The relation between the bounded point derivation and the fine derivation

Abstract

Let { be the complex plane, K c (; be a compact set with empty interior and R(K) be the function algebra whose elements can be approximated uniformly on K by functions holomorphic in the neighborhoods of K. It is proved by Hallstrom (see [4]) that R(K) has a bounded point derivation of an arbitrary order, Le., for any integer p = 0,1, ... and any non-peak point x (of order p) of R(K) there is a positive constant Mp(x) such that