Meromorphic continuation of functions and arbitrary distribution of interpolation points (Q394930)

The authors characterize the region of meromorphic continuation of an analytic function \(f\) in terms of the geometric rate of convergence on a compact set of sequences of multi-point rational interpolants of \(f\). Let \(\Pi_{n,m}=P_{n,m}/Q_{n,m}\) where \(Q_{n,m}\) and \(P_{n,m}\) are polynomials obtained eliminating all common zeros, and let \(\text{deg\,}P_{n,m}\leqslant n - m\), \(\text{deg\,}Q_{n,m}\leqslant m\) (\(Q_{n,m}\not\equiv0\)),NEWLINENEWLINE NEWLINE\[NEWLINE Q_{n,m}(z)=\prod_{|\zeta_k|\leqslant1}(z-\zeta_k)\prod_{|\zeta_k|>1}\left(1-\frac z{\zeta_k}\right)\,. NEWLINE\]NEWLINENEWLINENEWLINENEWLINELet \(\Sigma\) be a compact set of the complex plane \(\mathbb C\) with connected complement. Let \(\mu\) be a positive unit Borel measure supported on \(\Sigma\), and let \(P(\mu;z)\) be the logarithmic potential of \(\mu\). Set \(r_0 =\inf_{z\in\Sigma}\exp\{-P(\mu;z)\}\) and \(E_\mu(r)=\{z\in\mathbb{C}:\exp\{-P(\mu;z)\}<r\}\), \(r>r_0\). For a compact set \(K\) denote \(\rho_\mu(K) =\|\exp\{-P(\mu;\cdot)\}\|_K\).NEWLINENEWLINENEWLINENEWLINEThe main result of the paper is the following theorem.NEWLINENEWLINETheorem 4.1. Let the measure \(\mu\) be the asymptotic zero distribution of the sequence of interpolation points given by \(\{w_n\}_{n\in\mathbb N}\). Let \(K\) be a regular compact set for which the value \(\rho_\mu(K)\) is attained at a point that does not belong to the interior of \(\Sigma\). Suppose that the function \(f\) is defined on \(K\) and fulfills NEWLINE\[NEWLINE \limsup_{n\to\infty}\|f-\Pi_{n,m}\|_K^{1/n}\leqslant\frac{\rho_\mu(K)}{R}<1\,. NEWLINE\]NEWLINE Then \(f\) admits a meromorphic continuation with at most \(m\) poles on the set \(E_\mu(r)\).