Universal Taylor Series in several variables depending on parameters

TL;DR

This paper proves the generic existence of universal Taylor series in several variables across product domains, with approximation properties that depend on parameters and can be smooth up to the boundary.

Contribution

It extends the theory of universal Taylor series to multiple variables and parameter-dependent cases, establishing generic approximation results in product domains.

Findings

Universal Taylor series exist generically in product domains.

Approximation holds on compact sets disjoint from the domain.

Universal functions can be smooth up to the boundary.

Abstract

We establish generic existence of Universal Taylor Series on products $\Omega = \prod \Omega_i$ of planar simply connected domains $\Omega_i$ where the universal approximation holds on products $K$ of planar compact sets with connected complements provided $K \cap \Omega = \emptyset$. These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that $K \cap \overline{\Omega} = \emptyset$ and $\{\infty\} \cup [\mathbb{C} \setminus \overline{\Omega}_i]$ is connected for all $i$. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper. These universalities are topologically and algebraically generic.