Searcharxiv⌕ Search

arXiv subjects

Margit Rösler

Publications and source records attributed to Margit Rösler.

At least 19 recordsLinked to original sources
arXiv2026

Dunkl theory, convolution algebras, and related Markov processes

These lecture notes are intended as an introduction to the theory of rational Dunkl operators, the associated special functions and related Markov processes with an emphasis on examples which are related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones of positive semidefinite matrices. We start with a comprehensive introduction into Dunkl theory: Dunkl operators, the intertwining operator and its positivity, the Dunkl kernel and the Dunkl transform, the Dunkl Laplacian and the associated heat semigroup. We further give an outline of the connection with Calogero-Moser-Sutherland models and generalized Hermite polynomials. Moreover, of central interest will be product formulas, generalized translations and associated commutative hypergroup structures on closed Weyl chambers. In particular, we explain how Dunkl theory for particular multiplicities is related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones, and how this leads to a bunch of multiplicities for which the Weyl-group invariant Dunkl theory admits a probability preserving translation and an associated commutative hypergroup structure on the closed Weyl chamber. We finally discuss Markov processes on R^N which are related with Dunkl theory with an emphasis on connections to random walk on groups and hypergroups. In particular associated martingales, martingale characterizations, moment functions and Appell characters are studied, i.e. diffusion-reflection processes with the Dunkl Laplacians as generators.

math.CA↗
arXiv2026

Littlewood-Paley theory for orthogonal expansions associated with root systems

We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley $g$-functions in the spirit of Stein's work for compact Lie groups and prove their $L^{p}$-boundedness for $1<p\leq 2$ and for some of them also for $1<p<\infty$. As an application, we define associated Riesz transforms and imaginary powers and prove their $L^{p}$-continuity for $1<p<\infty$. Passing to the average with respect to the action of the associated reflection group, we obtain $L^{p}$-boundedness of $g$-functions for the symmetric Poisson semigroup for all $1<p<\infty$. In particular, our framework covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.

math.CA↗
arXiv2025

Multiresolution analysis on spectra of hermitian matrices

We establish a multiresolution analysis on the space $\text{Herm}(n)$ of $n\times n$ complex Hermitian matrices which is adapted to invariance under conjugation by the unitary group $U(n).$ The orbits under this action are parametrized by the possible ordered spectra of Hermitian matrices, which constitute a closed Weyl chamber of type $A_{n-1}$ in $\mathbb R^n.$ The space $L^2(\text{Herm}(n))^{U(n)}$ of radial, i.e. $U(n)$-invariant $L^2$-functions on $\text{Herm}(n)$ is naturally identified with a certain weighted $L^2$-space on this chamber. The scale spaces of our multiresolution analysis are obtained by usual dyadic dilations as well as generalized translations of a scaling function, where the generalized translation is a hypergroup translation which respects the radial geometry. We provide a concise criterion to characterize orthonormal wavelet bases and show that such bases always exist. They provide natural orthonormal bases of the space $L^2(\text{Herm}(n))^{U(n)}.$ Furthermore, we show how to obtain radial scaling functions from classical scaling functions on $\mathbb R^{n}$. Finally, generalizations related to the Cartan decompositions for general compact Lie groups are indicated.

math.CA↗
arXiv2024

Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated of root systems of type $A_{n-1}$ and type $B_n$ with positive multiplicities as the rank $n$ tends to infinity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type $A$ case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of $β$-ensembles in random matrix theory. These results generalize known facts about the approximation of the (positive-definite) Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over $\mathbb F = \mathbb R, \mathbb C, \mathbb H$ (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

math.CA↗
arXiv2022

The Dunkl-Laplace transform and Macdonald's hypergeometric series

We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type A. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type A and more generally, for the associated Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture in an unpublished manuscript of Macdonald (2013). Our proof for the Jack polynomials is based on Dunkl operator techniques and the raising operator of Knop and Sahi. Moreover, we use these results to establish Laplace transform identities between hypergeometric series in terms of Jack polynomials. Finally, we conclude with a Post-Widder inversion formula for the Dunkl-Laplace transform.

math.CA↗
arXiv2021

Elementary symmetric polynomials and martingales for Heckman-Opdam processes

We consider the generators $L_k$ of Heckman-Opdam diffusion processes in the compact and non-compact case in $N$ dimensions for root systems of type $A$ and $B$, with a multiplicity function of the form $k=κk_0$ with some fixed value $k_0$ and a varying constant $κ\in\,[0,\infty[$. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the $L_k$ for all $κ\in\,]0,\infty[$. This leads to martingales associated with the Heckman-Opdam diffusions $ (X_{t,1},\ldots,X_{t,N})_{t\ge0}$. As our results extend to the freezing case $κ=\infty$ with a deterministic limit after some renormalization, we find formulas for the expectations $\mathbb E(\prod_{j=1}^N(y-X_{t,j})),$ $y\in\mathbb C$.

math.PR↗
arXiv2020

Riesz distributions and Laplace transform in the Dunkl setting of type A

We study Riesz distributions in the framework of rational Dunkl theory associated with root systems of type A. As an important tool, we employ a Laplace transform involving the associated Dunkl kernel, which essentially goes back to Macdonald, but was so far only established at a formal level. We give a rigorous treatment of this transform based on suitable estimates of the type A Dunkl kernel. Our main result is a precise analogue in the Dunkl setting of a well-known result by Gindikin, stating that a Riesz distribution on a symmetric cone is a positive measure if and only if its exponent is contained in the Wallach set. For Riesz distributions in the Dunkl setting, we obtain an analogous characterization in terms of a generalized Wallach set which depends on the multiplicity parameter on the root system.

math.CA↗
arXiv2019

Positive intertwiners for Bessel functions of type B

Let $V_k$ denote Dunkl's intertwining operator for the root sytem $B_n$ with multiplicity $k=(k_1,k_2)$ with $k_1\geq 0, k_2>0$. It was recently shown that the positivity of the operator $V_{k^\prime\!,k} =V_{k^\prime}\circ V_k^{-1}$ which intertwines the Dunkl operators associated with $k$ and $k^\prime=(k_1+h,k_2)$ implies that $h\in[k_2(n-1),\infty[\,\cup\,(\{0,k_2,\ldots,k_2(n-1)\}-\mathbb Z_+)$. This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For $k_1 \geq 0, \,k_2\in\{1/2,1,2\}$ and $h>k_2(n-1)$, the operator $V_{k^\prime\!,k}$ is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type $B_n$. Moreover, the same positivity results hold for arbitrary $k_1\geq 0, k_2>0$ and $h\in k_2\cdot \mathbb Z_+.$ The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials.

math.CA↗
arXiv2019

Sonine formulas and intertwining operators in Dunkl theory

Let $V_k$ denote Dunkl's intertwining operator associated with some root system $R$ and multiplicity function $k$. For two multiplicities $k, k^\prime$ on $R$, we study the operator $V_{k^\prime,k} = V_{k^\prime}\circ V_k^{-1}$, which intertwines the Dunkl operators for multiplicity $k$ with those for multiplicity $k^\prime.$ While it is well-known that the operator $V_k$ is positive for nonnegative $k$, it has been a long-standing conjecture that its generalizations $V_{k^\prime,k}$ are also positive if $k^\prime \geq k \geq 0,$ which is known to be true in rank one. In this paper, we disprove this conjecture by constructing examples for root system $B_n$ with multiplicites $k^\prime \geq k \geq 0$ for which $V_{k^\prime, k}$ is not positive. This matter is closely related to the existence of integral representations of Sonine type between the Dunkl kernels and Bessel functions associated with the relevant multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine-type integral formulas for Heckman-Opdam hypergeometric functions of type $BC_n$ as well as conditions on the existence of positive branching coefficients between systems of multivariable Jacobi polynomials.

math.CA↗
arXiv2018

Beta distributions and Sonine integrals for Bessel functions on symmetric cones

There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index $μ+ ν$ with respect to such functions of lower index $μ.$ For Bessel functions on matrix cones, Sonine formulas involve beta densities $β_{μ,ν}$ on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges $\Reμ, \Re ν> μ_0$, where the involved Beta densities are probability measures and the limiting index $μ_0\geq 0$ depends on the rank of the cone. It is zero only in the one-dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of $ν$ below the critical limit $μ_0$. This is achieved by an analytic extension of the involved Beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric cones, we analyze for which indices the obtained Beta distributions are still measures. At the same time, we characterize the indices for which a Sonine formula between the related Bessel functions exists. As for Riesz distributions, there occur gaps in the admissible range of indices which are determined by the so-called Wallach set.

math.CA↗
arXiv2016

On the Green function and Poisson integrals of the Dunkl Laplacian

We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian $Δ_k$ in $\mathbb{R}^d$. As applications we derive the Poisson-Jensen formula for $Δ_k$-subharmonic functions and Hardy-Stein identities for the Poisson integrals of $Δ_k$. We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in $\mathbb{R}^d$. These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.

math.AP↗
arXiv2015

A multivariate version of the disk convolution

We present an explicit product formula for the spherical functions of the compact Gelfand pairs $(G,K_1)= (SU(p+q), SU(p)\times SU(q))$ with $p\ge 2q$, which can be considered as the elementary spherical functions of one-dimensional $K$-type for the Hermitian symmetric spaces $G/K$ with $K= S(U(p)\times U(q))$. Due to results of Heckman, they can be expressed in terms of Heckman-Opdam Jacobi polynomials of type $BC_q$ 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 $p\in]2q-1,\infty[$. 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 $q=1$ to higher rank.

math.CA↗
arXiv2015

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. 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 $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. 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.

math.CA↗
arXiv2014

Convolution algebras for Heckman-Opdam polynomials derived from compact Grassmannians

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 are linked to explicit positive product formulas for Heckman-Opdam polynomials of type BC, which occur for certain discrete multiplicities as the spherical functions of U/K. These results complement those of a recent paper by the second author for the non-compact case.

math.RT↗
arXiv2014

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.

math.RT↗
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↗