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

TL;DR

This paper characterizes which Bessel potential spaces admit the Mat extbackslash'ern kernel as a reproducing kernel, extending understanding beyond the Hilbert space case.

Contribution

It provides a novel characterization of pairs of Bessel potential spaces that share the Mat extbackslash'ern kernel as their reproducing kernel.

Findings

Identifies conditions under which Bessel potential spaces admit the Mat extbackslash'ern kernel.

Extends the theory of reproducing kernels from Hilbert spaces to certain Banach spaces.

Clarifies the structure of reproducing kernel Banach spaces related to Gaussian process covariance kernels.

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 / Mat\`ern 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.