Chebyshev polynomials on equipotential curves

TL;DR

This paper demonstrates that Chebyshev polynomials on certain equipotential curves converge to Faber polynomials associated with an analytic function, using approximation theory and Laurent series analysis.

Contribution

It establishes the convergence of Chebyshev polynomials on equipotential curves to Faber polynomials as the potential level tends to infinity.

Findings

Chebyshev polynomials converge to Faber polynomials as r→∞.

Zero is the unique best polynomial approximation to z^n on the unit circle.

The proof utilizes Laurent series and approximation theory techniques.

Abstract

For an analytic function $\phi(z)$ with a Laurent expansion at $\infty$ of the form \begin{equation*} \phi(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial $F_n$ of degree $n$ associated to $\phi$ is the polynomial part of the Laurent series at $\infty$ of $\phi(z)^n$. We prove that the $n$th Chebyshev polynomial $T_{n,L_r}$ for the equipotential curve $L_r=\{z\in \mathbb{C}:|\phi(z)|=r \}$ converges to $F_n$ as $r\to\infty$. The proof makes use of the fact that zero is the strongly unique best approximation to the monomial $z^n$ on the unit circle by polynomials of degree less than $n$.