具有分段解析边界区域的Faber多项式的渐近性
On the asymptotic behavior of Faber polynomials for domains with piecewise analytic boundary
摘要：
对于{w：1< |w| < \infty }中解析的单叶函数g（w），在\infty上有一个单极点，并且是{w：|w| \geq 1}的连续扩张，我们考虑Faber多项式F_n（z），n= 0，1，2，.，通过它们的生成函数g ′（w）/（g（w）-z）= \sum_{n=0}^\ infty F_n（z）w^{-（n+1）}与g（w）相关联。假设g（w）将单位圆T映射到一条外区域没有外尖点的分段解析曲线L上，并在关于g（w）关于T映射到L的角点的“Lehman展开式”的附加假设下，得到了F_n（z）的渐近公式，得到了关于Faber多项式零点的位置、极限分布和聚点的精确结果.渐近公式一致，并提供了确切的衰减率的误差项。
英文摘要：
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.
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