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

Miliczká, Eva

doi:10.1090/S0002-9939-08-09528-2

Constructive decomposition of a function of two variables as a sum of functions of one variable
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Proc. Amer. Math. Soc. 137 (2009), 607-614 Request permission

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 for 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.

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