The Existence of Rational Functions of Best Approximation

Abstract

t. Introduction. Some results have recently been proved on the approximation to arbitrary functions of a complex variable by rational functions: first, general results on the possibility of approximation with an arbitrarily small error;t and second, results on the degree of approximation, connecting the degree of approximation with the analytic or meromorphic character of the functioin approximated.: In connection with both types of results, it is of interest to know that a sequence of rational functions of best approximation exists, and the present paper undertakes to prove this existence in certain cases. The term best approximation may here be interpreted for error measured in the sense of Tchebycheff, or for error measured by an integral taken in any one of several ways. In the study of both possibility of arbitrarily close approximation and degree of approximation, much significance is attached to the position of the poles of the approximating rational functions, and consideration of such position is of central importance in the sequel. 2. Sequences of rational functions. By a rational function of z of degree n we mean a function which can be written in the form