Extremal polynomials on a Jordan arc

TL;DR

This paper establishes new bounds for Widom factors of extremal polynomials on Jordan arcs, improving previous results by considering Green function derivatives and the arc's analyticity.

Contribution

It provides concrete upper and lower bounds for Widom factors on Jordan arcs, refining earlier estimates by analyzing Green function derivatives and arc regularity.

Findings

Derived bounds for Widom factors on Jordan arcs.

Improved upper bounds when Green function derivatives differ.

Enhanced bounds for non-analytic arcs.

Abstract

Let $\Gamma$ be a $C^{2+}$ Jordan arc and let $\Gamma_0$ be the open arc which consists of interior points of $\Gamma$. We find concrete upper and lower bounds for the limit of Widom factors for $L_2(\mu)$ extremal polynomials on $\Gamma$ which was given in [18]. In addition, we show that the upper bound for the limit supremum of Widom factors for the weighted Chebyshev polynomials which was obtained in [18] can be improved once two normal derivatives of the Green function do not agree at one point $z\in \Gamma_0$. We also show that if $\Gamma_0$ is not analytic then we have improved upper bounds.