@unpublished{66550,
  abstract     = {{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       = {{Langen, Lukas}},
  booktitle    = {{arXiv:2607.02176}},
  pages        = {{31}},
  title        = {{{Uniform bounds on the Dunkl kernel}}},
  year         = {{2026}},
}

@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}},
}