Escaping the native space of Sobolev kernels by interpolation

TL;DR

This paper develops a framework for analyzing kernel interpolation convergence beyond the native space, allowing approximation of broader classes of functions using Sobolev kernels on Lipschitz domains.

Contribution

It introduces a general approach to kernel interpolation convergence outside the native space, including necessary and sufficient conditions and applications to Sobolev kernels.

Findings

Kernel interpolation converges for functions in larger Banach spaces beyond the native space.

Every continuous function can be approximated in L2 norm using Sobolev kernels with quasi-uniform centers.

Lebesgue constants are uniformly bounded for certain Sobolev kernels, ensuring uniform convergence.

Abstract

Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space $\mathcal{H}_k\!\left(\Omega\right)$ induced by a kernel on a domain $\Omega\subset\mathbb{R}^N$. 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 $A(\Omega)$ and $B(\Omega)$ be Banach spaces with continuous embeddings $\mathcal{H}_k\!\left(\Omega\right) \hookrightarrow A(\Omega)\hookrightarrow B(\Omega)$, assume point evaluation is continuous on $A(\Omega)$, and that $\mathcal{H}_k\!\left(\Omega\right)$ is dense in $A(\Omega)$. For a nested sequence of node sets $(X_n)_{n\ge1}\subset\Omega$ with $\bigcup_n X_n$ dense, we characterize convergence of the kernel interpolants in the $B(\Omega)$-norm for all target functions in $A(\Omega)$ via the uniform boundedness of the interpolation operators $\Pi^{\,n}_{A,B}:A(\Omega)\to B(\Omega)$. This yields a necessary and sufficient condition under which kernel interpolation extends beyond $\mathcal{H}_k\!\left(\Omega\right)$. Specializing to Sobolev kernels of order $\tau>N/2$ on bounded Lipschitz domains, we show that every $f \in C(\overline{\Omega})$ can be approximated in the $L^2(\Omega)$-norm by interpolation using quasi-uniform nested centers. Moreover, for a subclass of Sobolev kernels (including integer-order Mat\'ern 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 $f \in C([a,b])$.