Extremal polynomials on a Jordan arc

Abstract

Let Γ be a C2+ Jordan arc and let Γ0 be the open arc which consists of interior points of Γ. We find concrete upper and lower bounds for the limit of Widom factors for L2(μ) extremal polynomials on Γ which was given in Widom (1969). In addition, we show that the upper bound for the limit supremum of Widom factors for the weighted Chebyshev polynomials which was obtained in Widom (1969) can be improved once two normal derivatives of the Green function do not agree at one point z∈Γ0. We also show that if Γ0 is not analytic then we have improved upper bounds.