On the theorem of M.Golomb

TL;DR

This paper proves that M. Golomb's formula for the approximation error of functions on product spaces by sums of functions on individual factors is correct in a stronger form, resolving a long-standing open question.

Contribution

The paper establishes the validity of Golomb's approximation error formula in a stronger form, confirming its correctness despite previous doubts.

Findings

Golomb's formula is valid in a stronger form.

Resolved the open question about the formula's correctness.

Strengthened the theoretical foundation of approximation on product spaces.

Abstract

Let $X_{1},...,X_{n}$ be compact spaces and $X=X_{1}\times ... \times X_{n}.$ Consider the approximation of a function $f\in C(X)$ by sums $g_{1}(x_{1})+... g_{n}(x_{n}),$ where $g_{i}\in C(X_{i}),$ $i=1,...,n.$ In [8], M.Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of $X$, called "projection cycles". However, his proof had a gap, which was pointed out by Marshall and O'Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb's formula holds in a stronger form.