Chebyshev polynomials on equipotential curves

2025/05/06 by Erwin Miña‐Díaz, Miña-Díaz, Erwin, Olof Rubin +1
Mathematics · #30C10 #30C20 #30E10 #31A15 #41A50 #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2505.03967

openalex publication_date 2025/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an analytic function ϕ(z) with a Laurent expansion at ∞ of the form ϕ(z)=z+c₀+\fracc₁z+\fracc₂z²+⋯, the Faber polynomial F_n of degree n associated to ϕ is the polynomial part of the Laurent series at ∞ of ϕ(z)ⁿ. We prove that the nth Chebyshev polynomial T_n,Lr for the equipotential curve L_r=\z∈ ℂ:|ϕ(z)|=r \ converges to F_n as r→∞. The proof makes use of the fact that zero is the strongly unique best approximation to the monomial zⁿ on the unit circle by polynomials of degree less than n.

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