Pointwise bounded approximation and Dirichlet algebras

Those open sets U of S² for which A(U) is pointwise boundedly dense in H^∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in C_R(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H^∞(U).