Universal Taylor series on convex subsets of

Daras, Nicholas J.; Nestoridis, Vassili

doi:10.1080/17476933.2015.1036048

Cited by 7 publications

(4 citation statements)

References 27 publications

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“…In several complex variables approximation theory is less developed and there is no satisfactory Mergelyan's theorem. In consequence there are only a few theorems for the existence of some kind of universal Taylor series in several variables ( [2], [4]). Recently, during a Research in pairs program at the Institute of Oberwolfach [5] a new Mergelyan's type theorem was obtained for products of compact planar sets.…”

Section: Introductionmentioning

confidence: 99%

Universal Taylor Series on products of planar domains

Kioulafa

Kotsovolis

Nestoridis

2019

Preprint

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Using a recent Mergelyan type theorem for products of planar compact sets we establish generic existence of Universal Taylor Series on products of planar simply connected domains Ω i , i = 1, . . . , d. The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets K i , i = 1, . . . , d such that C − K i is connected and for at least one i 0 the set K i0 is disjoint from Ω i0 .

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Section: Introductionmentioning

confidence: 99%

Universal Taylor Series on products of planar domains

Kioulafa

Kotsovolis

Nestoridis

2019

Preprint

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“…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%

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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“…In [1] we 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], [11].…”

Section: Introductionmentioning

confidence: 99%

Universal Taylor Series in several variables depending on parameters

Giorgos¹,

Maronikolakis²,

Nestoridis³

2020

Preprint

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We establish generic existence of Universal Taylor Series on products Ω = Ω i of planar simply connected domains Ω i where the universal approximation holds on products K of planar compact sets with connected complements provided 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 ∩ Ω = ∅ and {∞} ∪ [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. These universalities are topologically and algebraically generic.

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Universal Taylor series on convex subsets of

Cited by 7 publications

References 27 publications

Universal Taylor Series on products of planar domains

Universal Taylor Series on products of planar domains

Universal Taylor Series in Several Variables Depending on Parameters

Universal Taylor Series in several variables depending on parameters

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