On the Diliberto-Straus algorithm for the uniform approximation by a sum of two algebras

TL;DR

This paper analyzes the convergence of the Diliberto-Straus algorithm for uniform approximation of functions by sums of two algebras, extending its applicability to a broad class of compact convex sets.

Contribution

The paper proves the convergence of the Diliberto-Straus algorithm for continuous functions approximated by sums of two closed algebras on compact Hausdorff spaces.

Findings

Convergence of the Diliberto-Straus algorithm under certain assumptions.

Extension of the original result to a large class of compact convex sets.

Applicability to approximation by sums of univariate functions.

Abstract

In 1951, Diliberto and Straus proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto-Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto-Straus's original result holds for a large class of compact convex sets.