An Extension of E. Bishop's Localization Theorem

Abstract

We prove that if f is a function belonging to Baire first class on a compact set K ⊂ C and each point of K has a (closed) neighborhood where f is the pointwise limit of some sequence of uniformly bounded rational functions, then f on the whole of K is the pointwise limit of a sequence of rational functions uniformly bounded on K . This is an extension of Bishop's localization theorem. As an application we establish a “pointwise” version of Mergelyan's classical theorem on uniform approximation by rational functions on compact sets for which the components of its complement have diameters greater than a fixed positive number.