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On the radial behavior of universal Taylor series. (English) Zbl 1079.30002
In this paper, the author deals with universal Taylor series in the open unit disk \(D\). Here the universality is related to the overconvergence phenomenon. Let \(H(D)\) denote the FrĂ©chet space of all holomorphic functions in \(D\). By definition, a function \(f \in H(D)\) is said to be a universal Taylor series in the sense of Nestoridis – its set is denoted by \(U(D,0)\) – if for every compact subset \(K \subset D^c\) with \(K^c\) connected and for every function \(h:K \to C\), continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((n_j)\) of natural numbers such that \(\lim_{n \to \infty} \sup_{z \in K} | S_{n_j}(f,0)(z) - h(z)| = 0\), where \(S_n(f,0)\) denotes the \(n\)th partial sum of the Taylor development of \(f\) with center \(0\). Nestoridis proved in 1996 that \(U(D,0)\) is in fact a residual set in \(H(D)\), so improving earlier results by Luh, Chui and Parnes. Katsoprinakis, Papadimitrakis, Melas, Nestoridis and Papadoperakis have shown that universal series are not \((C,k)\) summable at every boundary point for every positive integer \(k\). Nevertheless, using an approximation theorem of Nersesjan together with a result due to Herzog about induced universality in \(G_\delta\) sets, the author proves that universal Taylor series can have a good radial behavior –namely, they can be Abel summable– at some points of the unit circle. In fact, he is able to demonstrate the following stronger result: Let \(E\) be a closed and nowhere dense subset of the unit circle and \(S := \{rt: r \in [0,1)\), \(t \in E\}\), \(\varepsilon :[0,1) \to (0,+\infty )\) be a decreasing continuous function, and let \(g \in H(D)\). Consider the set \(A := \{f \in H(D): | f(z) - g(z)| < \varepsilon (| z| )\) \(\forall z \in S\}\). Then \(U(D,0) \cap A\) is residual in \(A\). An interesting question about the existence of universal Taylor series bounded on a sector is posed.

30B30 Boundary behavior of power series in one complex variable; over-convergence
30E10 Approximation in the complex plane
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