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arXiv · 1505.02604

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

Jacob S. Christiansen·Barry Simon·Maxim Zinchenko
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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.

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Jacob S. Christiansen, Barry Simon, Maxim Zinchenko. 2015-05-11. Asymptotics of Chebyshev Polynomials, I. Subsets of $\mathbb{R}$. https://doi.org/10.1007/s00222-016-0689-x

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