Unbounded Characteristic and Universal Kernels
arXiv (preprint)
arXiv
While we understand bounded kernels and their RKHSs, the same couldn't be said for the unbounded case: the classical definitions of characteristic, universal and i.s.p.d. kernels (all of whom play a key role in guaranteeing the utility/power of the downstream kernel method) and the established relations among them fail in the unbounded case. In this work we address this issue by studying the relation among different properties of RKHSs in the unbounded case, relating characteristicness, i.s.p.d. kernels, s.n.t. semimetric spaces and ($L_p$-)universality.
Why Is This Important?
We derived an interesting result we believe it can help broaden the understanding of RKHSs and their flexibility: All $L_p$-universalities [$\overline{\mathscr H_K} = L_p(\mathcal X,\mathbb P)$ for every law $\mathbb P$ for which the kernel is $p$-integrable] are equivalent for every value of $p\in[1,\infty)$, even if the kernel $K$ is unbounded.
This result was known to occur in locally compact 2nd countable spaces (LC2C) with $c_0$-kernels. This case could be explained as in LC2C spaces $\overline{\mathcal C_0(\mathcal X)}=L_p(\mathcal X, \mathbb P)$, for any law $\mathbb P$ and any $p\in[1,\infty)$; hence the equivalence of $L_p$-universalities was thought to come from the denseness of $\mathcal C_0$ rather than from any sort of intrinsic flexibility of the RKHS. However, we showed that the equivalence of $L_p$-universalities is completely independent from $\mathcal C_0$, and in fact doesn't even depend on the kernel being bounded.
Cite
@TECHREPORT{cribeiro26unbounded,
AUTHOR = {Jose Cribeiro-Ramallo and Florian Kalinke and Zolt{\'a}n Szab{\'o}},
TITLE = {Unbounded Characteristic and Universal Kernels},
YEAR = {2026},
note = {(\url{https://arxiv.org/abs/2610.09731})},
}