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Boundary behavior of universal Taylor series and their derivatives. (English) Zbl 1098.30003
Summary: For a given first category subset \(E\) of the unit circle and any given holomorphic function \(g\) on the open unit disk, we construct a universal Taylor series \(f\) on the open unit disk, such that, for every \(n = 0,1,2,\dots, f^{(n)}\) is close to \(g^{(n)}\) on a set of radii having endpoints in \(E\). Therefore, there is a universal Taylor series \(f\), such that \(f\) and all its derivatives have radial limits on all radii with endpoints in \(E\). On the other hand, we prove that if \(f\) is a universal Taylor series on the open unit disk, then there exists a residual set \(G\) of the unit circle, such that for every strictly positive integer \(n\), the derivative \(f^{(n)}\) is unbounded on all radii with endpoints in the set \(G\).

MSC:
30B30 Boundary behavior of power series in one complex variable; over-convergence
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