Abstract:
We present a conjecture that the 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 the asymptotics for Ahlfors extremal polynomials in the complement to a system of intervals on $\mathbb{R}$, arcs on $\mathbb{T}$, and the asymptotics of the extremal entire functions for the continuous counterpart of this problem.
Bibliography: 35 titles.
Keywords:Chebyshev polynomial, analytic capacity, hyperelliptic Riemann surface, Abel-Jacobi inversion, complex Green's and Martin functions, reproducing kernel.
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