On the Sundberg approximation theorem

Let H∞H∞ be the Banach algebra of bounded holomorphic functions on the open unit disk D⊂CD⊂C. We extend Sundberg's theorem on uniform approximation of functions in BMOA   by H∞H∞ functions to other classes of holomorphic functions on DD. In our proofs we use a new characterization of meromorphic functions on DD that extend to continuous maps of the maximal ideal space of H∞H∞ to the Riemann sphere.