  This paper derives error bounds for regression in continuous time over
subsets of certain types of Riemannian manifolds.The regression problem is
typically driven by a nonlinear evolution law taking values on the manifold,
and it is cast as one of optimal estimation in a reproducing kernel Hilbert
space (RKHS). A new notion of persistency of excitation (PE) is defined for the
estimation problem over the manifold, and rates of convergence of the
continuous time estimates are derived using the PE condition. We discuss and
analyze two approximation methods of the exact regression solution. We then
conclude the paper with some numerical simulations that illustrate the
qualitative character of the computed function estimates. Numerical results
from function estimates generated over a trajectory of the Lorenz system are
presented. Additionally, we analyze an implementation of the two approximation
methods using motion capture data.
