Universal Taylor Series in Several Variables Depending on Parameters

We establish generic existence of Universal Taylor Series on products $$\varOmega = \prod \varOmega _i$$ Ω = ∏ Ω i of planar simply connected domains $$\varOmega _i$$ Ω i where the universal approximation holds on products K of planar compact sets with connected complements provided $$K \cap \varOmega = \emptyset $$ K ∩ Ω = ∅ . 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 $$K \cap \overline{\varOmega } = \emptyset $$ K ∩ Ω ¯ = ∅ and $$\{\infty \} \cup [{\mathbb {C}} {\setminus } \overline{\varOmega }_i]$$ { ∞ } ∪ [ C \ Ω ¯ 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.

ON THE FREQUENT UNIVERSALITY OF UNIVERSAL TAYLOR SERIES IN THE COMPLEX PLANE

We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.