On the asymptotic behavior of Faber polynomials for domains with piecewise analytic boundary

TL;DR

This paper derives detailed asymptotic formulas for Faber polynomials associated with functions mapping the unit circle onto piecewise analytic curves, revealing their zero distribution and accumulation behavior.

Contribution

It provides new uniform asymptotic formulas for Faber polynomials for domains with piecewise analytic boundaries, including error decay rates and zero distribution insights.

Findings

Asymptotic formulas for Faber polynomials are established.

Results describe zero distribution and accumulation points.

Error terms decay at an explicitly characterized rate.

Abstract

For a function g(w) analytic and univalent in {w:1<|w|<\infty} with a simple pole at \infty and a continuous extension to {w:|w|\geq 1}, we consider the Faber polynomials F_n(z), n=0,1,2,..., associated to g(w) via their generating function g'(w)/(g(w)-z)=\sum_{n=0}^\infty F_n(z)w^{-(n+1)}. Assuming that g(w) maps the unit circle T onto a piecewise analytic curve L whose exterior domain has no outward-pointing cusps, and under an additional assumption concerning the "Lehman expansion" of g(w) about those points of T mapped onto corners of L, we obtain asymptotic formulas for F_n(z) that yield fine results on the location, limiting distribution and accumulation points of the zeros of the Faber polynomials. The asymptotic formulas are shown to hold uniformly and the exact rate of decay of the error terms involved is provided.