Universal Taylor Series On Convex Subsets Of $\Mathbb{C}^{N}$

Abstract

We prove the existence of holomorphic functions $f$ defined on any open convex subset $\mathrm{\Omega }\subset {\mathbb{C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from $\bar{\mathrm{\Omega }}$ and on denumerably many convex compact sets in ${\mathbb{C}}^n\backslash \mathrm{\Omega }$ which may meet the boundary $\partial \mathrm{\Omega }$. If the universal approximation is only required on convex compact sets disjoint from $\bar{\mathrm{\Omega }}$, then $f$ may be chosen to be smooth on $\partial \mathrm{\Omega }$, that is $f\in A^{\infty }(\mathrm{\Omega })$. Those are generic universalities.

16 pages