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Universal Taylor series on convex subsets of \(\mathbb{C}^n\). (English) Zbl 1336.32002

Let \(\mathbb{M}\) be an infinite subset of \(\mathbb{N}\), \(\Omega \subset \mathbb{C}^n\) be an open convex set, \(\zeta \in \Omega\) be fixed, \((K_m)_{m \in \mathbb{N}}\) be a family of compact convex subsets of \(\mathbb{C}^n\) disjoint from \(\Omega\) and \((N_j)_{j \in \mathbb{N}_0}\) be any enumeration of \(\mathbb{N}^n\).
Combining facts from the theory of functions of several complex variables with the abstract theory of universal series, the authors show that there exists a function \(f\) holomorphic in \(\Omega\), for short \(f \in {\mathcal O}(\Omega)\), such that the partial sums \(S_\lambda(f,\zeta)\) of the Taylor expansion of \(f\) around \(\zeta\) satisfy the following. For every compact convex set \(K \subset \mathbb{C}^n\) which is disjoint from \(\overline{\Omega}\) or is equal to \(K_m\) for some \(m \in \mathbb{N}\) and every analytic polynomial \(P\), there exists a sequence \((\lambda_N)_{N \in \mathbb{N}} \subset \mathbb{M}\) such that \(S_{\lambda_N}(f,\zeta) \to P\) uniformly on \(K\), and \(S_{\lambda_N}(f,\zeta) \to f\) uniformly on each compact subset of \(\Omega\), as \(N \to \infty\).
The set of all such \(f\) is a dense \(G_\delta\) subset of \({\mathcal O}(\Omega)\) and contains a dense vector space except 0, where \({\mathcal O}(\Omega)\) is endowed with the topology of uniform convergence on compacta.
If the universal approximation is only required on convex compact sets disjoint from \(\overline{\Omega}\), then \(f\) may chosen to be smooth on the boundary of \(\Omega\), that is \(f \in A^{\infty}(\Omega)\).

MSC:

32A05 Power series, series of functions of several complex variables
30K05 Universal Taylor series in one complex variable
32A30 Other generalizations of function theory of one complex variable
40A05 Convergence and divergence of series and sequences
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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