On approximation by rational functions to an arbitrary function of a complex variable

Abstract

Cauchy's Integral for the function and region (if the Jordan curves bounding R are rectifiable) in Theorem II splits up the function /(z) into n functions. If Theorem I is applied to each of these in turn (a simple transformation w = l/(z —a,) is first necessary for « —1 of them), we find Theorem II immediately. If the Jordan curves bounding R are not rectifiable, we readily obtain the same result. Consider an auxiliary variable region R' of the same connectivity as R, which lies within R, which is bounded by n rectifiable Jordan curves, and which approaches R as the bounding curves change continuously. For the region R', Cauchy's Integral for/(z) splits that function into « functions which do not change as R' varies. Each of these « functions is, when properly defined, analytic not merely in R but in a region bounded by a single Jordan curve and which contains R. Theorem II then follows as before from Theorem I. On the possibility of approximation of arbitrary functions by polynomials, no deeper lying results than Theorems I and II, except for approximation by certain restricted types of polynomials, appear to have been found until