Piecewise linear interpolation via kernels

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.

To appear in the proceedings of ENUMATH 2025