Approximation by polynomials with only real critical points

TL;DR

This paper enhances the Weierstrass approximation theorem by showing that any continuous real-valued function on an interval can be uniformly approximated by polynomials with critical points confined to that interval, using Chebyshev polynomials and Brouwer's fixed point theorem.

Contribution

It introduces a novel approximation method employing polynomials with critical points only within the interval, strengthening classical approximation results.

Findings

Polynomials with critical points only in the interval can approximate any continuous function.

The proof combines perturbed Chebyshev polynomials with Brouwer's fixed point theorem.

The result broadens the understanding of polynomial approximation constraints.

Abstract

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical points are contained in $I$. The proof uses a perturbed version of the Chebyshev polynomials and an application of the Brouwer fixed point theorem.