@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{53542, abstract = {{AbstractThis work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi-K-invariance.}}, author = {{Papageorgiou, Efthymia}}, issn = {{1424-3199}}, journal = {{Journal of Evolution Equations}}, keywords = {{Mathematics (miscellaneous)}}, number = {{2}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces}}}, doi = {{10.1007/s00028-024-00959-6}}, volume = {{24}}, year = {{2024}}, } @article{53300, author = {{Brennecken, Dominik}}, issn = {{0022-247X}}, journal = {{Journal of Mathematical Analysis and Applications}}, keywords = {{Applied Mathematics, Analysis}}, number = {{2}}, publisher = {{Elsevier BV}}, title = {{{Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting}}}, doi = {{10.1016/j.jmaa.2024.128125}}, volume = {{535}}, 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{56366, abstract = {{AbstractWe discuss in which cases the Dunkl convolution of distributions , possibly both with non‐compact support, can be defined and study its analytic properties. We prove results on the (singular‐)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root system we consider the Riesz distributions and prove that their Dunkl convolution exists and that  holds.}}, author = {{Brennecken, Dominik}}, issn = {{0025-584X}}, journal = {{Mathematische Nachrichten}}, publisher = {{Wiley}}, title = {{{Dunkl convolution and elliptic regularity for Dunkl operators}}}, doi = {{10.1002/mana.202300370}}, year = {{2024}}, } @article{53540, abstract = {{AbstractThis note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $$L^1$$ L 1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.}}, author = {{Papageorgiou, Efthymia}}, issn = {{0926-2601}}, journal = {{Potential Analysis}}, keywords = {{Analysis}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds}}}, doi = {{10.1007/s11118-023-10109-1}}, year = {{2023}}, } @article{53539, abstract = {{AbstractThe infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length T around a fixed origin when $$T \rightarrow +\infty $$ T → + ∞ . The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces G/K of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely $$L^1$$ L 1 asymptotic convergence without requiring bi-K-invariance for initial data, and strong $$L^{\infty }$$ L ∞ convergence.}}, author = {{Papageorgiou, Efthymia}}, issn = {{2296-9020}}, journal = {{Journal of Elliptic and Parabolic Equations}}, keywords = {{Applied Mathematics, Numerical Analysis, Analysis}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Asymptotics for the infinite Brownian loop on noncompact symmetric spaces}}}, doi = {{10.1007/s41808-023-00250-8}}, year = {{2023}}, } @article{53538, abstract = {{AbstractWe study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.}}, author = {{Polychrou, G. and Papageorgiou, Efthymia and Fotiadis, A. and Daskaloyannis, C.}}, issn = {{1139-1138}}, journal = {{Revista Matemática Complutense}}, keywords = {{General Mathematics}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation}}}, doi = {{10.1007/s13163-023-00476-z}}, year = {{2023}}, } @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{40053, author = {{Graczyk, P. and Luks, Tomasz and Sawyer, P.}}, issn = {{0008-414X}}, journal = {{Canadian Journal of Mathematics}}, number = {{4}}, pages = {{1005--1033}}, publisher = {{Canadian Mathematical Society}}, title = {{{Potential kernels for radial Dunkl Laplacians}}}, doi = {{10.4153/s0008414x21000195}}, volume = {{74}}, year = {{2022}}, } @article{36271, author = {{Brennecken, Dominik and Hilgert, Joachim and Ciardo, Lorenzo}}, journal = {{Journal of Lie Theory}}, number = {{2}}, pages = {{459----468}}, publisher = {{Heldermann Verlag}}, title = {{{Algebraically Independent Generators for the Algebra of Invariant Differential Operators on SLn(R)/SOn(R)}}}, doi = {{10.48550/arXiv.2008.07479}}, volume = {{31}}, 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{40051, author = {{Luks, Tomasz and Xiao, Yimin}}, issn = {{0894-9840}}, journal = {{Journal of Theoretical Probability}}, number = {{1}}, pages = {{153--179}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Multiple Points of Operator Semistable Lévy Processes}}}, doi = {{10.1007/s10959-018-0859-4}}, volume = {{33}}, 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{40050, author = {{Baeumer, Boris and Luks, Tomasz and Meerschaert, Mark M.}}, issn = {{0025-584X}}, journal = {{Mathematische Nachrichten}}, keywords = {{General Mathematics}}, number = {{17-18}}, pages = {{2516--2535}}, publisher = {{Wiley}}, title = {{{Space‐time fractional Dirichlet problems}}}, doi = {{10.1002/mana.201700111}}, volume = {{291}}, year = {{2018}}, } @article{40065, author = {{Luks, Tomasz and Xiao, Yimin}}, issn = {{0894-9840}}, journal = {{Journal of Theoretical Probability}}, number = {{1}}, pages = {{297--325}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On the Double Points of Operator Stable Lévy Processes}}}, doi = {{10.1007/s10959-015-0638-4}}, volume = {{30}}, year = {{2017}}, } @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}}, }