Rates of Ideal Convergence for Approximation Operators

Duman, Oktay; Özarslan, Mehmet Ali; Erkuş-Duman, Esra

doi:10.1007/s00009-010-0031-6

Cited by 10 publications

(8 citation statements)

References 13 publications

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“…Also the case of not necessarily positive operators is considered, following an approach given in [5]. Our results extend Korovkin-type theorems given in [8,10,17,18] in the context of modular spaces and in [16] in the setting of ideal convergence. Note that at least the results concerning positive operators can be extended to more general kinds of convergence, not necessarily generated by free filters or regular matrix methods: among them we recall almost convergence [25].…”

Section: Introductionsupporting

confidence: 67%

Abstract Korovkin-type theorems in modular spaces and applications

et al. 2013

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Abstract:We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones. MSC:40A35, 41A35, 46E30

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Section: Introductionsupporting

confidence: 67%

Abstract Korovkin-type theorems in modular spaces and applications

et al. 2013

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“…whenever s ∈ N. This contradicts (33), and thus (34) is proved. Set now, for all j, n ∈ N, a (j) n = m j ({n}), and for every j ∈ N, put a (j) = (a (j) n ) n .…”

Section: Proofmentioning

confidence: 82%

“…From (33) it follows that for any ϕ ∈ N N there exists j with the property that, for each j ≥ j and p ∈ N,…”

Section: Proofmentioning

confidence: 99%

“…These kinds of theorems have several functional analytic applications, and they are related with matrix theorems, which are powerful tools to give some results both in measure theory and in the context of operators (see for instance [1,21,40,51]). Some other applications to integration, control measures and signal processing can be found, for example, in [5,20,33,38].…”

Section: Introductionmentioning

confidence: 99%

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Limit Theorems in (<em>l</em>)-Groups with Respect to (<em>D</em>)-Convergence

Boccuto¹,

Dimitriou²,

Papanastassiou³

2012

Real Analysis Exchange

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Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for (l)-group-valued measures are given in the context of (D)-convergence. We consider both the σ-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measures is intended relatively to a common (D)-sequence. Among the tools, we use the Fremlin lemma, which allows us to replace a countable family of (D)-sequence with one regulator, and the Maeda-Ogasawara-Vulikh representation theorem for Archimedean lattice groups.

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“…There have been also several studies on Korovkin-type theorems related to convergence associated with summability methods, statistical and filter convergence (see e.g., [5,8,9,29,[32][33][34]49]).…”

Section: Introductionmentioning

confidence: 99%