  The reproducing kernel Hilbert space (RKHS) embedding method is a recently
introduced estimation approach that seeks to identify the unknown or uncertain
function in the governing equations of a nonlinear set of ordinary differential
equations (ODEs). While the original state estimate evolves in Euclidean space,
the function estimate is constructed in an infinite-dimensional RKHS that must
be approximated in practice. When a finite-dimensional approximation is
constructed using a basis defined in terms of shifted kernel functions centered
at the observations along a trajectory, the RKHS embedding method can be
understood as a data-driven approach. This paper derives sufficient conditions
that ensure that approximations of the unknown function converge in a Sobolev
norm over a submanifold that supports the dynamics. Moreover, the rate of
convergence for the finite-dimensional approximations is derived in terms of
the fill distance of the samples in the embedded manifold. Numerical simulation
of an example problem is carried out to illustrate the qualitative nature of
convergence results derived in the paper.
