Families of Universal Taylor Series Depending on a Parameter

Abakumov, Evgeny; Müller, Jürgen; Nestoridis, Vassili

doi:10.1007/978-1-4939-7543-3_10

Cited by 2 publications

(2 citation statements)

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“…In the present paper we obtain several extensions of the result in [1] and other results in the case of several complex variables. We divide the set of variables into two groups: one group, the set of parameters and a second group, the set of variables appearing in the Taylor expansion.…”

Section: Introductionsupporting

confidence: 55%

See 1 more Smart Citation

Universal Taylor Series in Several Variables Depending on Parameters

Giorgos

Maronikolakis

Nestoridis

2021

Comput. Methods Funct. Theory

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We establish generic existence of Universal Taylor Series on products $$\varOmega = \prod \varOmega _i$$ Ω = ∏ Ω i of planar simply connected domains $$\varOmega _i$$ Ω i where the universal approximation holds on products K of planar compact sets with connected complements provided $$K \cap \varOmega = \emptyset $$ K ∩ Ω = ∅ . These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that $$K \cap \overline{\varOmega } = \emptyset $$ K ∩ Ω ¯ = ∅ and $$\{\infty \} \cup [{\mathbb {C}} {\setminus } \overline{\varOmega }_i]$$ { ∞ } ∪ [ C \ Ω ¯ i ] is connected for all i. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper.

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Section: Introductionsupporting

confidence: 55%

“…In [1] the authors consider families of universal Taylor series depending on a parameter; then the function h to be approximated by the partial sums can depend on the same parameter. This led to functions of several complex variables; see also [4,5,12].…”

Section: Introductionmentioning

confidence: 99%