The Convergence of Rational Functions of Best Approximation to the Exponential Function. II

Abstract

Let ${W_{m,n}}(z)$ be a rational function of type (m, n) of best uniform approximation to the function ${e^z}$ on the closed unit disk. In this paper we show that any sequence $\{ {W_{m,n}}(z)\}$ for which $m + n \to \infty$ must converge to ${e^z}$ for all values of z. This is the first result which describes completely the regions of convergence of arbitrary sequences formed from a Walsh array.