Extremal Polynomials and Sets of Minimal Capacity

Abstract

AbstractThis article examines the asymptotic behavior of the Widom factors, denoted $${\mathcal {W}}_n$$ 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, $${\mathcal {W}}_n$$ 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 $$[-2,2]$$ [ - 2 , 2 ] , and we provide a complete description of the asymptotic behavior of $${\mathcal {W}}_n$$ W n for symmetric star graphs and quadratic preimages of $$[-2,2]$$ [ - 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.