On the decrease of Faber polynomials in domains with piecewise analytic boundary (Q2743913)

Let \(G\) be a bounded subset of the complex plane bounded by a Jordan curve \(\Gamma\), and let \(\psi\) be the conformal mapping of \(\Delta= \{w: |w|>1\}\) onto the exterior of \(\gamma\) so that \(\psi\) has the form NEWLINE\[NEWLINEz=\psi(w)= aw+b_0+ {b_1\over w}+\cdots,\;a>0,\;w\in\Delta.NEWLINE\]NEWLINE The Faber polynomials \(F_n\) associated with the domain \(G\) are defined by the equation NEWLINE\[NEWLINE{\psi'(w) \over\psi (w)-z}= \sum^\infty_{n=0} {F_n(z) \over w^{n+1}},\;z\in G,\;w\in\Delta.NEWLINE\]NEWLINE The author investigates the situation where the Jordan curve \(\Gamma\) is piecewise analytic, that is, \(\Gamma= \cup\Gamma_j\), where the \(\Gamma_j\) are analytic Jordan arcs meeting at corners \(z_j\) where \(\Gamma\) has an exterior angle \(\lambda_j\pi\), with \(0<\lambda_j <2\). The author shows that if \(\lambda= \min\{\lambda_j\}\), then NEWLINE\[NEWLINEF_n(z)= O\left({1\over n^\lambda} \right) n\to\infty \tag{*}NEWLINE\]NEWLINE for \(z\in G\), uniformly on compact subsets of \(G\). Further, for each fixed \(z\in G\), (*) holds for \(\lambda=2\) if \(\Gamma\) is smooth (that is, \(\lambda_j=1\) for each \(j)\), and \(\lambda=\min \{\lambda_j\neq 1\}\) if there is at least one \(\lambda_j\neq 1\), and both \(\lambda\) is the best possible, and \(O({1\over n^\lambda})\) cannot be replaced by \(o({1\over n^\lambda})\). The proofs make use of an asymptotic expansion of \(\varphi= \psi^{-1}\) near corners developed by \textit{R. S. Lehman} [Pac. J. Math. 7, 1437-1449 (1957; Zbl 0087.28902)].