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A Stone-Weierstrass approximation theorem for monotone functions

Ettore Minguzzi

TL;DR

The paper develops a Stone–Weierstrass–type theorem for approximating continuous isotone functions on compact preordered spaces using a nonempty convex cone S with a single φ-invariance property, under the condition that S generates the preorder. It provides an algebraic characterization of C^+(X,≤,ℝ) via a cone in a commutative unital C*-algebra and establishes a duality between compact ordered spaces and algebraic data (B,C) through Gelfand theory. A concrete Bernstein-type construction yields explicit rational approximants with non-negative coefficients that uniformly approximate all continuous isotone functions on compact intervals, along with a rigorous monotonicity and convergence proof. The work unifies algebraic and functional-analytic perspectives and points toward applications in spacetime causality and potential links to quantum gravity.

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.

A Stone-Weierstrass approximation theorem for monotone functions

TL;DR

The paper develops a Stone–Weierstrass–type theorem for approximating continuous isotone functions on compact preordered spaces using a nonempty convex cone S with a single φ-invariance property, under the condition that S generates the preorder. It provides an algebraic characterization of C^+(X,≤,ℝ) via a cone in a commutative unital C*-algebra and establishes a duality between compact ordered spaces and algebraic data (B,C) through Gelfand theory. A concrete Bernstein-type construction yields explicit rational approximants with non-negative coefficients that uniformly approximate all continuous isotone functions on compact intervals, along with a rigorous monotonicity and convergence proof. The work unifies algebraic and functional-analytic perspectives and points toward applications in spacetime causality and potential links to quantum gravity.

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.

Paper Structure

This paper contains 4 sections, 13 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Algebraic characterization of $C^+(X,\le,\mathbb{R})$
  3. Bernstein's type realization of the rational approximation
  4. Conclusions

Key Result

Theorem 1.1

Let $(X,\mathscr{T},\le)$ be a compact preordered space. Let $S\subset C(X, \le, \mathbb{R})$ be a uniformly closed type-I semi-algebra with identity such that $\le=\le_S$, then $S=C^+(X,\le, \mathbb{R})$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 1.8
  • proof
  • Lemma 1.9
  • ...and 21 more