Generalizations of Montessus's theorem on the row convergence of rational interpolations
The author studies multipoint Padé approximants on a point set \(E\) with following properties (called Property A): \(E\) is connected and equal to the closure of the interior of a finite number of mutually exterior analytic curves \(C_1,\dots,C_k\). \(G\) denotes Green's function for \(K: =\overline \mathbb{C} \backslash E\) with pole at infinity, \(\Delta\) is the logarithmic capacity of \(E\), \(\Gamma_\rho: =\{z\in\mathbb{C} \mid G(z) = \log\rho\}\), \(E_\rho:=\) interior of \(\Gamma_\rho\). Theorem. Suppose \(E\) has property \(A\). Let \(\rho\) be a fixed positive number, \(f(z)\) be analytic on \(E\), meromorphic with precisely \(\mu(\geq 0)\) poles (counting multiplicity) in \(E_\rho\), and analytic on \(\Gamma_\rho\) except for poles in the \(m\) distinct points \(\alpha_1\), \(\alpha_2, \dots, \alpha_m\) on \(\Gamma_\rho\) of respective orders \(r_1, \dots, r_m\) \((r_1\geq r_2 \geq \dots r_m)\). Let the points \(\beta_j^{(n)}\), \(j=1, \dots, n+1\); \(n=0, 1, \dots\) have no limit point exterior to \(E\) and satisfy the relations, \[ \bigl|\omega_n(z) \bigr|= \left|\prod^{n+1}_{j=1} \Bigl(z-\beta_j^{(n)}\Bigr) \right|\leq M \Delta^n \] \[ \biggl|G(z) + \log z-n^{-1} \log\bigl|\omega_n(z)\bigr|\biggr|\leq Mn^{-1} \] uniformly in \(z\) on each closed bounded subset of \(K\). If none of \(\{\alpha_i\}^m_{i=1}\) is a critical point of \(G(z)\), then for ``good'' \(\nu\) (to be given in the text) and for all \(n\) sufficiently large, there exists a unique multipoint Padé approximant \(R_{n,\mu+\nu}(z)\). The theorem contains also results on the situation and the behaviour of the poles and the rate of convergence of \(R_{n,\mu+\nu}(z)\) as \(n\to\infty\), which are too voluminous to be cited. Furthermore, applications to the zero distribution of certain orthogonal polynomials are given.
scientific article; zbMATH DE number 908800