On various types of universal taylor series

Abstract

We denote by SN (f, ζ)(z) the Nth partial sum of the Taylor development of a holomorphic function f on a domain Ω with center ζ ∊ Ω. Let Ω be a Jordan domain with rectifiable boundary, for example the unit disc. We prove the existence of a holomorphic function f ∊H(Ω) with the following property: For every compact set with and K c connected and every function h:, continuous on K and holomorphic in K 0, there exists a sequence λ n ∊ {0,1,2,…}n =1,2,…, such that . We also show that the set of functions f∊H(Ω) with the previous property is contained in A ∞(Ω) and is a Gρ-dense subset under the natural topology of A ∞(Ω).