Universal Taylor series on convex subsets of

Abstract

We prove the existence of holomorphic functions defined on any open convex subset, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from , and on denumerably many convex compact sets in which may meet the boundary . If the universal approximation is only required on convex compact sets disjoint from, then may be chosen to be smooth on , that is . Those are generic universalities.