A counterexample for H ∞ approximable functions

Abstract

Let D be the unit disk. We show that for some relatively closed set F ⊂ D there is a function f that can be uniformly approximated on F by functions of H∞, but such that f cannot be written as f = h + g, with h ∈ H∞ and g uniformly continuous on F. This answers a question of Stray.