A formula for a bounded point derivation on $R^p(X)$

Abstract

Let $X$ be a compact subset of the complex plane. It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$ and if $X$ contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to $x_0$. A similar result is proven for higher order bounded point derivations. These results extend a theorem of O'Farrell for $R(X)$, the closure of rational functions with poles off $X$ in the uniform norm.