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

TL;DR

This paper demonstrates the existence of holomorphic functions on convex subsets of complex space whose Taylor partial sums can uniformly approximate any polynomial on certain disjoint convex sets, including some meeting the boundary, with some functions being smooth on the boundary.

Contribution

It establishes the generic existence of universal Taylor series on convex subsets of , extending approximation capabilities to sets meeting the boundary and providing smooth boundary functions.

Findings

Existence of universal Taylor series on convex subsets of .

Approximation of polynomials on disjoint convex sets, including boundary meeting sets.

Functions can be chosen to be smooth on the boundary.

Abstract

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