Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick, which shows that functions having double the smoothness required by the RKHS (along with complicated, but well understood boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS. Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron, and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.

翻译：径向基函数（RBF）是再生核希尔伯特空间（RKHS）中再生核的典型示例。该空间中基于核的插值收敛理论已得到充分理解，且整个RKHS的最优收敛速率通常已知。Schaback引入了加倍技巧，表明具有RKHS所需光滑度两倍（以及复杂但易理解的边界行为）的函数，能够以高于整个空间最优速率的收敛速度被逼近。其他进展允许对光滑性较弱的目标函数进行插值，并采用不同范数度量插值误差。当前RBF插值误差分析的前沿，涉及光滑性不超过原生空间两倍的目标函数，但误差度量采用弱于RKHS成员资格要求的范数。鉴于核及其生成的逼近函数比原生空间所需更加光滑，本文将加倍技巧扩展到度量更高光滑性的误差。该扩展适用于满足本文所述易检验假设的一类核族，并包含多种常见RBF。在证明过程中，获得了Devore和Ron所考虑抽象算子的新收敛速率，并建立了将高阶光滑性范数与原生空间范数相联系的新Bernstein估计。