Uniform polynomial approximation

Abstract

A new proof is given of Mergelyan’s solution [l] of the problem of uniform polynomial approximation on compact subsets of the complex plane. Functions of the form f(z) + g(z) for polynomials f(z) and g(z) are called harmonic polynomials. The analogue of Mergelyan’s theorems for uniform approximation by harmonic polynomials is proved. The existence of solutions of the Dirichlet problem on compact sets with connected complement is a corollary. Proofs of special cases of the Riemann mapping theorem are also given. Proofs are based on functional annihilator arguments which relate uniform approximation to measure theory via the Riesz representation and Hahn- Banach theorem. Bounded nonnegative measures with compact support enter into the arguments because of a theory of bounded polynomial approximation due to L. de Branges [2]. This makes possible the application of extreme point methods to polynomial approximation. A characterization of extremal measures is an underlying principle in connection with the Stone-Weierstrass theorem [3]. A computation of the bounded closure of the polynomials is due to Samson [4]. Simplifications are given by Conway and Olin [5]. These results are contained in the de Branges theory [2]. Bounded nonnegative measures, with compact support, on Bore1 subsets of the complex plane are considered in the weak topology induced by the continuous functions. Two such measures p and v are said to be equivalent, denoted p N V, if the identity holds for every harmonic polynomial h(z). If TV is a bounded nonnegative measure with compact support, the weak closure of the set of bounded nonnegative measures which are absolutely continuous with respect to p and equivalent to CL, denoted J?(p), is a weakly compact convex set. By the Krein-Milman theorem, the set J%‘(P) is the closed convex span of its extreme points. A summary of the arguments given in [3] and [6] leads to a density characterization of extreme points. 81