Szego\H o-Widom asymptotics of Chebyshev Polynomials on Circular Arcs

Abstract

Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called $\textrm{Szeg\H o}$-Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian the analytic capacity in an arbitrary domain is the diagonal of the corresponding $\textrm{Szeg\H o}$ kernel. We don't know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question.