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

Abstract

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

29 pages

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