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Partially smooth universal Taylor series on products of simply connected domains. (English) Zbl 1451.30113

Summary: Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains \(\Omega_i\) that extend continuously on \(\prod\nolimits_{i=1}^d(\Omega_i\cup S_i)\), where \(S_i\) are subsets of \(\partial \,\Omega_i\), open in the relative topology. The universal approximation occurs on every product of compact sets \(K_i\) such that \(C-K_i\) are connected and for some \(i_0\) it holds \(K_{i_0}\cap(\Omega_{i_0}\cup\overline{S_{i_0}})=\varnothing\).

MSC:

30K05 Universal Taylor series in one complex variable
32A05 Power series, series of functions of several complex variables
32A17 Special families of functions of several complex variables
32A30 Other generalizations of function theory of one complex variable
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