Abstract
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.
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Acknowledgements
The authors would like to thank Professor Vassili Nestoridis for helpful communications and guidance throughout the creation of this paper. They would also like to thank the reviewers for helpful suggestions.
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Communicated by Adrian Constantin.
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Maronikolakis, K., Stamatiou, G. Universal power series of Seleznev with parameters in several variables. Monatsh Math 195, 477–488 (2021). https://doi.org/10.1007/s00605-020-01509-1
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DOI: https://doi.org/10.1007/s00605-020-01509-1