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

TL;DR

This paper investigates boundary smoothness conditions for functions in $R^p(X)$, a space of rational functions on a compact set in the complex plane, revealing relationships among three specific smoothness conditions.

Contribution

It introduces and analyzes three boundary smoothness conditions for functions in $R^p(X)$, establishing equivalences and implications among them.

Findings

Two conditions are equivalent and imply the third.

The third condition does not imply the other two.

Boundary smoothness can be characterized by these conditions.

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.