Chebyshev polynomials on a Jordan arc
Benedikt Buchecker, Benjamin Eichinger, Olof Rubin, Aron Wennman
Abstract
We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.
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