Escaping the native space of Sobolev kernels by interpolation
Tobias Ehring, Max-Paul Vogel, Bernard Haasdonk
TL;DR
The article develops a general framework to analyze kernel interpolation beyond the native RKHS by introducing k-admissible pairs (A(Ω),B(Ω)) and a criterion based on the uniform boundedness of extended interpolation operators Π^{n}_{A,B}. It establishes a necessary-and-sufficient condition for convergence on A(Ω) and demonstrates concrete results for Sobolev kernels: L^2-convergence for continuous targets under quasi-uniform sampling, and sup-norm convergence on intervals by proving uniformly bounded Lebesgue constants via exponential decay of Lagrange functions. These findings significantly broaden the applicability of kernel interpolation to functions outside the native space and provide a constructive pathway for convergence guarantees in practical settings. The work also identifies open questions for infinitely smooth kernels (e.g., Gaussian) and higher-dimensional domains, suggesting directions for future research and extensions.
Abstract
Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space $\mathcal{H}_k\!\left(Ω\right)$ induced by a kernel on a domain $Ω\subset\mathbb{R}^N$. 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 $A(Ω)$ and $B(Ω)$ be Banach spaces with continuous embeddings $\mathcal{H}_k\!\left(Ω\right) \hookrightarrow A(Ω)\hookrightarrow B(Ω)$, assume point evaluation is continuous on $A(Ω)$, and that $\mathcal{H}_k\!\left(Ω\right)$ is dense in $A(Ω)$. For a nested sequence of node sets $(X_n)_{n\ge1}\subsetΩ$ with $\bigcup_n X_n$ dense, we characterize convergence of the kernel interpolants in the $B(Ω)$-norm for all target functions in $A(Ω)$ via the uniform boundedness of the interpolation operators $Π^{\,n}_{A,B}:A(Ω)\to B(Ω)$. This yields a necessary and sufficient condition under which kernel interpolation extends beyond $\mathcal{H}_k\!\left(Ω\right)$. Specializing to Sobolev kernels of order $τ>N/2$ on bounded Lipschitz domains, we show that every $f \in C(\overlineΩ)$ can be approximated in the $L^2(Ω)$-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 $f \in C([a,b])$.