Best approximation of a three-variable function by sum of one-variable coordinate functions

TL;DR

This paper derives a formula to compute the best uniform approximation error of a continuous three-variable function by the sum of three single-variable continuous functions, under specific conditions.

Contribution

It provides a new explicit formula for calculating the approximation error in a three-variable function approximation problem.

Findings

Derived a formula for the approximation error E(f,Ω).

Applicable under certain conditions on the function f.

Enhances understanding of function approximation by sums of univariate functions.

Abstract

Consider the following approximation problem of a continuous function of three variables by the sum of three continuous functions of one variable: \[ E(f,\Omega)=\inf||f(x,y,z)-\phi(x)-\psi(y)-\omega(z)||_{\infty}=? \] where $f(x,y,z)$ is a given continuous function defined on $\Omega=[0,1]^3$ and the infimum runs over all triplets of continuous functions $\phi(x),\psi(y),\omega(z)$ defined on the unit interval $[0,1]$. In this paper, we will prove a formula to calculate the error $E(f,\Omega)$ under certain conditions.