Partially Smooth Universal Taylor Series on products of simply connected domains

Abstract

Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains Oi that extend continuously on the product of the union of Oi with Si , where Si are subsets of the boundary of Oi, open in the relative topology. The universal approximation occurs on every product of compact sets Ki such that C - Ki are connected and for some i0 it holds that Ki0 is contained in the complement of the union of Oi0 with the closure of Si0. Furthermore,we introduce some topological properties of universal Taylor series that lead to the voidance of some families of functions.