Universality of kernels on Riemannian symmetric spaces
Abstract
We investigate universality properties of continuous, positive-definite invariant kernels on Riemannian symmetric spaces, providing a unified harmonic-analytic characterization across compact and non-compact settings. In the compact symmetric case, we prove that a continuous, positive-definite invariant kernel is C-universal if and only if all of its spherical coefficients are strictly positive, a sharpening of classical Bochner-type results. This characterization is extended to compact homogeneous spaces, where universality is shown to be equivalent to the strict positive-definiteness of coefficient matrices arising from a representation-theoretic expansion. In contrast, for non-compact symmetric spaces, we establish that any continuous, positive-definite invariant kernel that is C0 and integrable is automatically C0-universal. Our analysis essentially relies on spectral decompositions using spherical functions and group Fourier transforms, in order to provide alternative, harmonic-analytic formulations of universality conditions. Several examples (including kernels on spheres and compact Lie groups, as well as integrable kernels on hyperbolic spaces and symmetric cones) both illustrate the theory and demonstrate practical criteria for constructing universal kernels.
Related
- Spherical maximal operators on rank one Riemannian symmetric spaces of noncompact type
- Grassmannian Realizations of Classical Pseudo-Riemannian Symmetric Spaces
- A Deformation Approach to Axially Symmetric Kernels on the Sphere: Harmonic Coupling and Regularity
- Geodesic Models for Loop Spaces on Riemannian Manifolds
- Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings
BibTeX
@misc{said2026universality,
title = {{Universality of kernels on Riemannian symmetric spaces}},
author = {Salem Said and Nathaël da Costa and Franziskus Steinert and Cyrus Mostajeran},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05745254v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universality of kernels on Riemannian symmetric spaces
AU - Salem Said
AU - Nathaël da Costa
AU - Franziskus Steinert
AU - Cyrus Mostajeran
PY - 2026
DA - 2026/09/10
PB - HAL
UR - https://hal.science/hal-05745254v1
AB - We investigate universality properties of continuous, positive-definite invariant kernels on Riemannian symmetric spaces, providing a unified harmonic-analytic characterization across compact and non-compact settings. In the compact symmetric case, we prove that a continuous, positive-definite invariant kernel is C-universal if and only if all of its spherical coefficients are strictly positive, a sharpening of classical Bochner-type results. This characterization is extended to compact homogeneous spaces, where universality is shown to be equivalent to the strict positive-definiteness of coefficient matrices arising from a representation-theoretic expansion. In contrast, for non-compact symmetric spaces, we establish that any continuous, positive-definite invariant kernel that is C0 and integrable is automatically C0-universal. Our analysis essentially relies on spectral decompositions using spherical functions and group Fourier transforms, in order to provide alternative, harmonic-analytic formulations of universality conditions. Several examples (including kernels on spheres and compact Lie groups, as well as integrable kernels on hyperbolic spaces and symmetric cones) both illustrate the theory and demonstrate practical criteria for constructing universal kernels.
ER -