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.