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

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.