The convergence of sequences of rational functions of best approximation

Abstract

Unless the poles of a rational function of best approximation are prescribed in advance, the determination of those poles even approximately may be difficult, yet may be necessary for a complete study of approximation or convergence. Certain cases where this determination is possible have recently been considered [1] by the present author; the object of the present note is to enlarge the category of those cases, especially to study approximation on a set having no interior points, and on a set with several components. Montessus de Ballore considered in 1902 an analogous situation related to the Pade table, with analogous results so far as concerns regions of convergence, but his methods relate to the special properties of the Taylor development and do not apply to the hypotheses of the present paper. Here our methods are based primarily on the deeper use of geometric degree of convergence, in various forms, including methods developed [2] in the theory of approximating polynomials. A rational function of the form