Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spaces H-p(omega) of analytic functions in the open unit disc D of the complex plane C. We prove that if X is a relatively closed subset of D, the class of uniform limits on X of functions in H-p(omega) coincides, module H-p(omega), with the space of uniformly continuous functions on a certain hull of X which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets for H-p(omega), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spaces H-p(D), 1 less than or equal to p < infinity.