Kernel approximation beyond the native space -- with applications to approximation on manifolds

Abstract

This article treats kernel approximation and interpolation on embedded manifolds of $\mathbb{R}^N$using restrictions of positive and conditionally positive definite kernels. The main challenge is to develop an approximation theory that treats error measured in highly regular smoothness spaces relative to the kernel. This means that the order of smoothness is higher than that of the kernel's associated native space (in the positive definite case, the reproducing kernel Hilbert space generated by the kernel). This prevents the use of standard techniques for controlling error in this setting, especially RKHS space arguments like orthogonality of the interpolation projector, or bounds using the {\em power function}. We generalize an approximation scheme introduced by DeVore and Ron which treats target functions that are in the range of the kernel's integral operator. In the case of embedded manifolds, this generalization is now feasible due to recently developed local polynomial reproductions for certain submanifolds of $\mathbb{R}^N$. Furthermore, we give sufficient conditions on kernel and manifold which allow the range of the integral operator to be precisely identified: in particular, guaranteeing that the range is a Sobolev space. Finally, we provide new kernel-based Bernstein inequalities for embedded manifolds which lead to estimates for interpolation in Sobolev spaces compactly contained in the native space.