Generic approach to multiply universal functions

Abstract

Let {z _n } be a sequence of complex numbers. We use a generic approach to show the existence of universal entire functions  ϕ having several universal properties, i.e. for any pre-assigned compact set K  ⊂ ℂ with connected complement and any pre-assigned function f  ∈ C(K), holomorphic in the interior of K, there exist:

•	a suitable sequence of additive translates,
•	a suitable sequence of multiplicative translates (if 0  ∉ K) and
•	a suitable sequence of derivatives

of  ϕ that all converge to f uniformly on K. By a so-called constructive approach, Luh (W. Luh, Entire functions with various universal properties, Complex Var. Theory Appl. 31 (1996), pp. 87–96) obtained functions of this type. In addition, our functions  ϕ join two non-universal properties, namely having zeros at certain prescribed points of prescribed order and solving a certain interpolation problem.

Keywords:

• universality
• approximation in the complex domain
• zeros
• interpolation
• Baire spaces

AMS Subject Classifications:

• 30C15
• 30D15
• 30E05
• 30E10
• 54E52