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Speaker: Benjamin Eichinger
Johannes Kepler University Linz

Geometry-Analysis Seminar: Chebyshev Problems on a Circular Arc

Wednesday, February 15, 2017
4:00 PM  to 5:00 PM

227  Herman Brown Hall
Rice University
6100 Main St
Houston, Texas, USA

We consider the Chebyshev polynomials, $T_n$, on a circular arc $A_\alpha$, i.e., the monic polynomials of degree at most $n$ minimizing the sup-norm, $ \|T_n\|_{A_\alpha}. $ Thiran and Detaille found an explicit formula for the asymptotics of $\|T_n\|_{A_\alpha}$, which disproved a conjecture of Widom. We give the Szeg\H o-Widom asymptotics of the domain explicitly. That is, the limit of the properly normalized extremal functions $T_n$. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on $A_\alpha$. Our computations show that in the proper normalization the limit of the upper envelope is the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Similarly, we study the \textit{polynomial Ahlfors problem}, i.e., we maximize $|P'(z)|$ at a fixed point $z\in\mathbb C\setminus A_\alpha$ in the class of polynomials of degree at most $n$ that are uniformly bounded on $A_\alpha$ and vanish at the given point. The corresponding asymptotics can again be given in terms of an explicit reproducing kernel.

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