---
_id: '67147'
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."
author:
- first_name: Lukas
  full_name: Langen, Lukas
  id: '73664'
  last_name: Langen
- first_name: Margit
  full_name: Rösler, Margit
  id: '37390'
  last_name: Rösler
citation:
  ama: Langen L, Rösler M. Littlewood-Paley Theory For Orthogonal Expansions Associated
    With Root Systems. <i>arXiv:260912722</i>. Published online 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} }'
  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.
  mla: Langen, Lukas, and Margit Rösler. “Littlewood-Paley Theory For Orthogonal Expansions
    Associated With Root Systems.” <i>ArXiv:2609.12722</i>, 2026.
  short: L. Langen, M. Rösler, ArXiv:2609.12722 (2026).
date_created: 2026-09-15T07:55:23Z
date_updated: 2026-09-15T08:03:26Z
external_id:
  arxiv:
  - '2609.12722'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/pdf/2609.12722v1
oa: '1'
page: '29'
publication: arXiv:2609.12722
status: public
title: Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems
type: preprint
user_id: '73664'
year: '2026'
...
---
_id: '66550'
abstract:
- lang: eng
  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 $k>1/2$.
author:
- first_name: Lukas
  full_name: Langen, Lukas
  id: '73664'
  last_name: Langen
citation:
  ama: Langen L. Uniform bounds on the Dunkl kernel. <i>arXiv:260702176</i>. Published
    online 2026.
  apa: Langen, L. (2026). Uniform bounds on the Dunkl kernel. In <i>arXiv:2607.02176</i>.
  bibtex: '@article{Langen_2026, title={Uniform bounds on the Dunkl kernel}, journal={arXiv:2607.02176},
    author={Langen, Lukas}, year={2026} }'
  chicago: Langen, Lukas. “Uniform Bounds on the Dunkl Kernel.” <i>ArXiv:2607.02176</i>,
    2026.
  ieee: L. Langen, “Uniform bounds on the Dunkl kernel,” <i>arXiv:2607.02176</i>.
    2026.
  mla: Langen, Lukas. “Uniform Bounds on the Dunkl Kernel.” <i>ArXiv:2607.02176</i>,
    2026.
  short: L. Langen, ArXiv:2607.02176 (2026).
date_created: 2026-07-21T13:08:08Z
date_updated: 2026-09-15T08:02:59Z
external_id:
  arxiv:
  - '2607.02176'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/pdf/2607.02176
oa: '1'
page: '31'
publication: arXiv:2607.02176
status: public
title: Uniform bounds on the Dunkl kernel
type: preprint
user_id: '73664'
year: '2026'
...
---
_id: '56717'
abstract:
- lang: eng
  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."
article_type: original
author:
- first_name: Lukas
  full_name: Langen, Lukas
  id: '73664'
  last_name: Langen
- first_name: Margit
  full_name: Rösler, Margit
  id: '37390'
  last_name: Rösler
citation:
  ama: Langen L, Rösler M. Multiresolution analysis on spectra of hermitian matrices.
    <i>Indagationes Mathematicae</i>. 2025;36(6):1671-1694.
  apa: Langen, L., &#38; Rösler, M. (2025). Multiresolution analysis on spectra of
    hermitian matrices. <i>Indagationes Mathematicae</i>, <i>36</i>(6), 1671–1694.
  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} }'
  chicago: 'Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra
    of Hermitian Matrices.” <i>Indagationes Mathematicae</i> 36, no. 6 (2025): 1671–94.'
  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.
  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.
  short: L. Langen, M. Rösler, Indagationes Mathematicae 36 (2025) 1671–1694.
date_created: 2024-10-22T09:31:19Z
date_updated: 2026-02-19T14:16:43Z
ddc:
- '510'
department:
- _id: '555'
external_id:
  arxiv:
  - '2410.10364'
file:
- access_level: closed
  content_type: application/pdf
  creator: llangen
  date_created: 2026-02-19T14:14:39Z
  date_updated: 2026-02-19T14:14:39Z
  file_id: '64288'
  file_name: MSA_hermitsch_published.pdf
  file_size: 443262
  relation: main_file
  success: 1
file_date_updated: 2026-02-19T14:14:39Z
has_accepted_license: '1'
intvolume: '        36'
issue: '6'
language:
- iso: eng
main_file_link:
- url: https://doi.org/10.1016/j.indag.2025.03.009
page: 1671-1694
project:
- _id: '357'
  name: TRR 358 - Ganzzahlige Strukturen in Geometrie und Darstellungstheorie
publication: Indagationes Mathematicae
publication_status: published
publisher: Elsevier
related_material:
  link:
  - relation: research_paper
    url: https://arxiv.org/abs/2410.10364
status: public
title: Multiresolution analysis on spectra of hermitian matrices
type: journal_article
user_id: '73664'
volume: 36
year: '2025'
...