A note on the approximation by a sum of two algebras

TL;DR

This paper investigates how well continuous functions on compact metric spaces can be approximated by sums of two algebras, providing bounds and formulas for the approximation error.

Contribution

It introduces a de la Vallée Poussin type theorem and a duality formula for the approximation error by sums of two algebras.

Findings

Establishes a lower bound for approximation error.

Derives a duality formula for exact error computation.

Extends approximation theory to sums of two algebras.

Abstract

We consider the problem of approximation of a continuous function $f$ defined on a compact metric space $X$ by elements from a sum of two algebras. We prove a de la Vall\'{e}e Poussin type theorem, which estimates the approximation error $E(f)$ from below. We also obtain a duality formula for the precise computation of $E(f)$.