On the zeros of asymptotically extremal polynomial sequences in the plane

TL;DR

This paper establishes geometric conditions on a compact set in the complex plane that ensure the zeros of asymptotically extremal polynomials distribute according to the equilibrium measure, enhancing understanding of zero distributions in polynomial approximation.

Contribution

It provides new sufficient geometric conditions, such as inward corners on the boundary, guaranteeing the weak-star convergence of zero counting measures to the equilibrium measure.

Findings

Zeros of extremal polynomials converge to equilibrium measure under specified geometric conditions.

Inward corners and non-convex singularities on the boundary influence zero distribution.

Results improve existing theorems on zero distribution of special polynomial sequences.

Abstract

Let $E$ be a compact set of positive logarithmic capacity in the complex plane and let $\{P_n(z)\}_{1}^{\infty}$ be a sequence of asymptotically extremal monic polynomials for $E$ in the sense that \begin{equation*}%\label{} \limsup_{n\to\infty}\|P_n\|_E^{1/n}\le\mathrm{cap}(E). \end{equation*} The purpose of this note is to provide sufficient geometric conditions on $E$ under which the (full) sequence of normalized counting measures of the zeros of $\{P_n\}$ converges in the weak-star topology to the equilibrium measure on $E$, as $n\to\infty.$ Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of $E$ has a single component and the boundary of this component has an "inward corner" (more generally, a "non-convex singularity"). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences.