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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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