The Borel map for compact sets in the plane

Abstract

We discuss the Borel map at a point p∈Kˆ, where Kˆ is the polynomially convex hull of the compact set K⊂C, associating to every function f∈A∞(K) its formal Taylor series at p, and show that it satisfies an alternative (provided that K fulfills a mild condition): It is either surjective or injective, and the latter is the case if and only if p∈Kˆ∘. We also discuss variants of this result for approximation by rational functions and a smooth version of the Mergelyan theorem. © 2019 The Authors