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Universal power series of Seleznev with parameters in several variables. (English) Zbl 1469.30119

Summary: We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products \(K = \displaystyle \prod \nolimits_{i = 1}^d K_i\), where \(K_i \subseteq \mathbb{C}\) are compact sets and \(\mathbb{C} {\setminus } K_i\) are connected, \(i = 1, \ldots , d\) and \(0 \notin K\). On such \(K\) the partial sums approximate uniformly any polynomial. Finally, the partial sums may be replaced by more general expressions. The phenomenon is topologically and algebraically generic.

MSC:

30K05 Universal Taylor series in one complex variable
32A05 Power series, series of functions of several complex variables
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References:

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