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

Abstract

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

The study of <i>R</i>(<i>K</i>) 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 <i>K</i> be a compact subset of real <i>N</i>-dimensional space, where <i>N</i> ≥ 2. Let harm<i>K</i> denote those functions harmonic in a neighborhood of <i>K</i>. Let <i>h</i>(<i>K</i>) denote the closure of harm<i>K</i> in <i>C</i>(<i>K</i>). We also study the problem: What is <i>h</i>(<i>K</i>)?

The study of <i>h</i>(<i>K</i>) 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.