Partially Smooth Universal Taylor Series on products of simply connected domains

TL;DR

This paper proves the existence of universal Taylor series on products of simply connected domains, extending previous results using a Mergelyan type theorem and exploring topological properties that restrict certain function families.

Contribution

It introduces new universal Taylor series on product domains and analyzes their topological properties, extending the scope of approximation theory.

Findings

Universal Taylor series exist on product domains with boundary extensions.

Approximation occurs on specific compact sets with connected complements.

Certain families of functions are shown to be voided by topological properties.

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.