Szeg\H{o}'s Condition on Compact subsets of $\mathbb{C}$

TL;DR

This paper investigates Szeg"H{o} conditions on compact subsets of the complex plane and their implications for orthogonal polynomials, linking measure properties to geometric conditions like Parreau-Widom.

Contribution

It establishes a Szeg"H{o} condition criterion that guarantees a positive lower bound for normalized polynomial norms and relates unboundedness of these norms to the Parreau-Widom condition.

Findings

Szeg"H{o} condition implies positive lower bound for polynomial norms

Norms are at least 1 for equilibrium measure on any non-polar compact set

Unbounded norms imply the Parreau-Widom condition under additional assumptions

Abstract

Let $K$ be a non-polar compact subset of $\mathbb{C}$ and $\mu_K$ be its equilibrium measure. Let $\mu$ be a unit Borel measure supported on a compact set which contains the support of $\mu_K$. We prove that a Szeg\H{o} condition in terms of the Radon-Nikodym derivative of $\mu$ with respect to $\mu_K$ implies that $$\inf_n \frac{\|P_n(\cdot;\mu)\|_{L^2(\mathbb{C};\mu)}}{\mathrm{Cap}(K)^n}>0.$$ We show that $\frac{\|P_n(\cdot;\mu_K)\|_{L^2(\mathbb{C};\mu_K)}}{\mathrm{Cap}(K)^n}\geq 1$ for any compact non-polar set $K$. We also prove that under an additional assumption, unboundedness of the sequence $\left(\frac{\|P_n(\cdot;\mu_K)\|_{L^2(\mathbb{C};\mu_K)}}{\mathrm{Cap}(K)^n}\right)$ implies that $K$ satisfies the Parreau-Widom condition.