A STRONG NOTION OF UNIVERSAL TAYLOR SERIES

Abstract

For a holomorphic function f in the open unit disc D, the Nth partial sum of its Taylor series with center ζ ∈ D is denoted by SN(f,ζ)(z)= ∑ n = 0 N ( f ( n ) ( ζ ) / n ! ) ( z − ζ ) n . Generically, all functions f in H(D) satisfy the following. For every compact set K⊂ C with K∩D=Ø and Kc connected and every polynomial h, there exists a sequence of positive integers { λ n } ∞ such that, for every 1 ∈ {0,1,2,…}, sup z ∈ K | ∂ l ∂ z l S λ n ( f , 0 ) ( z ) − h ( l ) ( z ) | → 0 as n → + ∞ .