Boundary Smoothness conditions for functions in $R^p(X)$

Abstract

Let $X$ be a compact subset of the complex plane and let $R^p(X)$, $2<p<\infty$, denote the closure of the rational functions with poles off $X$ in the $L^p$ norm. In this paper we consider three conditions that show how the functions in $R^p(X)$ can have a greater degree of smoothness at the boundary of $X$ than might otherwise be expected. We will show that two of the conditions are equivalent and imply the third but the third does not imply the other two.

arXiv comment15 pages 1 figure; Minor changes made to improve exposition. Accepted for publication by Computational Methods and Function Theory