Approximation by rational functions on compact nowhere dense subsets of the complex plane

Abstract

Let \(X\) be a compact nowhere dense subset of the complex plane, and let \(dA\) denote two-dimensional or area measure on \(X\). Let \(R(X)\) denote the uniform closure of the rational functions having no poles on \(X\), and for each \(p,\, 1\le p<\infty \), let \(R^p(X)\) be the closure of \(R(X)\) in the \(L^p(X, dA)\)-norm. Since \(X\) has no interior \(R^p(X)=L^p(X)\) whenever \(1\le p <2\), but for \(p=2\) a kind of phase transition occurs that can be quite striking at times. Our main goal here is to study the manner in which similar phase transitions can occur at any value of \(p, \, 2\le p < \infty \).