A Stone-Weierstrass approximation theorem for monotone functions

TL;DR

This paper establishes a Stone-Weierstrass type theorem for continuous non-decreasing functions on compact preordered spaces, providing new algebraic characterizations and explicit approximation formulas for such functions.

Contribution

It introduces a novel approximation theorem for monotone functions and characterizes their function spaces algebraically, extending classical approximation results.

Findings

Positive non-decreasing rational functions can uniformly approximate all continuous non-decreasing functions on compact intervals.

An explicit approximation formula for these functions is provided.

The theorem applies to functions on compact preordered spaces, broadening classical approximation theory.

Abstract

We present an approximation theorem for continuous non-decreasing functions on compact preordered spaces, leading to an algebraic characterization of their corresponding function spaces. As an application, we prove that the family of positive non-decreasing rational functions with non-negative coefficients can uniformly approximate all continuous non-decreasing functions on compact intervals. An explicit approximation formula of this type is provided.