ON THE SHARPNESS OF JENTZSCH'S THEOREM GENERIC PROPERTIES

Gehlen/Luh [5] proved the existence of a function f holomorphic in the unit disk D, such that for every closed set C subset of C, partial derivative D subset of c subset of D(c), there exists a sequence {N(k)} of natural numbers such that the limit points of zeros of the N(k)(th) partial sum of the Taylor series expansion of f around 0 in D(c) are exactly given by the set C. During the workshop "Complex Approximation and Universality" [3] in Oberwolfach, Luh posed the following open problems: Do generic proofs of this and related theorems exist? How many of such functions f, measured in Baire categories, do exist? Is a combination with other universalities possible? We give positive answers on these questions. In particular, we show that the set of all these functions f form a dense G(delta)-subset in the space of all functions holomorphic in D endowed with the topology of uniform convergence on compact subsets.