Title:  On Bounded Point Derivations and Analytic Capacity

Abstract:  Let X be a compact subset of the complex plane and let R(X) denote the uniform closure of the space of rational functions whose poles lie off X. We say that there is a bounded point derivation on R(X) at x if and only if there exists a constant k such that |f t(x)| ≤ k||f ||X for all f ∈ H(X), where H(X) is the space of all functions that are holomorphic in some neighborhood of X. In this talk we will give necessary and sufficient conditions for the existence of a bounded point derivation on R(X) at x.