[{"user_id":"73664","_id":"67147","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://arxiv.org/pdf/2609.12722v1","open_access":"1"}],"page":"29","date_updated":"2026-09-15T08:03:26Z","author":[{"id":"73664","full_name":"Langen, Lukas","last_name":"Langen","first_name":"Lukas"},{"full_name":"Rösler, Margit","last_name":"Rösler","first_name":"Margit","id":"37390"}],"year":"2026","title":"Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems","status":"public","oa":"1","type":"preprint","date_created":"2026-09-15T07:55:23Z","external_id":{"arxiv":["2609.12722"]},"abstract":[{"lang":"eng","text":"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 $1<p<\\infty$. As an application, we define associated Riesz transforms and Riesz potentials and prove their $L^{p}$-continuity for $1<p<\\infty$. \r\n  Passing to the average with respect to the action of the associated reflection group, our framework in particular covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases."}],"citation":{"short":"L. Langen, M. Rösler, ArXiv:2609.12722 (2026).","chicago":"Langen, Lukas, and Margit Rösler. “Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems.” <i>ArXiv:2609.12722</i>, 2026.","ieee":"L. Langen and M. Rösler, “Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems,” <i>arXiv:2609.12722</i>. 2026.","apa":"Langen, L., &#38; Rösler, M. (2026). Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems. In <i>arXiv:2609.12722</i>.","bibtex":"@article{Langen_Rösler_2026, title={Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems}, journal={arXiv:2609.12722}, author={Langen, Lukas and Rösler, Margit}, year={2026} }","ama":"Langen L, Rösler M. Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems. <i>arXiv:260912722</i>. Published online 2026.","mla":"Langen, Lukas, and Margit Rösler. “Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems.” <i>ArXiv:2609.12722</i>, 2026."},"publication":"arXiv:2609.12722"},{"external_id":{"arxiv":["2607.02176"]},"date_created":"2026-07-21T13:08:08Z","type":"preprint","oa":"1","publication":"arXiv:2607.02176","citation":{"chicago":"Langen, Lukas. “Uniform Bounds on the Dunkl Kernel.” <i>ArXiv:2607.02176</i>, 2026.","short":"L. Langen, ArXiv:2607.02176 (2026).","apa":"Langen, L. (2026). Uniform bounds on the Dunkl kernel. In <i>arXiv:2607.02176</i>.","ieee":"L. Langen, “Uniform bounds on the Dunkl kernel,” <i>arXiv:2607.02176</i>. 2026.","ama":"Langen L. Uniform bounds on the Dunkl kernel. <i>arXiv:260702176</i>. Published online 2026.","bibtex":"@article{Langen_2026, title={Uniform bounds on the Dunkl kernel}, journal={arXiv:2607.02176}, author={Langen, Lukas}, year={2026} }","mla":"Langen, Lukas. “Uniform Bounds on the Dunkl Kernel.” <i>ArXiv:2607.02176</i>, 2026."},"abstract":[{"text":"For an arbitrary reduced root system, we give upper bounds for the Dunkl kernel with regular spectral parameter and its derivatives, which are uniform in the spatial variable. These estimates generalize well-known sharp upper bounds for classical one-variable Bessel functions and for spherical functions of Cartan motion groups. As a consequence, we prove that the representing measure of Dunkl's intertwining operator is absolutely continuous with respect to the Lebesgue measure for multiplicities $k> 1/2$ and generic spectral parameter. This settles a conjecture posed in [RdJ02] at least for $k>1/2$.","lang":"eng"}],"page":"31","main_file_link":[{"open_access":"1","url":"https://arxiv.org/pdf/2607.02176"}],"language":[{"iso":"eng"}],"_id":"66550","user_id":"73664","year":"2026","title":"Uniform bounds on the Dunkl kernel","status":"public","author":[{"id":"73664","full_name":"Langen, Lukas","first_name":"Lukas","last_name":"Langen"}],"date_updated":"2026-09-15T08:02:59Z"},{"date_created":"2024-10-22T09:31:19Z","file":[{"relation":"main_file","date_updated":"2026-02-19T14:14:39Z","file_name":"MSA_hermitsch_published.pdf","file_size":443262,"access_level":"closed","file_id":"64288","content_type":"application/pdf","success":1,"creator":"llangen","date_created":"2026-02-19T14:14:39Z"}],"department":[{"_id":"555"}],"type":"journal_article","publication":"Indagationes Mathematicae","issue":"6","abstract":[{"text":"We establish a multiresolution analysis on the space $\\text{Herm}(n)$ of\r\n$n\\times n$ complex Hermitian matrices which is adapted to invariance under\r\nconjugation by the unitary group $U(n).$ The orbits under this action are\r\nparametrized by the possible ordered spectra of Hermitian matrices, which\r\nconstitute a closed Weyl chamber of type $A_{n-1}$ in $\\mathbb R^n.$ The space\r\n$L^2(\\text{Herm}(n))^{U(n)}$ of radial, i.e. $U(n)$-invariant $L^2$-functions\r\non $\\text{Herm}(n)$ is naturally identified with a certain weighted $L^2$-space\r\non this chamber.\r\n  The scale spaces of our multiresolution analysis are obtained by usual dyadic\r\ndilations as well as generalized translations of a scaling function, where the\r\ngeneralized translation is a hypergroup translation which respects the radial\r\ngeometry. We provide a concise criterion to characterize orthonormal wavelet\r\nbases and show that such bases always exist. They provide natural orthonormal\r\nbases of the space $L^2(\\text{Herm}(n))^{U(n)}.$\r\n  Furthermore, we show how to obtain radial scaling functions from classical\r\nscaling functions on $\\mathbb R^{n}$. Finally, generalizations related to the\r\nCartan decompositions for general compact Lie groups are indicated.","lang":"eng"}],"related_material":{"link":[{"relation":"research_paper","url":"https://arxiv.org/abs/2410.10364"}]},"language":[{"iso":"eng"}],"main_file_link":[{"url":"https://doi.org/10.1016/j.indag.2025.03.009"}],"author":[{"id":"73664","first_name":"Lukas","last_name":"Langen","full_name":"Langen, Lukas"},{"last_name":"Rösler","first_name":"Margit","full_name":"Rösler, Margit","id":"37390"}],"year":"2025","title":"Multiresolution analysis on spectra of hermitian matrices","intvolume":"        36","article_type":"original","date_updated":"2026-02-19T14:16:43Z","publication_status":"published","external_id":{"arxiv":["2410.10364"]},"citation":{"bibtex":"@article{Langen_Rösler_2025, title={Multiresolution analysis on spectra of hermitian matrices}, volume={36}, number={6}, journal={Indagationes Mathematicae}, publisher={Elsevier}, author={Langen, Lukas and Rösler, Margit}, year={2025}, pages={1671–1694} }","ama":"Langen L, Rösler M. Multiresolution analysis on spectra of hermitian matrices. <i>Indagationes Mathematicae</i>. 2025;36(6):1671-1694.","mla":"Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra of Hermitian Matrices.” <i>Indagationes Mathematicae</i>, vol. 36, no. 6, Elsevier, 2025, pp. 1671–94.","chicago":"Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra of Hermitian Matrices.” <i>Indagationes Mathematicae</i> 36, no. 6 (2025): 1671–94.","short":"L. Langen, M. Rösler, Indagationes Mathematicae 36 (2025) 1671–1694.","ieee":"L. Langen and M. Rösler, “Multiresolution analysis on spectra of hermitian matrices,” <i>Indagationes Mathematicae</i>, vol. 36, no. 6, pp. 1671–1694, 2025.","apa":"Langen, L., &#38; Rösler, M. (2025). Multiresolution analysis on spectra of hermitian matrices. <i>Indagationes Mathematicae</i>, <i>36</i>(6), 1671–1694."},"file_date_updated":"2026-02-19T14:14:39Z","project":[{"_id":"357","name":"TRR 358 - Ganzzahlige Strukturen in Geometrie und Darstellungstheorie"}],"_id":"56717","publisher":"Elsevier","page":"1671-1694","volume":36,"ddc":["510"],"user_id":"73664","status":"public","has_accepted_license":"1"}]