Approximation with Conditionally Positive Definite Kernels on Deficient Sets

TL;DR

This paper explores how conditionally positive definite kernels can be used for function approximation on non-determining sets, ensuring optimal recovery properties and enabling applications like sparse differentiation formulas and accurate function approximation.

Contribution

It demonstrates that polynomial consistency suffices for kernel-based approximation on deficient sets, extending the applicability of kernel methods beyond determining sets.

Findings

Polynomial consistency ensures kernel-based approximation on deficient sets.

Application of kernels to generate sparse numerical differentiation formulas.

Accurate function approximation on complex geometries like ellipses.

Abstract

Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define kernel-based numerical approximation of the functional with usual properties of optimal recovery. Application examples include generation of sparse kernel-based numerical differentiation formulas for the Laplacian on a grid and accurate approximation of a function on an ellipse.