On Borel Map for Compact Sets in Plane of Entire Functions

We follow the well build method of a discussion shown by Paulo D. Cordaro, Giuseppe Della Sala and Bernhard Lamel [12] on the Borel map at a point p^2∈K, where K is the polynomially convex hull of the compact set K ⊂ C, associating to every sequence of functions f_r∈A^∞ (K) its formal Taylor series be at p^2. We show that it satisfies an alternative condition (provided that K fulfills a mild metric condition that): It is either surjective or injective, and the latter is the case if and only if p^2∈K. We also discuss variants of this results for approximation by rational functions and a smooth version of the Mergelyan theorem.