Ahlfors problem for polynomials

TL;DR

This paper conjectures that asymptotics of Chebyshev polynomials in complex domains relate to reproducing kernels of specific Hilbert spaces, supported by asymptotic analysis of Ahlfors extremal polynomials in various geometric settings.

Contribution

It introduces a conjecture linking Chebyshev polynomial asymptotics to reproducing kernels, supported by new asymptotic results for Ahlfors extremal polynomials.

Findings

Asymptotics for Ahlfors extremal polynomials are studied in various geometric configurations.

Supports the conjecture relating polynomial asymptotics to Hilbert space kernels.

Provides new insights into polynomial approximation in complex domains.

Abstract

We raise a conjecture that asymptotics for Chebyshev polynomials in a complex domain can be given in terms of the reproducing kernels of a suitable Hilbert space of analytic functions in this domain. It is based on two classical results due to Garabedian and Widom. To support this conjecture we study asymptotics for Ahlfors extremal polynomials in the complement to a system of intervals on $\mathbb R$, arcs on $\mathbb T$, and its continuous counterpart.