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Universal Taylor series with respect to a prescribed subsequence. (English) Zbl 1461.30132

Summary: For a holomorphic function \(f\) in the open unit disc \(\mathbb{D}\) and \(\zeta \in \mathbb{D}\), \(S_n(f, \zeta)\) denotes the \(n\)-th partial sum of the Taylor development of \(f\) at \(\zeta \). Given an increasing sequence of positive integers \(\mu = ( \mu_n)\), we consider the classes \(\mathcal{U}(\mathbb{D}, \zeta)\) and \(\mathcal{U}^{( \mu )}(\mathbb{D}, \zeta)\) of holomorphic functions \(f\) in \(\mathbb{D}\) such that subsequences of the partial sums \(\{ S_n(f, \zeta) : n = 1, 2, \ldots \}\) and \(\{ S_{\mu_n}(f, \zeta) : n = 1, 2, \ldots \}\) respectively approximate all polynomials uniformly on the compact sets \(K \subset \{z \in \mathbb{C} : | z | \geq 1 \}\) with connected complement. We show that these two classes of universal Taylor series coincide if and only if \(\limsup_{n \to + \infty} \left( \frac{ \mu_{n + 1}}{ \mu_n}\right ) < + \infty \). In the same spirit, we prove that, for \(\zeta \neq 0\), the equality \(\mathcal{U}^{( \mu )}(\mathbb{D}, \zeta) = \mathcal{U}^{( \mu )}(\mathbb{D}, 0)\) holds if and only if \(\limsup_{n \to + \infty} \left( \frac{ \mu_{n + 1}}{ \mu_n} \right) < + \infty \). Finally we deal with the case of real universal Taylor series.

MSC:

30K05 Universal Taylor series in one complex variable
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