Error bounds for numerical differentiation using kernels of finite smoothness

O.V. Davydov

doi:10.15421/242514

Error bounds for numerical differentiation using kernels of finite smoothness

O.V. Davydov (Justus Liebig University Giessen), https://orcid.org/0000-0001-8813-9485

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 Hölder 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.

Keywords

numerical differentiation; meshless methods; kernel-based method; growth function; deficient sets; polyharmonic splines; thin plate splines; Wendland kernels

MSC 2020

Pri 65D25, Sec 65D12, 41A30, 41A80, 41A63

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