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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 \(A_n(f,\zeta)=\sum a_{nk} S_k(f,\zeta)\).

30E10 Approximation in the complex plane
30B10 Power series (including lacunary series) in one complex variable
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