Universal power series of Seleznev with parameters in several variables

TL;DR

This paper extends Seleznev's universal power series to multiple variables with parameter-dependent coefficients, demonstrating generic approximation properties on product sets in the complex plane.

Contribution

It introduces a multivariable generalization of Seleznev's universal power series with parameter dependence, expanding the scope of universal approximation in complex analysis.

Findings

Universal approximation on product sets in complex plane.

Approximate any polynomial uniformly with partial sums.

The results are topologically and algebraically generic.

Abstract

We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products $K = \displaystyle\prod_{i = 1}^d K_i$, where $K_i \subseteq \mathbb{C}$ are compact sets and $\mathbb{C} \setminus K_i$ are connected, $i = 1, \dots, d$ and $0 \notin K$. On such $K$ the partial sums approximate uniformly any polynomial. Finally, the partial sums may be replaced by more general expressions. The phenomenon is topologically and algebraically generic.