Universality of kernels on Riemannian symmetric spaces

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 $C_0$ and integrable is automatically $C_0$-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.