A distance formula for algebras on the disk

Suppose H^∞(double-struck D sign)[ℱ] is the closed algebra on the disk generated by H^∞(double-struck D sign) and a countable collection ℱ of bounded harmonic functions. Given g ∈ L^∞(double-struck D sign) we give a method for calculating the distance from g to H^∞(double-struck D sign)[ℱ] (in the L^∞ norm). As applications we compute the Bourgain closures of such algebras and give a new proof of a result of Axler and Shields.