Simultaneous Approximation of a Multivariate Function and its Derivatives by Multilinear Splines

TL;DR

This paper investigates the simultaneous approximation of multivariate functions and their derivatives using multilinear splines, providing formulas for uniform approximation errors within specific function classes.

Contribution

It introduces formulas for uniform approximation errors of functions and derivatives by multilinear splines, advancing understanding of spline approximation accuracy.

Findings

Derived formulas for approximation errors

Applicable to functions with bounded moduli of continuity

Enhances spline approximation theory

Abstract

In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation error on classes of functions with moduli of continuity bounded above by certain majorants.