Singular points of analytic functions expanded in series of Faber polynomials (Q1384870)

Let \(E\subset\mathbb C\) be a compact simply connected set such that \(\partial E\) is an analytic Jordan curve and \(\text{Cap}(E)=1\). Suppose that \(\Phi\) maps conformally the exterior of \(E\) onto the exterior of the closed unit disc so that \(\Phi(\infty)=\infty\) and let \(z_0\in\partial E\) be such that \(\Phi(z_0)=1\). Suppose that \((\Phi_{n})_{n=0}^\infty\) is the sequence of Faber polynomials for \(E\). Let \((a_{n})_{n=0}^\infty\subset[0,\infty)\) be such that \(\limsup_{n\to+\infty}a_{n}^{1/n}=1\). Define \(f(z):=\sum_{n=0}^\infty a_{n}\Phi_{n}(z)\). The main results of the paper are the following two theorems: (1) The point \(z_0\) is singular for \(f\) (it is a generalization of the Pringsheim theorem). (2) Assume moreover that \(a_{n}=O(1/n^{k})\) for \(k=1,2,\dots\), and \(a_{n}\neq 0 \Longleftrightarrow n\in\{n_1, n_2,\dots\}\), where \((n_{k})_{k}\) is such that \(\lim_{k\to+\infty}n_{k}/k=\infty\). Then \(\partial E\) is singular for \(f\) and \(f\in\mathcal C^\infty(\partial E)\).