Paper detail

Universal Taylor Series in several variables depending on parameters

We establish generic existence of Universal Taylor Series on products $Ω= \prod Ω_i$ of planar simply connected domains $Ω_i$ where the universal approximation holds on products $K$ of planar compact sets with connected complements provided $K \cap Ω= \emptyset$. 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 $K \cap \overlineΩ = \emptyset$ and $\{\infty\} \cup [\mathbb{C} \setminus \overlineΩ_i]$ 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.