1 - Approximation on compact sets

Summary

Introduction

If Ω is an open set in C or Rⁿ(n ≥ 2), then we will use Ω* to denote the Alexandroff, or one-point, compactification of Ω, and will use A to denote the ideal point. Thus Ω* = Ω ∪ {A}, and a set A is open in Ω* if either A is an open subset of Ω or A = Ω*\K, where K is a compact subset of Ω. In the special case where Ω is C or Rⁿ we continue to write ∞ for A.

If A is a subset of C, we denote by Hol(A) the collection of all functions which are holomorphic on an open set containing A. Historically the following result (essentially in [Run]; cf. [Con, pp.198, 201]) can be regarded as the starting point of the theory of holomorphic approximation.

Runge's Theorem (1885).Let Ω be an open subset ofCand K be a compact subset of Ω. The following are equivalent:

1. (a) for each f in Hol(K) and each positive number ∈, there exists g in Hol(Ω) such that |g – f| < ∈ on K;

2. (b) Ω*\K is connected.

Condition (b) above is equivalent to asserting that no component of Ω\K is relatively compact in Ω. Also, when Ω = C, this condition is clearly equivalent to saying that C\K is connected.

We record below one further important development in the theory of holomorphic approximation, which deals with approximation of a much larger class of functions on a given compact set K.