@article{56717,
abstract = {{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.}},
author = {{Langen, Lukas and Rösler, Margit}},
journal = {{Indagationes Mathematicae}},
number = {{6}},
pages = {{1671--1694}},
publisher = {{Elsevier}},
title = {{{Multiresolution analysis on spectra of hermitian matrices}}},
volume = {{36}},
year = {{2025}},
}
@article{54820,
abstract = {{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
$\beta$-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.}},
author = {{Brennecken, Dominik and Rösler, Margit}},
journal = {{Indagationes Mathematicae}},
publisher = {{Elsevier}},
title = {{{Limits of Bessel functions for root systems as the rank tends to infinity}}},
doi = {{10.1016/j.indag.2024.05.004}},
year = {{2024}},
}
@inbook{56001,
author = {{Brennecken, Dominik and Rösler, Margit}},
booktitle = {{Women in Analysis and PDE}},
editor = {{Chatzakou, Marianna and Ruzhansky, Michael and Stoeva, Diana}},
isbn = {{978-3-031-57004-9}},
pages = {{425}},
publisher = {{Birkhäuser Cham}},
title = {{{The Laplace transform in Dunkl theory}}},
volume = {{5}},
year = {{2024}},
}
@article{36294,
author = {{Brennecken, Dominik and Rösler, Margit}},
journal = {{Transactions of the American Mathematical Society}},
number = {{4}},
pages = {{2419--2447}},
publisher = {{ American Mathematical Society}},
title = {{{The Dunkl-Laplace transform and Macdonald’s hypergeometric series}}},
doi = {{10.1090/tran/8860}},
volume = {{376}},
year = {{2023}},
}
@article{38039,
abstract = {{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 $\mathbb E(\prod_{j=1}^N(y-X_{t,j})),$ $y\in\mathbb C$.}},
author = {{Rösler, Margit and Voit, Michael}},
journal = {{Contemporary Mathematics}},
number = {{780}},
pages = {{243--262}},
title = {{{Elementary symmetric polynomials and martingales for Heckman-Opdam processes}}},
doi = {{10.48550/ARXIV.2108.03228}},
year = {{2022}},
}
@article{37649,
abstract = {{Abstract
Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$, we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.}},
author = {{Rösler, Margit and Voit, Michael}},
issn = {{1073-7928}},
journal = {{International Mathematics Research Notices}},
keywords = {{General Mathematics}},
number = {{17}},
pages = {{13202--13230}},
publisher = {{Oxford University Press (OUP)}},
title = {{{Sonine Formulas and Intertwining Operators in Dunkl Theory}}},
doi = {{10.1093/imrn/rnz313}},
volume = {{2021}},
year = {{2021}},
}
@article{37659,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{0002-9939}},
journal = {{Proceedings of the American Mathematical Society}},
keywords = {{Applied Mathematics, General Mathematics}},
number = {{3}},
pages = {{1151--1163}},
publisher = {{American Mathematical Society (AMS)}},
title = {{{Positive intertwiners for Bessel functions of type B}}},
doi = {{10.1090/proc/15312}},
volume = {{149}},
year = {{2021}},
}
@article{37660,
author = {{Rösler, Margit}},
issn = {{0022-1236}},
journal = {{Journal of Functional Analysis}},
keywords = {{Analysis}},
number = {{12}},
publisher = {{Elsevier BV}},
title = {{{Riesz distributions and Laplace transform in the Dunkl setting of type A}}},
doi = {{10.1016/j.jfa.2020.108506}},
volume = {{278}},
year = {{2020}},
}
@article{37661,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{0022-2526}},
journal = {{Studies in Applied Mathematics}},
keywords = {{Applied Mathematics}},
number = {{4}},
pages = {{474--500}},
publisher = {{Wiley}},
title = {{{Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones}}},
doi = {{10.1111/sapm.12217}},
volume = {{141}},
year = {{2018}},
}
@article{37662,
author = {{Rösler, Margit and Graczyk, Piotr and Luks, Tomasz}},
issn = {{0926-2601}},
journal = {{Potential Analysis}},
keywords = {{Analysis}},
number = {{3}},
pages = {{337--360}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{On the Green Function and Poisson Integrals of the Dunkl Laplacian}}},
doi = {{10.1007/s11118-017-9638-6}},
volume = {{48}},
year = {{2018}},
}
@article{38032,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{1088-6850}},
journal = {{Transactions of the American Mathematical Society}},
number = {{8}},
pages = {{6005--6032}},
publisher = {{ American Mathematical Society}},
title = {{{Integral representation and sharp asymptotic results for some Heckman-Opdam hypergeometric functions of type BC}}},
doi = {{10.48550/ARXIV.1402.5793}},
volume = {{368}},
year = {{2016}},
}
@article{37663,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{0022-247X}},
journal = {{Journal of Mathematical Analysis and Applications}},
keywords = {{Applied Mathematics, Analysis}},
number = {{1}},
pages = {{701--717}},
publisher = {{Elsevier BV}},
title = {{{A multivariate version of the disk convolution}}},
doi = {{10.1016/j.jmaa.2015.10.062}},
volume = {{435}},
year = {{2016}},
}
@article{38037,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{1815-0659}},
journal = {{Symmetry, Integrability and Geometry: Methods and Applications}},
keywords = {{Geometry and Topology, Mathematical Physics, Analysis}},
number = {{013}},
pages = {{18pp}},
publisher = {{SIGMA (Symmetry, Integrability and Geometry: Methods and Application)}},
title = {{{A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian}}},
doi = {{10.3842/sigma.2015.013}},
volume = {{11}},
year = {{2015}},
}
@article{37667,
author = {{Rösler, Margit and Remling, Heiko}},
issn = {{0021-9045}},
journal = {{Journal of Approximation Theory}},
keywords = {{Applied Mathematics, General Mathematics, Numerical Analysis, Analysis}},
pages = {{30--48}},
publisher = {{Elsevier BV}},
title = {{{Convolution algebras for Heckman–Opdam polynomials derived from compact Grassmannians}}},
doi = {{10.1016/j.jat.2014.07.005}},
volume = {{197}},
year = {{2014}},
}
@article{37672,
abstract = {{AbstractLet ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{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.}},
author = {{Rösler, Margit and Koornwinder, Tom and Voit, Michael}},
issn = {{0010-437X}},
journal = {{Compositio Mathematica}},
keywords = {{Algebra and Number Theory}},
number = {{8}},
pages = {{1381--1400}},
publisher = {{Wiley}},
title = {{{Limit transition between hypergeometric functions of type BC and type A}}},
doi = {{10.1112/s0010437x13007045}},
volume = {{149}},
year = {{2013}},
}
@article{38038,
author = {{Rösler, Margit and Voit, Michael}},
journal = {{Journal of Lie Theory 23}},
number = {{4}},
pages = {{899----920}},
publisher = {{Heldermann }},
title = {{{Olshanski spherical functions for infinite dimensional motion groups of fixed rank}}},
doi = {{10.48550/ARXIV.1210.1351}},
year = {{2013}},
}
@article{39911,
author = {{Rösler, Margit and Remling, H.}},
issn = {{1073-7928}},
journal = {{International Mathematics Research Notices}},
keywords = {{General Mathematics}},
number = {{18}},
pages = {{4200–4225}},
publisher = {{Oxford University Press (OUP)}},
title = {{{The Heat Semigroup in the Compact Heckman-Opdam Setting and the Segal-Bargmann Transform}}},
doi = {{10.1093/imrn/rnq239}},
year = {{2011}},
}
@article{39921,
author = {{Rösler, Margit and Voit, Michael}},
issn = {{0025-584X}},
journal = {{Mathematische Nachrichten}},
keywords = {{General Mathematics}},
number = {{1}},
pages = {{87--104}},
publisher = {{Wiley}},
title = {{{Limit theorems for radial random walks on p × q-matrices as p tends to infinity}}},
doi = {{10.1002/mana.200710235}},
volume = {{284}},
year = {{2011}},
}
@article{39924,
author = {{Rösler, Margit}},
issn = {{0022-1236}},
journal = {{Journal of Functional Analysis}},
keywords = {{Analysis}},
number = {{8}},
pages = {{2779--2800}},
publisher = {{Elsevier BV}},
title = {{{Positive convolution structure for a class of Heckman–Opdam hypergeometric functions of type BC}}},
doi = {{10.1016/j.jfa.2009.12.007}},
volume = {{258}},
year = {{2010}},
}
@inproceedings{39950,
author = {{Rösler, Margit}},
booktitle = {{Infinite Dimensional Harmonic Analysis IV}},
pages = {{ 255–271}},
publisher = {{World Scientific}},
title = {{{Convolution algebras for multivariable Bessel functions}}},
doi = {{10.1142/9789812832825_0017}},
year = {{2009}},
}