Bases for conditionally positive definite kernels

摘要

This paper extends a previous one by Pazouki and Schaback (2011) [2] to the important case of conditionally positive kernels such as thin-plate splines or polyharmonic kernels. The goal is to construct well-behaving bases for interpolation on a finite set X⊂Rd by translates K(⋅,x) for x∈X of a fixed kernel K:Ω×Ω→R which is conditionally positive definite of order m>0. Particularly interesting cases are bases of Lagrange or Newton type, and bases which are orthogonal or orthonormal either discretely (i.e. via their function values on X) or as elements of the underlying “native” space for the given kernel, which is a direct sum of a Hilbert space with the space Pmd of d-variate polynomials of order up to m. All of these cases are considered, and relations between them are established. It turns out that there are many more possibilities for basis construction than in the unconditionally positive definite situation m=0, and these possibilities are sorted out systematically. Some numerical examples are provided.