Rational and harmonic approximation on F.P.A. sets

Let K be a compact subset of complex N-dimensional space, where N ≥ 1. Let H(K) denote the functions analytic in a neighborhood of K. Let R(K) denote the closure of H(K) in C(K). We study the problem: What is R(K)?

The study of R(K) is contained in the field of rational approximation. In a set of lecture notes, T. Gamelin [6] has shown a certain operator to be essential to the study of rational approximation. We study properties of this operator.

Now let K be a compact subset of real N-dimensional space, where N ≥ 2. Let harmK denote those functions harmonic in a neighborhood of K. Let h(K) denote the closure of harmK in C(K). We also study the problem: What is h(K)?

The study of h(K) is contained in the field of harmonic approximation. As well as obtaining harmonic analogues to our results in rational approximation, we also produce a harmonic analogue to the operator studied in Gamelin's notes.