Universal power series of Seleznev with parameters in several variables

2020/08/09 by Maronikolakis, Konstantinos, Stamatiou, Giorgos
#30K05 #32A05 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2008.03780

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 = ∏_{i = 1}^d K_i, where K_i ⊆ ℂ are compact sets and ℂ ∖ K_i are connected, i = 1, …, d and 0 ∉ 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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