Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals

TL;DR

This paper introduces a new method to derive Bernstein asymptotics for the best polynomial approximation of |x|^p and Laurent polynomial approximation of sgn(x) on symmetric intervals, providing explicit asymptotics for the polynomials.

Contribution

A novel approach that directly proves Bernstein asymptotics and extends results to Laurent polynomial approximation of sgn(x).

Findings

Derived Bernstein asymptotics for |x|^p approximation.

Obtained asymptotics for approximating Laurent polynomials of sgn(x).

Provided explicit asymptotic formulas for the polynomials.

Abstract

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of $|x|^p$ on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of $\sgn(x)$ on $[-1,-a]\cup[a,1]$ by Laurent polynomials.