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Chebyshev polynomials on equipotential curves

Erwin Miña-Díaz, Olof Rubin

TL;DR

The paper analyzes the asymptotics of the $n$th Chebyshev polynomial for equipotential curves $L_r$ defined by $|\phi(z)|=r$, where $\phi$ has a Laurent expansion $\phi(z)=c z+c_0+c_{-1}/z+\cdots$. It proves that $T_{n,L_r}$ converges to the monic Faber polynomial $\hat{F}_n=c^{-n}F_n$ as $r\to\infty$, with a quantitative rate $O(r^{-1})$, leveraging the fact that $0$ is the strongly unique best approximation to $z^n$ on the unit circle by polynomials of degree $<n$. The proof uses a Rivlin-type inequality and Parseval-based coefficient bounds to show that the deviation $|T_{n,L_r}-\hat{F}_n|$ vanishes at rate $r^{-1}$ uniformly on $L_r$, and moreover establishes a stronger convergence that extends to the entire growing family of curves. This work connects Chebyshev and Faber polynomials in the context of potential theory and equipsotential curves, providing a precise asymptotic description of polynomial minimizers in this geometric setting.

Abstract

For an analytic function $φ(z)$ with a Laurent expansion at $\infty$ of the form \begin{equation*} φ(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 $φ$ is the polynomial part of the Laurent series at $\infty$ of $φ(z)^n$. We prove that the $n$th Chebyshev polynomial $T_{n,L_r}$ for the equipotential curve $L_r=\{z\in \mathbb{C}:|φ(z)|=r \}$ 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 $n$.

Chebyshev polynomials on equipotential curves

TL;DR

The paper analyzes the asymptotics of the th Chebyshev polynomial for equipotential curves defined by , where has a Laurent expansion . It proves that converges to the monic Faber polynomial as , with a quantitative rate , leveraging the fact that is the strongly unique best approximation to on the unit circle by polynomials of degree . The proof uses a Rivlin-type inequality and Parseval-based coefficient bounds to show that the deviation vanishes at rate uniformly on , and moreover establishes a stronger convergence that extends to the entire growing family of curves. This work connects Chebyshev and Faber polynomials in the context of potential theory and equipsotential curves, providing a precise asymptotic description of polynomial minimizers in this geometric setting.

Abstract

For an analytic function with a Laurent expansion at of the form \begin{equation*} φ(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial of degree associated to is the polynomial part of the Laurent series at of . We prove that the th Chebyshev polynomial for the equipotential curve converges to as . The proof makes use of the fact that zero is the strongly unique best approximation to the monomial on the unit circle by polynomials of degree less than .
Paper Structure (3 sections, 7 theorems, 48 equations, 2 figures)

This paper contains 3 sections, 7 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Proofs

Key Result

Theorem A

Let $P$ be a monic polynomial of degree $m$ and let $R>0$. If we set then for all $n\in\mathbb{N}$ and $r>1$.

Figures (2)

  • Figure 1: The Bernoulli lemniscate $K = \{z: |z^2-1| = 1\}$ as a solid curve along with samples of associated equipotential curves $L_r = \{z:|z^2-1| = r^2\}$ for $r>1$.
  • Figure 2: The orange lines illustrate the trajectories of the zeros of the Chebyshev polynomials $T_{21,L_r}$ corresponding to the level curves $L_r = \{ z \in \mathbb{C} : |z^2 - 1| = r^2 \}$ as $r$ increases from $0$. The red dots mark the zeros of the corresponding Faber polynomial. The Bernoulli lemniscate $L_1\coloneqq \{z\in \mathbb{C}: |z^2-1| = 1\}$ is marked with a solid blue line. The computations were carried out using the algorithm presented in tang87tang88.

Theorems & Definitions (9)

  • Theorem A: Faber 1919 faber19
  • Theorem B: Faber 1919 faber19
  • Theorem C: Christiansen, Simon and Zinchenko 2020 christiansen-simon-zinchenko-IV
  • Theorem D: Stawiska stawiska96
  • Theorem 2.1
  • Remark 2.3
  • Theorem 2.4: Rivlin 1984 rivlin84
  • Theorem 3.1
  • proof