Zeros of Faber polynomials for Joukowski airfoils

TL;DR

This paper investigates the asymptotic distribution of zeros of Faber polynomials for Joukowski airfoils, revealing unique limit measures and implications for electrostatic skeletons and Chebyshev quadrature.

Contribution

It determines the weak-* limit of zeros of Faber polynomials for Joukowski airfoils, showing it differs from the equilibrium measure and highlighting new electrostatic properties.

Findings

The zeros' limit measure is explicitly identified and differs from the equilibrium measure.

Many Joukowski airfoils admit an electrostatic skeleton.

The results explain examples related to Chebyshev quadrature.

Abstract

Let $K$ be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for $K$ are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized counting measures of the Faber polynomials for Joukowski airfoils; it is never equal to the potential-theoretic equilibrium measure of $K$. This implies that many of these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of Ullman related to Chebyshev quadrature.