Approximation in the mean by rational functions

John B. Conway, Liming Yang

Published 2019-04-12Version 1

For $1\le t < \infty$, a compact subset $K$ of the complex plane $\mathbb C$, and a finite positive Borel measure $\mu$ supported on $K$, $R^t(K, \mu)$ denotes the closure in $L^t(\mu)$ of the rational functions with poles off $K$. Let $\text{abpe}(R^t(K, \mu))$ denote the set of analytic bounded point evaluations for $R^t(K, \mu)$. The purpose of this paper is to describe the structure of $R^t(K, \mu)$. In the work of Thomson on describing the closure in $L^t(\mu)$ of analytic polynomials, $P^t(\mu)$, the existence of analytic bounded point evaluations plays critical roles, while $\text{abpe}(R^t(K, \mu))$ may be empty. We introduce the concept of non-removable boundary $\mathcal F$ for $R^t(K, \mu)$ such that $\mathcal R = K\setminus \mathcal F$ contains $\text{abpe}(R^t(K, \mu))$. The sets $\mathcal F$ and $\mathcal R$ are essential for us to describe $R^t(K, \mu)$.