Universal Taylor series

Nestoridis, Vassili

doi:10.5802/aif.1549

Cited by 165 publications

(186 citation statements)

References 13 publications

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“…The set U (D) of universal Taylor series in D has been proved to be non-void and especially G δ (countable intersection of open sets) and dense in H(D) endowed with the topology of uniform convergence on compacta [11].…”

Section: Introductionmentioning

confidence: 99%

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Universal Laurent Series

Costakis

Nestoridis²,

Papadoperakis³

2005

Proceedings of the Edinburgh Mathematical Society

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Section: Introductionmentioning

confidence: 99%

“…Furthermore, some properties of such universal Taylor series have been investigated in [6][7][8]11]. …”

Section: Introductionmentioning

confidence: 99%

Universal Laurent Series

Costakis

Nestoridis²,

Papadoperakis³

2005

Proceedings of the Edinburgh Mathematical Society

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“…This definition is due to V. Nestoridis who proved the existence and the basic properties of such series in the framework of a project studying the behaviour of partial sums of Taylor series (see [7], [8]). …”

Section: An Applicationmentioning

confidence: 99%

“…Now we shall make some comments for the class of Universal Taylor Series. As we mentioned before, several questions arise naturally about this class of series (see [3], [4], [6] and [7]). Although some of them have been answered, there are others which remain open.…”

Section: Remarks and Commentsmentioning

confidence: 99%

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Extensions of a theorem of Marcinkiewicz-Zygmund and of Rogosinski’s formula and an application to Universal Taylor series

Katsoprinakis

Papadimitrakis

1999

Proc. Amer. Math. Soc.

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Simultaneous approximation by universal series

Tsirivas

2010

Mathematische Nachrichten

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We show that, if individual universal series exist, then we can choose a sequence of universal series performing simultaneous universal approximation with the same sequence of indices. As an application we derive the existence of universal Laurent Series on an annulus using only the existence of universal Taylor Series on discs. Our results are generic.In the early 70's W. Luh [12] and independently Chui and Parnes [6] proved the existence of universal Taylor series in simply connected domains Ω with respect to a fixed center ζ in Ω. Namely there exists a holomorphic function f ∈ H(Ω) such that the partial sums of its Taylor expansion with center ζ uniformly approximate all polynomials on any compact set K ⊂ (Ω) c with connected complement.In 1986 W. Luh [13] proved the existence of a holomorphic function f ∈ H(Ω) which is a universal Taylor series with respect to every center ζ ∈ Ω. Furthermore, if a polynomial h and a compact set K as previously are given, then the indices λ n defining the subsequence of the partial sums which approximate h on K can be taken as the same for all centers ζ ∈ Ω. Thus we have a first example of simultaneous universal approximation.In the previous example the universal approximation is not requested to be valid on the boundary of Ω. However in 1996 V. Nestoridis [15] strengthened the notion of the universal Taylor series allowing the compact set K to meet ∂Ω; that is K ⊂ Ω c and the universal approximation is also valid on ∂Ω. Furthermore he obtained simultaneous universal approximation with the same sequence of indices and uniformly when the center ζ varies on compact subsets of the simply connected domain Ω [14,16]. The last gives us an example of simultaneous universal approximation.In [14] another generic property of holomorphic functions f on a simply connected domain Ω is found. Namely subsequences of the partial sums of the Taylor development of f approximate f uniformly on compacta not only of a disc but of the whole simply connected domain Ω.The two generic approximations, one inside Ω towards f and the other outside Ω are performed simultaneously by the same subsequence of partial sums [14].In [8] Costakis and Vlachou have proven a theorem where 5 generic approximations are performed simultaneously by the same sequence of indices.Recently an abstract theory of universal series has been obtained [2,18]. This abstract framework allows us to view in a unified way most of the known results and provides extremely simple and short proofs. Further the simplicity of this point of view allows us to obtain several new results. The purpose of the present paper is to establish a generic result for simultaneous universal approximation for a finite or infinite denumerable set of universalities in the form of universal series according to [18] and [2]. We also obtain algebraic genericity (see [1]), that is, the existence of dense linear manifolds of series satisfying such universalities. *

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Universal Taylor series

Cited by 165 publications

References 13 publications

Universal Laurent Series

Universal Laurent Series

Extensions of a theorem of Marcinkiewicz-Zygmund and of Rogosinski’s formula and an application to Universal Taylor series

Simultaneous approximation by universal series

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