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On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

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Constructive Approximation Aims and scope

Abstract

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F n (z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)]n. Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂ Ω, we obtain asymptotic formulas for F n that yield fine results on the limiting distribution of the zeros of Faber polynomials.

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  1. Department of Mathematics, University of Mississippi, Hume Hall 305, P.O. Box 1848, University, MS, 38677-1848, USA

    Erwin Miña-Díaz

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  1. Erwin Miña-Díaz
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Correspondence to Erwin Miña-Díaz.

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Communicated by Arno Kuijlaars.

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Miña-Díaz, E. On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary. Constr Approx 29, 421–448 (2009). https://doi.org/10.1007/s00365-008-9033-z

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  • DOI: https://doi.org/10.1007/s00365-008-9033-z

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