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Thiran and Detaille give an explicit formula for the asymptotics of thesup-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., explicitexpressions for the asymptotics of the corresponding extremal polynomials.Moreover, we solve a similar problem with respect to the upper envelope of afamily of polynomials uniformly bounded on this arc. That is, we give explicitformulas for the asymptotics of the error of approximation as well as of theextremal functions. Our computations show that in the proper normalization thelimit of the upper envelope represents the diagonal of a reproducing kernel ofa certain Hilbert space of analytic functions. Due to Garabedian the analyticcapacity 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 respectto upper envelopes of polynomials. If this is a general fact or a specificproperty of the given domain, we rise as an open question.

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