Error bounds for numerical differentiation using kernels of finite smoothness

TL;DR

This paper derives improved error bounds for kernel-based numerical differentiation methods that account for the smoothness of kernels and relax standard assumptions on point sets, enhancing accuracy estimates.

Contribution

It introduces new error bounds that incorporate kernel derivative smoothness and apply to deficient point sets, advancing the theoretical understanding of kernel-based differentiation.

Findings

Enhanced error bounds considering kernel smoothness

Applicability to deficient point sets

Improved order of estimate compared to previous results

Abstract

We provide improved error bounds for kernel-based numerical differentiation in terms of growth functions when kernels are of a finite smoothness, such as polyharmonic splines, thin plate splines or Wendland kernels. In contrast to existing literature, the new estimates take into account the H\"older class smoothness of kernel's derivatives, which helps to improve the order of the estimate. In addition, the new estimates apply to certain deficient point sets, relaxing a standard assumption that an approximation with conditionally positive definite kernels must rely on determining sets for polynomials.