Universal Taylor series with respect to a prescribed subsequence

Abstract

For a holomorphic function f in the open unit disc D and ζ∈D, Sn(f,ζ) denotes the n-th partial sum of the Taylor development of f at ζ. Given an increasing sequence of positive integers μ=(μn), we consider the classes U(D,ζ) and U(μ)(D,ζ) of holomorphic functions f in D such that subsequences of the partial sums {Sn(f,ζ):n=1,2,…} and {Sμn(f,ζ):n=1,2,…} respectively approximate all polynomials uniformly on the compact sets K⊂{z∈C:|z|≥1} with connected complement. We show that these two classes of universal Taylor series coincide if and only if lim supn→+∞(μn+1μn)<+∞. In the same spirit, we prove that, for ζ≠0, the equality U(μ)(D,ζ)=U(μ)(D,0) holds if and only if lim supn→+∞(μn+1μn)<+∞. Finally we deal with the case of real universal Taylor series.