On the Widom factors for $L_p$ extremal polynomials

TL;DR

This paper advances the understanding of Widom factors for $L_p$ extremal polynomials by characterizing sets where bounds are saturated, analyzing measure continuity, and exploring invariance and limits for specific measures and geometric configurations.

Contribution

It provides new characterizations of sets saturating bounds, proves continuity of Widom factors with respect to measures, and investigates invariance under polynomial pre-images and specific geometric cases.

Findings

Characterized sets saturating lower bounds for Widom factors.

Proved continuity of Widom factors with respect to the measure.

Analyzed Widom factors for orthogonal polynomials on circular arcs, including limits and monotonicity.

Abstract

We continue our study of the Widom factors for $L_p(\mu)$ extremal polynomials initiated in [4]. In this work we characterize sets for which the lower bounds obtained in [4] are saturated, establish continuity of the Widom factors with respect to the measure $\mu$, and show that despite the lower bound $[W_{2,n}(\mu_K)]^2\geq 2S(\mu_K)$ for the equilibrium measure $\mu_K$ on a compact set $K\subset\mathbb R$ the general lower bound $[W_{p,n}(\mu)]^p\geq S(\mu)$ is optimal even for measures $d\mu=wd\mu_K$ with polynomial weights $w$ on $K\subset\mathbb R$. We also study pull-back measures under polynomial pre-images introduced in [16, 23] 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.