Constructive decomposition of a function of two variables as a sum of functions of one variable

TL;DR

This paper provides a constructive method to decompose continuous functions on certain planar sets into sums of functions of one variable, advancing understanding of function decomposition and basic embeddings in topology.

Contribution

It offers a constructive proof for decomposing functions on specific planar sets, extending Sternfeld's theorem with an explicit approximation approach.

Findings

Decomposition of functions into sums of one-variable functions on special sets

Constructive approximation method for function decomposition

Advancement in understanding basic embeddings in the plane

Abstract

Given a compact set $K$ in the plane, which does not contain any triple of points forming a vertical and a horizontal segment, and a map $f\in C(K)$, we give a construction of functions $g,h\in C(\mathbb R)$ such that $f(x,y)=g(x)+h(y)$ for all $(x,y)\in K$. This provides a constructive proof of a part of Sternfeld's theorem on basic embeddings in the plane. In our proof the set $K$ is approximated by a finite set of points.