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

Abstract

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

30 pages