Refined error estimates for Green kernel-based interpolation

Abstract

Positive-definite kernels are probably best known for their application in many problems driven by scattered data interpolation. Fasshauer and Ye introduced constructive theory of reproducing kernels of generalized Sobolev spaces in 2011 to provide insight into the types of functions being well approximated by these kernels on a set of scattered points. In this approach, the reproducing kernel is viewed as the Green kernel of a suitable differential operator with some boundary conditions. Sampling inequalities and the minimum norm property in reproducing kernel Hilbert spaces (RKHSs) bring out the standard error bound; however, this estimate is valid only when the target functions belong to the native spaces of the Green kernels. In this paper we provide Sobolev-type error estimates for cases in which the target functions are smoother than functions in the native space. The results are useful and effective for the error analysis of Green kernel-based interpolation problems.