Introduction

The following is Arakelyan's generalization of Mergelyan's Theorem (see §1.1) to non-compact sets. It can be found in [Ara1] or [Ara2].

Arakelyan's Theorem (1968).Let Ω be an open set in C and E be a relatively closed subset of Ω. The following are equivalent:

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

2. (b) Ω*\E is connected and locally connected.

The above local connectedness condition will be discussed in §3.2. Its first appearance (at least, in an equivalent form) in the context of holomorphic approximation occurs in early work of Alice Roth which is not as well known as it should be. It is remarkable that, as early as 1938, Roth [Rot1] had shown that (when Ω = C) condition (b) above is sufficient for uniform approximation of functions in Hol(E) by entire holomorphic functions. (See [Rot2] for the generalization to other choices of Ω.) Of course, Arakelyan's Theorem is an improvement of Roth's result.

This chapter presents corresponding results for uniform approximation by harmonic functions on relatively closed sets. In fact, we will obtain generalizations of Theorems 1.3, 1.7, 1.10, 1.15, and Corollary 1.16. Further, it will be shown that, whenever uniform approximation is possible, something rather better is also true (at least, in most cases). The main results are Theorems 3.15, 3.17 and 3.19.