Extremal Polynomials and Sets of Minimal Capacity

Abstract

This article examines the asymptotic behavior of the Widom factors, denoted \({\mathcal {W}}_n\), for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom’s proposal in Widom (Adv Math 3:127–232, 1969), when dealing with a single smooth Jordan arc, \({\mathcal {W}}_n\) converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval \([-2,2]\), and we provide a complete description of the asymptotic behavior of \({\mathcal {W}}_n\) for symmetric star graphs and quadratic preimages of \([-2,2]\). We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential extension of the conjecture posed in Christiansen et al. (Oper Theory Adv Appl 289:301–319, 2022). Lastly, we propose a possible connection between the S-property and Widom factors converging to 2.