Quantitative and constructive aspects of the generalized Koenig's and de Montessus's theorems for Padé approximants

摘要

The generalized Koenig's theorem and de Montessus's theorem are two classical results concerning the convergence of the rows of the Padé table for meromorphic functions. Employing a technique that was recently developed for the analysis of vector extrapolation methods, refined versions of these theorems are proved in the present work. Specifically, complete expansions for the numerators and denominators of Padé approximants are derived. These expansions are then used to obtain (1) precise asymptotic rates of convergence of the poles of the Padé approximants to the corresponding poles, simple or multiple, of the meromorphic function in question, and (2) the precise asymptotic behavior of the error in the relevant Padé approximants. One important feature of the asymptotic results derived in this work is that these are expressed in terms of a very small number of parameters. Approximations of optimal accuracy to multiple poles and the principal parts of the corresponding Laurent expansions are also constructed. In addition, the convergence problem for the case in which the only singularities on the circle of meromorphy are poles is solved completely through the solution of a nonlinear integer programming problem.