Universal Taylor series and summability

Charpentier, Stéphane; Mouze, Augustin

doi:10.1007/s13163-014-0156-4

Cited by 13 publications

(18 citation statements)

References 17 publications

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“…We say that it has Ostrowski-gaps Gehlen, Müller and Luh proved that every universal series possesses Ostrowski-gaps [16]. Recently the authors of [13] introduced the notion of large Ostrowski-gaps where large refers to how fast q m /p m tends to ∞. More precisely, let us call a weight, a strictly increasing We denote by U (µ,ϕ) (D) the set of such power series.…”

Section: From Universal Series To Padé Universal Seriesmentioning

confidence: 99%

“…More precisely, let us call a weight, a strictly increasing We denote by U (µ,ϕ) (D) the set of such power series. It has been proved in [13] that, for any weight ϕ, the set U (µ,ϕ) (D) is residual in H(D). Actually it can be checked that the proof of Proposition 5.3 easily implies the following one stating the density of a very particular class of universal series.…”

Section: From Universal Series To Padé Universal Seriesmentioning

confidence: 99%

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Generic properties of Padé approximants and Padé universal series

2015

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Abstract. We establish properties concerning the distribution of poles of Padé approximants, which are generic in Baire category sense. We also investigate Padé universal series, an analog of classical universal series, where Taylor partial sums are replaced with Padé approximants. In particular, we complement previous studies on this subject by exhibiting dense or closed infinite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Padé universal series. We also show how Padé universal series can be built from classical universal series with large Ostrowski-gaps.

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Section: From Universal Series To Padé Universal Seriesmentioning

confidence: 99%

Section: From Universal Series To Padé Universal Seriesmentioning

confidence: 99%

Generic properties of Padé approximants and Padé universal series

2015

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“…We can compare this result with Theorem 3.8 of [10], where the authors give topologically generic subsets of U(D) with large Ostrowski-gaps whose universal property is preserved under regular summability methods.…”

Section: Remark 323 (1) It Is Easy To Check That the Functionmentioning

confidence: 71%

“…We would like to mention that the fact that Cesàro means preserve the universality is connected to a more general problem (see [12]): which maps preserve universal functions? In [10] the authors relate the size of the aforementioned Ostrowski-gaps to the invariance of the universality properties under general matrix summability methods. Therefore they obtain classes of universal Taylor series with large Ostrowski-gaps that are automatically preserved under summability methods.…”

Section: Introductionmentioning

confidence: 99%

Polynomial inequalities and universal Taylor series

Mouze¹,

Munnier

2016

Math. Z.

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We derive several new properties concerning both universal Taylor series and Fekete universal series from classical polynomial inequalities. In particular, we study some density properties of their approximating subsequences. Moreover we exhibit summability methods which preserve or imply the universality of Taylor series in the complex plane. Likewise we show that the partial sums of the Taylor expansion around zero of a C ∞ function is universal if and only if the sequence of its Cesàro means satisfies the same universal approximation property.

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“…Then Theorem 3.3 of [16] ensures that there exists a subsequence (λ n ) of positive integers with dens(λ n ) = 1 such that sup z∈K |S λn (f )(z) − h(z)| → 0, as n → +∞. According to Section 4 of [6] we have sup z∈K |σ λn (f )(z) − h(z)| → 0. We end as in the proof of [16,Theorem 3.3].…”

Section: Further Development and Remarkmentioning

confidence: 98%

On doubly universal functions

Mouze

2018

Journal of Approximation Theory

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Abstract. Let (λn) be a strictly increasing sequence of positive integers. Inspired by the notions of topological multiple recurrence and disjointness in dynamical systems, Costakis and Tsirivas have recently established that there exist power series k≥0 a k z k with radius of convergence 1 such that the pairs of partial sums {(. . } approximate all pairs of polynomials uniformly on compact subsets K ⊂ {z ∈ C : |z| > 1}, with connected complement, if and only if lim sup n λn n = +∞. In the present paper, we give a new proof of this statement avoiding the use of advanced tools of potential theory. It allows to obtain the algebraic genericity of the set of such power series and to study the case of doubly universal infinitely differentiable functions. Further we show that the Cesàro means of partial sums of power series with radius of convergence 1 cannot be frequently universal.

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Universal Taylor series and summability

Abstract: We introduce classes of universal Taylor series, both topologically and algebraically generic, whose image under some regular matrix summability methods are automatically universal.

Cited by 13 publications

References 17 publications

Generic properties of Padé approximants and Padé universal series

Generic properties of Padé approximants and Padé universal series

Polynomial inequalities and universal Taylor series

On doubly universal functions

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