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The reproducing kernel Hilbert space (RKHS) embedding method is a recentlyintroduced estimation approach that seeks to identify the unknown or uncertainfunction in the governing equations of a nonlinear set of ordinary differentialequations (ODEs). While the original state estimate evolves in Euclidean space,the function estimate is constructed in an infinite-dimensional RKHS that mustbe approximated in practice. When a finite-dimensional approximation isconstructed using a basis defined in terms of shifted kernel functions centeredat the observations along a trajectory, the RKHS embedding method can beunderstood as a data-driven approach. This paper derives sufficient conditionsthat ensure that approximations of the unknown function converge in a Sobolevnorm over a submanifold that supports the dynamics. Moreover, the rate ofconvergence for the finite-dimensional approximations is derived in terms ofthe fill distance of the samples in the embedded manifold. Numerical simulationof an example problem is carried out to illustrate the qualitative nature ofconvergence results derived in the paper.

Approximations of the Reproducing Kernel Hilbert Space (RKHS) Embedding Method over Manifolds