Elongating the partial sums of Faber series (Q691815)

Let \(K\subset\mathbb{C}\) be a compact set, \(\#K>1\), and assume that \(K\) and \(\mathbb{\widehat{C}}\setminus K\) are connected. Further, let \(\Phi:\mathbb{\widehat{C}}\setminus K\to\{w\in\mathbb{C}: |w|>\rho\}\) be the unique conformal mapping such that \(\lim_{z\to\infty}\Phi(z)/z=1\). These assumptions uniquely determine the sequence \((F_n(z))_{n=0,1,\dots}\) of Faber polynomials with respect to \(K\), and for each real number \(R>\rho\), \(C_R:=\Phi^{-1}(\{w\in\mathbb{C}: |w|=R\})\) is a closed Jordan curve with inner and outer domain \(I(C_R)\) and \(O(C_R)\), respectively. A Faber series \[ f(z):=\sum_{n\geq0}a_nF_n(z)\tag{*} \] with \(\limsup_{n\to\infty}|a_n|^{1/n}=1/R\), \(R>\rho\), converges in \(I(C_R)\) locally uniformly and diverges in \(O(C_R)\). The authors are particularly interested in the sequence of partial sums \((s_n(z))_{n=0,1,\dots}\) of series as in \((\ast)\). For subsets \(S\) of \(O(C_R)\) they prove that the series \((\ast)\) is over-convergent in \(S\) (i.e. a suitable subsequence \((s_{n_k}(z))_k\) of \((s_n(z))_n\) converges in \(S\)) if and only if there is a sequence \(m_0,m_1,\dots\) of natural numbers such that the sequence of the arithmetic mean of the `elongated' sequence \((s_0,\dots, s_0,s_1,\dots,s_1,\dots)\) converges in \(S\); here, for each \(n=0,1,\dots\), \(s_n:= s_n(z)\) appears exactly \(m_n\) times. \textit{T. L. Gharibyan} and \textit{W. Luh} [Comput. Methods Funct. Theory 11, 59--70 (2011; Zbl 1260.30003)] have proved an analog equivalence for Taylor series.