Convergence rates for $L_2$ approximation in a Hilbert space $H$ are a central theme in numerical analysis. The present work is inspired by Schaback (Math. Comp., 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the general rate for functions in the "native space" $H$. Motivated by this, we obtain a general result for $H$-orthogonal projection onto a finite dimensional subspace of $H$: namely, that any known $L_2$ convergence rate for all functions in $H$ translates into a doubled $L_2$ convergence rate for functions in a smoother normed space $B$, along with a similarly improved error bound in the $H$-norm, provided that $L_2$, $H$ and $B$ are suitably related. As a special case we improve the known $L_2$ and $H$-norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the $L_2$ convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space $B$.

翻译：希尔伯特空间$H$中$L_2$逼近的收敛速率是数值分析的核心主题。本文受Schaback（Math. Comp., 1999）的启发，该文在径向基函数插值的最佳点态逼近背景下证明，对于充分光滑的函数，其收敛速率可相较于“原生空间”$H$中函数的一般速率实现加倍。受此驱动，我们获得了$H$中有限维子空间上$H$-正交投影的一般结论：即，若已知$L_2$、$H$和$B$满足适当关系，则空间$H$中所有函数的任意已知$L_2$收敛速率均可转化为更光滑赋范空间$B$中函数的双倍$L_2$收敛速率，同时$H$-范数下的误差界也得到类似改进。作为特例，我们改进了再生核希尔伯特空间中核插值的已知$L_2$和$H$-范数收敛速率，特别关注了近期关于参数化偏微分方程格点周期核插值的研究（Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022）。第二个应用涉及一般条件正定基函数的径向基函数插值：对于更光滑赋范空间$B$中的所有函数，其$L_2$收敛速率再次实现加倍，原生空间范数下的收敛速率亦获得类似改进。