Abstract theory of universal series and applications

Bayart, Frédéric; Grosse‐Erdmann, Karl‐Goswin; Nestoridis, Vassili; Papadimitropoulos, C.

doi:10.1112/plms/pdm043

Cited by 126 publications

(216 citation statements)

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“…Let Y be the set of all such subsets Γ of Ω. If Γ ∈ Y and K, h are as in Definition 1.1, then we say that a holomorphic function f ∈ H(Ω) belongs to the class U (Ω, Γ), if there exists a sequence (λ n ) of natural numbers such that the subsequence S λn (f, Γ)(z) of the partial sums of the Faber series of f , converges to h(z) uniformly on K. On the other hand we say that f belongs to the class U F ab (Ω), if for any compact set M ⊂ Y , where Y is endowed with the metric topology introduced in [3], section B.5 (see also [4]), we have that sup Γ∈M sup z∈K |S λn (f, Γ)(z) − h(z)| → 0, as n → ∞.…”

Section: Lemma 32mentioning

confidence: 99%

Coincidence of some classes of universal functions

Katsoprinakis¹

2009

Rev. Mat. Complut.

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Let Ω be a domain in the complex plane such that Ω satisfies appropriate geometrical and topological properties. We prove that if f is a holomorphic function in Ω, then its Taylor series, with center at any ξ ∈ Ω, is universal with respect to overconvergence if and only if its Cesàro (C, k)-means are universal for any real k > −1. This is an extension of the same result, proved recently by F. Bayart, for any integer k ≥ 0. As a consequence, several classes of universal functions introduced in the related literature are shown to coincide.

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Section: Lemma 32mentioning

confidence: 99%

Coincidence of some classes of universal functions

Katsoprinakis¹

2009

Rev. Mat. Complut.

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“…In accordance with the abstract theory of universal series [2,5], we now state the following result.…”

Section: Introductionmentioning

confidence: 84%

“…Arguing as in [2], we construct inductively a sequence {α ν } ν≥1 ⊂ A + and sequences μ ν j , j, ν ≥ 1, of subsets of μ such that for every j, ν ≥ 1 the following conditions hold true:…”

mentioning

confidence: 99%

Approximation of probability distributions by convex mixtures of Gaussian measures

Bacharoglou

2010

Proc. Amer. Math. Soc.

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Abstract. Let A + = {a = (a n ) ∈ p>1 p : a n > 0, ∀n ∈ N} and let {φ j } ∞ j=1 be an enumeration of all normal distributions with mean a rational number and variance 1 n 2 , n = 1, 2 . . . . We prove that there exists an a ∈ A + such that every probability density function, continuous, with compact support in R, can be approximated in L 1 and L ∞ norm simultaneously by the averagesThe set of such sequences is a dense G δ set in A + and contains a dense positive cone.

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“…Let us recall the abstract theory of universal series [12], [1]. Let X be a metrizable topological vector space over the field K = R or C. Let ρ be a translation invariant metric compatible with the vector space operations of X.…”

Section: Universality In a Space Of Formal Power Series Of Beurling Typementioning

confidence: 99%

“…Furthermore it appears that the proofs need to exhibit a polynomial which approximates both a given function in the space where the universal function should live and a given function in the space where the universal property holds. Combining this fact with Baire's Theorem, Bayart, Grosse-Erdmann, Nestoridis and Papadimitropoulos in [1] develop an abstract theory of universal series, from which they deduce easily and in a unified way the existing results as well as new statements. In particular they obtain results about universal expansions of C ∞ functions on arbitrary open subsets of R l (l ≥ 1), as well as an extension of Fekete's Theorem at the level of formal power series in R l .…”

Section: Introductionmentioning

confidence: 99%