Doubly universal Taylor series (Q2637500)

For a holomorphic function \(f\) in the unit disk, \(S_n( f )\) denotes the \(n\)th partial sum of the Taylor expansion of \(f\) with center at \(0\). The authors show, that for a given strictly increasing sequence of positive integers \((\lambda_n),\) there exists a holomorphic function \(f\) on the unit disk such that the pairs of partial sums \(\{(S_n( f ), S_{\lambda_n} ( f )) : n = 1, 2, \ldots\}\) approximate all plausibly approximable functions uniformly on suitable compact subsets \(K\) of the complex plane if and only if \(\lim \sup_n \frac{\lambda_n}{n}=+\infty\).