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Piecewise linear interpolation via kernels

Toni Karvonen, Gabriele Santin, Tizian Wenzel

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.

Piecewise linear interpolation via kernels

Abstract

We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space 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 for . The rates coincide with classical rates for linear splines.
Paper Structure (9 sections, 10 theorems, 43 equations)

This paper contains 9 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Piecewise linear kernels as reproducing kernels of $W_2^1(0,1)$
  3. Examples of piecewise linear kernels covered by \ref{['thm:sobolev-1-space']}
  4. Examples of piecewise linear kernels as limits of \ref{['thm:sobolev-1-space']}
  5. Piecewise linear kernels as Green kernels
  6. Convergence theory
  7. Existing convergence theory
  8. Piecewise linear interpolation
  9. Trapezoidal rule

Key Result

Theorem 2

Let $0 = x_0 < x_1 < \cdots x_{n-1} < x_n = 1$. If $K$ is a 2-piecewise linear positive-semidefinite kernel on $[0, 1]$, then $P_n f = L_n f$ for every $f \in \mathcal{H}$.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Corollary 5
  • proof
  • Theorem 6
  • Proposition 7
  • ...and 7 more