  We consider the problem of learning an $\varepsilon$-optimal policy in a
general class of continuous-space Markov decision processes (MDPs) having
smooth Bellman operators. Given access to a generative model, we achieve
rate-optimal sample complexity by performing a simple, \emph{perturbed} version
of least-squares value iteration with orthogonal trigonometric polynomials as
features. Key to our solution is a novel projection technique based on ideas
from harmonic analysis. Our~$\widetilde{\mathcal{O}}(\epsilon^{-2-d/(\nu+1)})$
sample complexity, where $d$ is the dimension of the state-action space and
$\nu$ the order of smoothness, recovers the state-of-the-art result of
discretization approaches for the special case of Lipschitz MDPs $(\nu=0)$. At
the same time, for $\nu\to\infty$, it recovers and greatly generalizes the
$\mathcal{O}(\epsilon^{-2})$ rate of low-rank MDPs, which are more amenable to
regression approaches. In this sense, our result bridges the gap between two
popular but conflicting perspectives on continuous-space MDPs.
