Escaping the native space of Sobolev kernels by interpolation

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 $f \in C([a,b])$.