On Convergence Properties for a Class of Kantorovich Discrete Operators

Bardaro, Carlo; Mantellini, Ilaria

doi:10.1080/01630563.2011.652270

Cited by 38 publications

(23 citation statements)

References 14 publications

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“…In view of the above equality, since all the sequences of the form π w y , y ∈ R, w > 0, satisfy the conditions required in assumption (8), and Φ 1 < +∞, using (15) we finally have:…”

Section: Inverse Resultsmentioning

confidence: 99%

An Inverse Result of Approximation by Sampling Kantorovich Series

Costarelli¹,

Vinti²

2018

Proceedings of the Edinburgh Mathematical Society

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In the present paper, an inverse result of approximation, i.e., a saturation theorem for the sampling Kantorovich operators is derived, in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, here we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by P.L. Butzer. At the end, some other applications of such representation formula are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

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“…In view of the above equality, since all the sequences of the form π w y , y ∈ R, w > 0, satisfy the conditions required in assumption (8), and Φ 1 < +∞, using (15) we finally have:…”

Section: Inverse Resultsmentioning

confidence: 99%

An Inverse Result of Approximation by Sampling Kantorovich Series

Costarelli¹,

Vinti²

2018

Proceedings of the Edinburgh Mathematical Society

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“…The problem of the order of approximation for the sampling Kantorovich series, has been largely studied in various papers, see e.g. [8,27,28,29], both for functions of one and several variables. The approach used in the papers [27,28,29] involves functions belonging to suitable Lipschitz classes (see e.g., [9]) in the space of continuous functions and in Orlicz spaces.…”

Section: Linear Prediction and Order Of Approximationmentioning

confidence: 99%

“…The approach used in the papers [27,28,29] involves functions belonging to suitable Lipschitz classes (see e.g., [9]) in the space of continuous functions and in Orlicz spaces. While, the estimates established in [8] were given for continuous functions only, and by employing the modulus of continuity of the function being approximated. We recall that, the modulus of continuity of a given uniformly continuous function f : R → R is defined by:…”

Section: Linear Prediction and Order Of Approximationmentioning

confidence: 99%

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Approximation of discontinuous signals by sampling Kantorovich series

Costarelli

Minotti

Vinti

2017

Journal of Mathematical Analysis and Applications

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In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous signals are considered in the above sampling series. Moreover, the rate of approximation for the family of the above operators is estimated, when uniformly continuous and bounded signals are considered. Further, also the problem of the linear prediction by sampling values from the past is analyzed. At the end, the role of durationlimited kernels in the previous approximation processes has been treated, and several examples have been provided.

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“…In particular, Kantorovich-type operators have been successfully used in order to approximate/reconstruct not necessarily continuous data (see e.g., [5,24]), and consequently they revealed to be suitable for applications to image processing, see e.g., [2].…”

Section: Introductionmentioning

confidence: 99%

Approximation Results in Orlicz Spaces for Sequences of Kantorovich Max-Product Neural Network Operators

Costarelli

Sambucini

2018

Results Math

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In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces L ϕ . The results here proved, extend those given by the authors in Result Math., 2016, to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies in the possibility to approximate not necessarily continuous functions (data) belonging to different function spaces by a unique general approach. Further, in order to derive quantitative estimates in this context, we introduce a suitable K-functional in L ϕ and use it to provide an upper bound for the approximation error of the above operators. Finally, examples of sigmoidal activation functions have been considered and studied.1991 Mathematics Subject Classification. Primary: 41A25, 41A05, 41A30, 47A58.

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On Convergence Properties for a Class of Kantorovich Discrete Operators

Cited by 38 publications

References 14 publications

An Inverse Result of Approximation by Sampling Kantorovich Series

An Inverse Result of Approximation by Sampling Kantorovich Series

Approximation of discontinuous signals by sampling Kantorovich series

Approximation Results in Orlicz Spaces for Sequences of Kantorovich Max-Product Neural Network Operators

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