Error Estimates for Approximate Operator Inversion via Kernel-Based Methods

摘要

In this paper we investigate error estimates for the approximate solution of operator equations Af = u, where u needs not to be a function on the same domain as f. We use the well-established theory of generalized interpolation, also known as optimal recovery in reproducing kernel Hilbert spaces, to generate an approximation to / from finitely many samples u(x_1),...,u(x_N). To derive error estimates for this approximation process we will show sampling inequalities on fairly general Riemannian manifolds.