Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective, and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the $P$-greedy target-data-independent selection rule, and can additionally be proven to be optimal when they fully exploit adaptivity ($f$-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in Reproducing Kernel Hilbert Spaces, as they allow us to compare adaptive interpolation with non-adaptive best nonlinear approximation.

翻译：核插值是一种从数据逼近函数的多功能工具，当与某些Sobolev空间相关的核一起使用时，可证明其具有某些最优性质。在插值背景下，最优函数采样位置的选择是一个核心问题，既具有实践意义，又是有趣的理论课题。贪婪插值算法为此任务提供了可行的解决方案，其运行高效且逼近精度可证。本文利用关于一般贪婪算法的最新结果，填补了这些算法收敛理论中存在的空白。这一改进产生了新的收敛速率，当限制于与目标数据无关的$P$-贪婪选择规则时，该速率与最优速率相匹配；当完全利用自适应性（$f$-贪婪）时，可进一步证明其最优性。除填补空白外，这些新结果在再生核希尔伯特空间中一般逼近算法最优性的更广泛背景下具有重要意义，因为它们使我们能够比较自适应插值与非线性最优逼近。