Szegő-Widom asymptotics of Chebyshev polynomials on circular arcs

We review the main results of the seminal paper of Widom [2] on asymptotics of or-thogonal and Chebyshev polynomials associated with a set E (i.e., the monic polynomialsof degree at most n that minimize the sup-norm‖Tn‖E), where E is a system of Jordanregions and arcs. Thiran and Detaille [1], considered the Chebyshev polynomials Tn on acircular arc Aα and managed to find an explicit formula for the asymptotics of the extremalvalue‖Tn‖Aα , disproving a conjecture of Widom stated in the aforementioned paper. Wegive the Szegő-Widom asymptotics of the domain C \ Aα explicitly, i.e., the limit of theproperly normalized extremal functions Tn. Moreover, we solve a similar problem withrespect to the upper envelope of a family of polynomials uniformly bounded on Aα. Ourcomputations show that in the proper normalization the limit of the upper envelope isthe diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. References1. J.-P. Thiran and C. Detaille, Chebyshev polynomials on circular arcs in the complex plane, Progressin approximation theory, Academic Press, Boston, MA, 1991, pp. 771–786.2. H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Advancesin Math. 3 (1969), 127–232. Benjamin Eichinger,Johannes Kepler University, Linz,A-4040 Linz, [email protected]