  This paper studies convergence rates for some value function approximations
that arise in a collection of reproducing kernel Hilbert spaces (RKHS)
$H(\Omega)$. By casting an optimal control problem in a specific class of
native spaces, strong rates of convergence are derived for the operator
equation that enables offline approximations that appear in policy iteration.
Explicit upper bounds on error in value function and controller approximations
are derived in terms of power function $\mathcal{P}_{H,N}$ for the space of
finite dimensional approximants $H_N$ in the native space $H(\Omega)$. These
bounds are geometric in nature and refine some well-known, now classical
results concerning convergence of approximations of value functions.
