On the Widom factors for [formula omitted] extremal polynomials

Abstract

We continue our study of the Widom factors for Lp(μ) extremal polynomials initiated in (Alpan and Zinchenko, 2020). In this work we characterize sets for which the lower bounds obtained in (Alpan and Zinchenko, 2020) are saturated, establish continuity of the Widom factors with respect to the measure μ, and show that despite the lower bound [W2,n(μK)]2≥2S(μK) for the equilibrium measure μK on a compact set K⊂R the general lower bound [Wp,n(μ)]p≥S(μ) is optimal even for measures dμ=wdμK with polynomial weights w on K⊂R. We also study pull-back measures under polynomial pre-images introduced in (Geronimo and Van Assche,1988), (Peherstorfer and Steinbauer, 2001) and obtain invariance of the Widom factors for such measures. Lastly, we study in detail the Widom factors for orthogonal polynomials with respect to the equilibrium measure on a circular arc and, in particular, find their limit, infimum, and supremum and show that they are strictly monotone increasing with the degree and strictly monotone decreasing with the length of the arc.