  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.
