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Radial basis functions (RBFs) are prominent examples for reproducing kernelswith associated reproducing kernel Hilbert spaces (RKHSs). The convergencetheory for the kernel-based interpolation in that space is well understood andoptimal rates for the whole RKHS are often known. Schaback added the doublingtrick, which shows that functions having double the smoothness required by theRKHS (along with complicated, albeit complicated boundary behavior) can beapproximated with higher convergence rates than the optimal rates for the wholespace. Other advances allowed interpolation of target functions which are lesssmooth, and different norms which measure interpolation error. The currentstate of the art of error analysis for RBF interpolation treats targetfunctions having smoothness up to twice that of the native space, but errormeasured in norms which are weaker than that required for membership in theRKHS.  Motivated by the fact that the kernels and the approximants they generate aresmoother than required by the native space, this article extends the doublingtrick to error which measures higher smoothness. This extension holds for afamily of kernels satisfying easily checked hypotheses which we describe inthis article, and includes many prominent RBFs. In the course of the proof, newconvergence rates are obtained for the abstract operator considered by Devoreand Ron, and new Bernstein estimates are obtained relating high ordersmoothness norms to the native space norm.

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms