Characterization of the reproducing structure of the Bessel potential spaces beyond $p=2$

Abstract

Reproducing kernel Hilbert spaces are uniquely characterized by their kernel, but reproducing kernel Banach spaces (RKBS) are not. However, a characterization of which RKBS admit a given kernel as reproducing kernel is lacking. This work provides such a characterization for the well-known Bessel potential / Matèrn kernel, a widely used covariance kernel for Gaussian processes which is the reproducing kernel of the Bessel potential space $H^{s,2}({\mathbb{R}}^d)$ when $s>d/2$. Concretely, this work characterizes the pairs of Bessel potential spaces $H^{u,p}({\mathbb{R}}^d),H^{v,q}({\mathbb{R}}^d)$ which have this kernel.

15 pages