Extremal polynomials and polynomial preimages

TL;DR

This paper investigates the asymptotic behavior of Widom factors for Chebyshev polynomials on complex sets, revealing new conditions for convergence and connections to polynomial preimages and potential theory.

Contribution

It provides a complete description of Widom factor asymptotics for polynomial preimages of $[-2,2]$, especially for star graphs and quadratic preimages, and explores their relation to the $S$-property.

Findings

Widom factors converge to 2 only for straight line segments.

Asymptotics of Chebyshev and orthogonal polynomials coincide on star graphs.

Proposes a link between the $S$-property and Widom factors converging to 2.

Abstract

This article examines the asymptotic behavior of the Widom factors, denoted $\mathcal{W}_n$, for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom's proposal, when dealing with a single smooth Jordan arc, $\mathcal{W}_n$ converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval $[-2,2]$, and we provide a complete description of the asymptotic behavior of $\mathcal{W}_n$ for symmetric star graphs and quadratic preimages of $[-2,2]$. We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential extension of a conjecture posed by Christiansen, Simon and Zinchenko. Lastly, we propose a possible connection between the $S$-property and Widom factors converging to $2$.