Asymptotics of Chebyshev Polynomials, I. Subsets of $\mathbb{R}$

TL;DR

This paper proves a long-standing conjecture about the asymptotic behavior of Chebyshev polynomials on finite gap subsets of the real line and establishes new upper bounds for infinite component sets satisfying the Parreau-Widom condition.

Contribution

It resolves Widom's conjecture for finite gap sets and provides the first upper bounds for Chebyshev norms on certain infinite gap sets.

Findings

Confirmed Szeg\

Widom asymptotics for finite gap sets.

Established upper bounds for Chebyshev norms on infinite sets with Parreau-Widom condition.

Abstract

We consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets $\frak{e} \subset \mathbb{R}$ (that is, the monic polynomials minimizing the sup-norm, $\Vert T_n \Vert_{\frak{e}}$, on $\frak{e}$). We resolve a $45+$ year old conjecture of Widom that for finite gap subsets of $\mathbb{R}$, his conjectured asymptotics (which we call Szeg\H{o}-Widom asymptotics) holds. We also prove the first upper bounds of the form $\Vert T_n \Vert_{\frak{e}} \leq Q C({\frak{e}})^n$ (where $C(\frak{e})$ is the logarithmic capacity of $\frak{e}$) for a class of $\frak{e}$'s with an infinite number of components, explicitly for those $\frak{e} \subset \mathbb{R}$ that obey a Parreau-Widom condition.