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, albeit complicated 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）是再生核及其关联再生核希尔伯特空间（RKHS）的典型范例。该空间中基于核的插值收敛理论已得到充分发展，且全域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（虽伴随复杂的边界行为）的函数，能以高于全域空间最优速率的速度逼近。其他进展允许对光滑度较低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插值误差分析的先进水平可处理光滑度不超过原生空间两倍的目标函数，但误差度量采用弱于RKHS隶属条件要求的范数。鉴于核及其生成的逼近函数比原生空间所需更光滑，本文将此加倍技巧推广至度量更高光滑度的误差。该推广适用于满足本文所述易验证假设的一类核族，涵盖多种主流RBF。在证明过程中，针对Devore与Ron考虑的抽象算子获得了新的收敛速率，并建立了联系高阶光滑性范数与原生空间范数的新Bernstein估计。