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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 Fréchet 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.

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