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Universality of Taylor series as a generic property of holomorphic functions. (English) Zbl 0985.30023
In this paper the authors study several questions concerning universal Taylor series. In particular, for a given domain $$\Omega$$ in $$\mathbb{C}$$ and a point $$\zeta\in \Omega$$ they derive properties of the class $$U(\Omega, \zeta)$$ consisting of all functions $$f$$ holomorphic in $$\Omega$$ such that the partial sums $$S_n(f,\zeta)$$ of the Taylor development of $$f$$ around $$\zeta$$ have the following universality property: For every compact set $$K$$ in $$\mathbb{C} \setminus \Omega$$ with connected complement and for every function $$h$$ which is continuous on $$K$$ and holomorphic in the interior of $$K$$ a sequence $$(n_j)$$ exists such that $$S_{n_j} (f,\zeta)(z)\to h(z)$$ uniformly on $$K$$. Among others it is shown that
(1) If $$\Omega$$ is the unit disk, every $$f\in U(\Omega,0)$$ is a universal trigonometric series in the sense of Menchoff.
(2) $$U(\Omega, \zeta)$$ does not contain rational functions.
(3) The sequence $$(S_n(f,\zeta) (z))$$ is not $$(C,k)$$-summable for any $$z\in\partial\Omega$$ and any $$k=1,2,\dots.$$
(4) No $$f\in U(\Omega,\zeta)$$ extends continuously to $$\overline\Omega$$.
Moreover, the class $$U(\Omega)$$, is roughly speaking, the class of all $$f$$ holomorphic in $$\Omega$$ such that the above universality property holds for all $$\zeta\in\Omega$$ locally uniformly in $$\Omega$$. The authors prove that every function $$f$$ holomorphic in $$\Omega$$ is the difference of two functions in $$U (\Omega)$$ and that no $$f\in U(\Omega)$$ extends holomorphically across the boundary of $$\Omega$$. In addition, they generalize the known fact that $$U(\Omega)$$ is $$G_\delta$$-dense in the space $$H(\Omega)$$ of all functions holomorphic in $$\Omega$$ with the topology of locally uniform convergence to classes of functions where the partial sums $$S_n(f,\zeta)$$ are replaced by matrix transforms $$A_n(f,\zeta)=\sum a_{nk} S_k(f,\zeta)$$.

##### MSC:
 30E10 Approximation in the complex plane 30B10 Power series (including lacunary series) in one complex variable
##### Keywords:
universal functions; universal Taylor series
Full Text:
##### References:
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