Escaping the native space of Sobolev kernels by interpolation

Abstract

Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space ${\mathcal{H}}_k\left(\hat{I}\text{c◯}\right)$ induced by a kernel on a domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$. For many applications, however, this assumption is overly restrictive. We develop a general framework for analyzing the convergence of kernel interpolation {beyond the native space}. Let $A(\hat{I}\text{c◯})$ and $B(\hat{I}\text{c◯})$ be Banach spaces with continuous embeddings ${\mathcal{H}}_k\left(\hat{I}\text{c◯}\right)\hookrightarrow A(\hat{I}\text{c◯})\hookrightarrow B(\hat{I}\text{c◯})$, assume point evaluation is continuous on $A(\hat{I}\text{c◯})$, and that ${\mathcal{H}}_k\left(\hat{I}\text{c◯}\right)$ is dense in $A(\hat{I}\text{c◯})$. For a nested sequence of node sets $(X_n{)}_{n\ge 1}\subset \hat{I}\text{c◯}$ with ${\bigcup }_n{X}_n$ dense, we characterize convergence of the kernel interpolants in the $B(\hat{I}\text{c◯})$-norm for all target functions in $A(\hat{I}\text{c◯})$ via the uniform boundedness of the interpolation operators ${\hat{I}}_{A,B}^n:A(\hat{I}\text{c◯})\to B(\hat{I}\text{c◯})$. This yields a necessary and sufficient condition under which kernel interpolation extends beyond ${\mathcal{H}}_k\left(\hat{I}\text{c◯}\right)$. Specializing to Sobolev kernels of order $\ddot{I}„>N/2$ on bounded Lipschitz domains, we show that every $f\in C(\overline{\hat{I}}\text{c◯})$ can be approximated in the $L^2(\hat{I}\text{c◯})$-norm by interpolation using quasi-uniform nested centers. Moreover, for a subclass of Sobolev kernels (including integer-order Matérn kernels), we prove that the Lebesgue constant is uniformly bounded on $[a,b]\subset \mathbb{R}$ under quasi-uniform centers; within our framework this implies supremum norm convergence of the interpolants for every target functions $f\in C([a,b])$.