A Chebyshev type alternation theorem for best approximation by a sum of two algebras

TL;DR

This paper extends the Chebyshev alternation theorem to the problem of best approximation of continuous functions on compact spaces by sums of two subalgebras, providing a theoretical foundation for such approximation problems.

Contribution

It establishes a Chebyshev type alternation theorem for best approximation by sums of two closed subalgebras of continuous functions, generalizing classical results.

Findings

Proves a Chebyshev type alternation theorem for sums of two algebras.

Characterizes best approximations in the sum of two subalgebras.

Provides theoretical conditions for optimal approximation in this setting.

Abstract

Let $X$ be a compact metric space, $C(X)$ be the space of continuous real-valued functions on $X$, and $A_1$, $A_2$ be two closed subalgebras of $C(X)$ containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_1+A_2$. We prove a Chebyshev type alternation theorem for a function $u_0\in A_1+A_2$ to be a best approximation to $f$.