A simpler proof of Sternfeld’s Theorem

Abstract

In Sternfeld’s work on Kolmogorov’s Superposition Theorem appeared the combinatorial–geometric notion of a basic set and a certain kind of arrays. A subset $X\subset {ℝ}^n$ is basic if any continuous function $X\to ℝ$ could be represented as the sum of compositions of continuous functions $ℝ\to ℝ$ and projections to the coordinate axes.

The definition of a Sternfeld array is presented in this paper.

Sternfeld’s Arrays Theorem.If a closed bounded subset$X\subset {ℝ}^{2n}$ contains Sternfeld arrays of arbitrary large size then$X$ is not basic.

The paper provides a simpler proof of this theorem.