On the Convergence of Sequences of Rational Functions

Abstract

There are numerous kinds of sequences of rational functions, such as sequences of Pad6 approximants and sequences of rational functions of best approximation on a given point set, where the poles of the individual functions are not known even asymptotically, and no effective methods for their determination are at hand. Yet regions of convergence and degree of convergence depend heavily on the location of those poles. If assumptions on the location of poles are made, it may be possible to deduce results on regions and degree of convergence, as has been shown by Baker, Chisholm, Walsh, and others. The present note on these topics seems appropriate because a number of previous results, as set forth in a recent report by Baker [1], can be materially improved, especially with reference to degree of convergence. We emphasize Pad6 approximants and rational functions of best approximation. A rational function is said to be of type (m, n) if it can be expressed in the form