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$L_2$逼近的收敛速度是数值分析的核心主题。本研究受Schaback（Math. Comp., 1999）工作的启发，该工作揭示在径向基函数插值的最佳逐点逼近框架下，充分光滑函数的收敛速度可相较于"原生空间"$H$中函数的一般收敛速度实现翻倍。基于此，我们获得了$H$中有限维子空间上的$H$-正交投影的一般性结论：即只要$L_2$、$H$与$B$满足适当关联关系，任何已知的$H$中全体函数的$L_2$收敛速度可转化为更光滑赋范空间$B$中函数的双倍$L_2$收敛速度，同时$H$-范数的误差界亦获得类似改进。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$L_2$及$H$-范数收敛速度，特别关注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）。第二个应用是针对一般条件正定基函数的径向基函数插值，此时同样实现了$L_2$收敛速度的翻倍，且对更光滑赋范空间$B$中的所有函数，原生空间范数下的收敛速度也得到类似改进。