Generalizations ofde Montessus de Ballore's theorem on the row convergence of rational approximants

Let $\Pi\sb{m}$ denote the collection of all algebraic polynomials of degree at most m. A rational function $r\sb{m,n}(z)$ is said to be of type $(m,n)$ if it is of the form $r\sb{m,n}(z) = p\sb{m}(z)/q\sb{n}(z),\ q\sb{n} (z) \not\equiv 0,$ where $p\sb{m} \in \Pi\sb{m}$ and $q\sb{n} \in \Pi\sb{n}.$ Let E be a compact set whose complement K (with respect to the extended plane) is connected and possesses a classical Green's function $G(z)$ with a pole at infinity. Let $\Gamma\sb\sigma(\sigma > 1)$ denote generically the locus $G(z) = \log\sigma$ and let $E\sb\sigma$ be the interior of $\Gamma\sb\sigma.$ Suppose the function $f(z)$ is analytic on E and meromorphic with precisely $\mu$ poles (counting multiplicity) on $E\sb\rho.$ The classical de Montessus Theorem and analogous theorems guarantee the convergence of the $(\mu + 1)$th row sequence $\{ r\sb{n,\mu}(z)\}\sbsp{n=0}{\infty}$ of certain rational approximation arrays to $f(z),$ where the permissible degree of the denominators is the same as the number of poles. When the degree of the denominators is not the same as the number of poles of $f(z),$ the situation becomes very complicated. In this paper we thoroughly investigate the latter situation for the case of meromorphic functions f; namely, the convergence of row sequences $\{r\sb{n,\nu}(z)\}\sbsp{n=0}{\infty}$ from the best rational approximation array, the Pade array, and general interpolation arrays for $\nu \ne \mu.$ We give a straightforward criterion for convergence of all row sequences. In contrast to previously known theorems, this criterion is in explicit form so it is easy to apply. Also we give estimates for the rate of convergence in the appropriate cases. Furthermore, we have proved that $\{r\sb{n,\nu}(z)\}\sbsp{n=0}{\infty}$ is divergent outside $\Gamma\sb\rho$ when 0 $\le \nu < \mu$ and show some applications of our results to the zero distribution of orthogonal polynomials.