Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces (RKHS) usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we leverage an analysis of greedy kernel algorithms to prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $\Omega \subset \mathbb{R}^d$, thus allowing for non-Lipschitz domains including e.g. cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tilde{\Omega} \subset \Omega \subset \mathbb{R}^d$, the convergence rate does not deteriorate - i.e. the convergence rates are stable with respect to going to a subset. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness like the Gaussian kernel. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $\Omega$ has an impact on the approximation properties. Numerical experiments illustrate and confirm the experiments.

翻译：再生核希尔伯特空间（RKHS）中核插值的误差估计通常对域的形状施加相当严格的限制，尤其是在无穷光滑核（如常用的高斯核）情形下。本文通过利用贪婪核算法的分析，证明对于任意域 $\Omega \subset \mathbb{R}^d$（包括尖点和不规则边界等非利普希茨域），核插值在插值点数量上的收敛结果是可以获得的。特别地，我们证明当过渡到更小的域 $\tilde{\Omega} \subset \Omega \subset \mathbb{R}^d$ 时，收敛速率不会恶化——即收敛速率对子集选取具有稳定性。本文以有限光滑核和无穷光滑核（如高斯核）为例阐述了该结果的影响，并与索伯列夫空间中的逼近进行了对比——在索伯列夫空间中，域 $\Omega$ 的形状会对逼近性质产生影响。数值实验验证并确认了理论结果。