  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 general 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 $B$.
