Some Sharp Error Bounds for Multivariate Linear Interpolation and Extrapolation

TL;DR

This paper investigates the error bounds of multivariate linear interpolation and extrapolation, providing sharp bounds under Lipschitz gradient assumptions and analyzing specific cases like quadratic functions and bivariate extrapolation.

Contribution

It introduces new sharp error bounds for multivariate linear interpolation and extrapolation under Lipschitz gradient conditions, with detailed analysis of quadratic and bivariate cases.

Findings

Derived upper bounds for interpolation and extrapolation errors.

Identified conditions for bounds to be sharp.

Analyzed errors for quadratic functions and bivariate extrapolation.

Abstract

We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the function has Lipschitz continuous gradient and is interpolated on an affinely independent sample set. Errors for quadratic functions and errors of bivariate linear extrapolation are analyzed in depth.