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 best 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：Hilbert空间$H$中$L_2$逼近的收敛速率是数值分析的核心课题。本研究受Schaback (Math. Comp., 1999) 工作的启发，该研究在径向基函数插值的最佳逐点逼近背景下指出，对于充分光滑的函数，其收敛速率相较于"原生空间"$H$中函数的最佳速率可提高一倍。受此启发，我们得到了$H$正交投影到其有限维子空间的一般性结果：即只要$L_2$、$H$与$B$空间满足适当关系，则对$H$中所有函数已知的$L_2$收敛速率，可转化为更光滑赋范空间$B$中函数加倍的$L_2$收敛速率，同时在$H$范数下获得类似的改进误差界。作为特例，我们改进了再生核Hilbert空间中核插值的已知$L_2$与$H$范数收敛速率，特别关注近期关于格点上周期核基插值应用于参数偏微分方程的研究 (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022)。第二个应用针对一般条件正定基函数的径向基函数插值，同样对更光滑赋范空间$B$中的所有函数，其$L_2$收敛速率可提高一倍，且原生空间范数下的收敛速率获得类似改进。