Piecewise linear interpolation via kernels

TL;DR

This paper explores piecewise linear interpolation through kernel methods, revealing its connection to Green kernels of PDEs and establishing convergence rates matching classical spline results.

Contribution

It introduces a kernel perspective on piecewise linear interpolation, linking it to PDE Green kernels and deriving convergence rates for functions in Sobolev spaces.

Findings

Kernel-based superconvergence yields classical spline convergence rates

Piecewise linear kernels are Green kernels for specific PDEs

Rates of approximation depend on Sobolev space smoothness

Abstract

We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise to piecewise linear interpolation. We show that such kernels are Green kernels for certain second-order partial differential equations and use kernel-based superconvergence theory to obtain rates of convergence for approximation of functions lying in $W_2^s(0, 1)$ for $s \in [1, 2]$. The rates coincide with classical rates for linear splines.