On the approximation properties of Stieltjes polynomials

TL;DR

This paper introduces $g$-polynomials based on Stieltjes integrals, proving their density in uniformly $g$-continuous functions when $g$ has finitely many discontinuities, thus establishing a Weierstrass-type approximation theorem.

Contribution

It extends approximation theory to the framework of Stieltjes calculus by defining $g$-polynomials and proving their density under certain conditions.

Findings

$g$-polynomials are dense in uniformly $g$-continuous functions with finitely many discontinuities in $g$

Established a Weierstrass-type approximation theorem in the context of Stieltjes calculus

Discussed open problems related to the general case of more complex $g$ behaviors

Abstract

We introduce and study the approximation properties of $g$-polynomials, defined as linear combinations of iterated Stieltjes integrals of a constant function. Focusing on the case where the derivator $g$ has finitely many discontinuities, we prove that the space of $g$-polynomials is dense in the space of uniformly $g$-continuous functions. This result establishes a Weierstrass-type approximation theorem within the framework of Stieltjes calculus. The characterization of the closure of the space of $g$-polynomials in the general case, where the derivator may exhibit more complex behavior, remains an open and challenging problem, which we briefly discuss.