On doubly universal functions (Q1697430)

Let \((\lambda_n)\) be a strictly increasing sequence of positive integers. A function \(f \in H(\mathbb D)\), i.e., \(f\) is holomorphic on the unit disc \(\mathbb D\), belongs to the class \(\mathcal U(\mathbb D,(\lambda_n),0)\) if for every compact set \(K \subset \mathbb C\setminus\mathbb D\) with connected complement and for every pair of functions \((g_1,g_2) \in A(K) \times A(K)\), i.e., \(g_1, g_2\) are continuous on \(K\) and holomorphic in the interior of \(K\), there exists a subsequence of positive integers \((\mu_n)\) such that \[ \sup_{z \in K} \left| \sum_{k=0}^{\mu_n} \frac{f^{(k)}(0)}{k!} z^k - g_1(z) \right| \to 0 \;\text{ and } \; \sup_{z \in K} \left| \sum_{k=0}^{\lambda_{\mu_n}} \frac{f^{(k)}(0)}{k!} z^k - g_2(z) \right| \to 0, \] as \(n \to \infty\). In [\textit{G. Costakis} and \textit{N. Tsirivas}, J. Approx. Theory 180, 21--31 (2014; Zbl 1290.41009)] these \textit{doubly universal Taylor series} were introduced and the following theorem was established \[ {\mathcal U}(\mathbb{D},(\lambda_n),0) \not= \emptyset \; \Longleftrightarrow \; \limsup_{n \to \infty} \frac{\lambda_n}{n} = +\infty. \] The author gives an alternative proof of the implication ``\(\Longrightarrow\)'' using elementary Remez-type polynomial inequalities instead of potential theory. Moreover, the author studies \textit{doubly universal (real) infinitely differentiable functions}, these are functions \(f \in C^\infty(\mathbb{R})\) with \(f(0)=0\) satisfying that for every compact set \(K \subset \mathbb{R}\) and for every pair \((h_1, h_2)\) of continuous, real functions, vanishing at zero, there exists a subsequence of positive integers \((\mu_n)\) such that \[ \sup_{x \in K} \left| \sum_{k=0}^{\mu_n} \frac{f^{(k)}(0)}{k!} x^k - h_1(x) \right| \to 0 \;\text{ and } \; \sup_{x \in K} \left| \sum_{k=0}^{\lambda_{\mu_n}} \frac{f^{(k)}(0)}{k!} x^k - h_2(x) \right| \to 0, \] as \(n \to \infty\). He proves that the set \({\mathcal U}(C^\infty_0(\mathbb{R}),(\lambda_n))\) of these doubly universal functions is non-empty if and only if the same condition as above, namely \(\limsup_{n \to \infty} \frac{\lambda_n}{n} = +\infty\), holds. In this case, \({\mathcal U}(\mathbb{D},(\lambda_n),0)\) and \({\mathcal U}(C^\infty_0(\mathbb{R}),(\lambda_n))\) are \(G_\delta\) and dense subsets of \(H(\mathbb{D})\) and \(C^\infty_0(\mathbb{R})\), respectively, and their union with \(0\) contains a dense vector subspace of \(H(\mathbb{D})\) and \(C^\infty_0(\mathbb{R})\), respectively. \newline Furthermore, it is shown that the arithmetic means of partial sums of power series with radius of convergence 1 cannot be frequently universal.