A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Duman, Oktay

doi:10.1007/s10587-007-0065-5

Cited by 14 publications

(12 citation statements)

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“…This notion has been several applications in the very recent literature. Among them we recall weak ideal compactness in measure spaces (see also Boccuto, Das, Dimitriou, & Papanastassiou, 2012;Boccuto, Dimitriou, & Papanastassiou, 2010b;Dimitriou, 2011), Approximation Theory, signal sampling and reconstruction of images (see also Bardaro, Boccuto, Dimitriou, & Mantellini, 2012, 2013, 2013cDuman, 2007;Higgins & Stens, 1999), and in particular Brooks-Jewett, Dieudonné, Nikodým convergence and Vitali-Hahn-Saks-type theorems, with which we deal in this paper.…”

Section: Introductionmentioning

confidence: 99%

Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

Boccuto¹,

Dimitriou²

2013

JMR

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We prove some equivalence results between limit theorems for sequences of ( )-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same (O)-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems.

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Section: Introductionmentioning

confidence: 99%

Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

Boccuto¹,

Dimitriou²

2013

JMR

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“…[2], [3], [4], [5], [9], [15]) have been obtained by replacing the classical convergence with new methods of convergence, such as statistical and A-statistical convergence.…”

Section: Introductionmentioning

confidence: 99%

Rates of Ideal Convergence for Approximation Operators

2010

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In this paper, we study a general Korovkin-type approximation theory by using the notion of ideal convergence which includes many convergence methods, such as, the usual convergence, statistical convergence, A-statistical convergence, etc. We mainly compute the rate of ideal convergence of sequences of positive linear operators.Mathematics Subject Classification (2010). Primary 41A10; Secondary 41A25.

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“…It is known that, if I 1 is the class of all finite subsets of N then I -convergence coincides with the classical convergence (see [4], [17]). Therefore Theorem 3.1 reduces to the result obtained by Ç akar and Gadjiev (see [3]).…”

Section: Korovkin Type Approximation Theoremmentioning

confidence: 99%

“…Also Duman gave a Korovkin type approximation theorem via I -convergence by using the usual test functions f i (y) = y i , i = 0, 1, 2 (see [4]). …”

Section: Introductionmentioning

confidence: 99%