SU(d)$-biinvariant compactly supported measures on $SL(d,\comp)$ and $SU(d)$-conjugation invariant measures on the Euclidean space $H_d^0$ of all Hermitian $d\times d$-matrices with trace 0. This isomorphism concisely explains a close connection between the spectral problem for sums of Hermititan matrices on one hand and the singular spectral problem for products of matrices from $SL(d,\comp)$ on the other, which has recently been observed by Klyachko \cite{Kl2}. From this deformation we further obtain an explicit, probability preserving and isometric isomorphism between the Banach algebra of bounded $SU(d)$-biinvariant measures on $SL(d,\comp)$ and a certain (non-invariant) subalgebra of the bounded signed measures on $H_d^0$. We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of $SU(d)$-biinvariant random walks on $SL(d,\comp)$ in a simple way. Our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple Lie groups.

math.RT↗

arXiv2004

Positivity of Dunkl's intertwining operator via the trigonometric setting

In this note, a new proof for the positivity of Dunkl's intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their non-symmetric counterparts.

math.CA↗

arXiv2004

Deformations of convolution semigroups on commutative hypergroups

It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive semicharacters can be used to explain probabilistic connections between the Gelfand pairs (SL(d,C), SU(d)) and Hermitian matrices. We here study connections between general convolution semigroups on commutative hypergroups and their deformations. We are able to develop a satisfying theory, if the underlying positive semicharacter has some growth property. We present several examples which indicate that this growth condition holds in many interesting cases.

math.PR↗

arXiv1997

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.

q-alg↗