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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第1期48卷 1124-1141页

作者： Aktuglu, Huseyin Kara, Mustafa Baytunc, Erdem Eastern Mediterranean Univ Fac Art & Sci Dept Math Mersin Turkiye

In this article, we introduce a modified class of bernstein-kantorovich operators dependingonan integrable function psi(alpha) and investigate their approximation properties. By choosing an appropriate function f, the order of approxi-mation of our operators to a function psi(alpha) is at least as good as the classical bernstein-kantorovich operators on the interval[0,1]. We compared the operators defined in this study not only with bernstein-kantorovich operators butalso with some other bernstein-kantorovich type operators. In this paper, wealso study the results on the uniform convergence and rate of convergenceof these operators in terms of the first- and second-order moduli of continuity, and we prove that our operators have shape-preserving properties. Finally,some numerical examples which support the results obtained in this paper areprovided.

关键词： bernstein operators bernstein-kantorovich operators polynomial approximation rate of convergence modulus of continuity shape-preserving properties uniform convergence

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Approximation by Stancu-type α-bernstein-Schurer-kantorovich operators

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2025年第1期2025卷 1-16页

作者： Nasiruzzaman, Md. Univ Tabuk Fac Sci Dept Math POB 4279 Tabuk 71491 Saudi Arabia

In the present article, we study the approximation properties of constructed operators based on the shape parameter alpha. We construct the Stancu-type operators of alpha-bernstein-Schurer-kantorovich operators. Here the shape parameter alpha is an element of[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in [0,1]$\end{document}. We obtain the convergence of proposed operators in terms of Lipschitz-continuous functions and Peetre's K-functional by using the modulus of continuity of orders one and two.

关键词： bernstein-operators bernstein-Schurer operators alpha-bernstein-operators alpha-bernstein-Schurer operators bernstein-kantorovich operators Modulus of continuity Lipschitz-spaces and Peetre's K-functional

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bernstein-kantorovich operators, approximation and shape preserving properties

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2024年第3期118卷 1-18页

作者： Acu, Ana-Maria Rasa, Ioan Steopoaie, Ancuta Emilia Lucian Blaga Univ Sibiu Dept Math & Informat Sibiu Romania Tech Univ Cluj Napoca Fac Automat & Comp Sci Dept Math Str Memorandumului 28 Cluj Napoca 400114 Romania

Inspired by the construction of bernstein and kantorovich operators, we introduce a family of positive linear operators K n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}_n$$\end{document} preserving the affine functions. Their approximation properties are investigated and compared with similar properties of other operators. We determine the central moments of all orders of K n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {K}}}_n$$\end{document} and use them in order to establish Voronovskaja type formulas. A special attention is paid to the shape preserving properties. The operators K n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {K}}}_n$$\end{document} preserve monotonicity, convexity, strong convexity and approximate concavity. They have also the property of monotonic convergence under convexity. All the established inequalities involving convex functions can be naturally interpreted in the framework of convex stochastic ordering.

关键词： bernstein-kantorovich operators Shape preserving properties Inequalities for convex functions

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Approximation of functions by Stancu variant of bernstein-kantorovich operators based on shape parameter α

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2020年第2期114卷 1-17页

作者： Mohiuddine, S. A. - Ozger, Faruk King Abdulaziz Univ Fac Appl Studies Dept Gen Required Courses Math Jeddah 21589 Saudi Arabia King Abdulaziz Univ Dept Math Operator Theory & Applicat Res Grp Jeddah 21589 Saudi Arabia Izmir Katip Celebi Univ Dept Engn Sci TR-35620 Izmir Turkey

We construct the Stancu variant of bernstein-kantorovich operators based on shape parameter alpha. We investigate the rate of convergence of these operators by means of suitable modulus of continuity to any continuous functions f(x) on x is an element of and Voronovskaja-type approximation theorem. Moreover, we study other approximation properties of our new operators such as weighted approximation as well as pointwise convergence. Finally, some illustrative graphics are provided here by our new Stancu-type bernstein-kantorovich operators in order to demonstrate the significance of our operators.

关键词： bernstein-kantorovich operators Rate of convergence Weighted approximation Pointwise convergence

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Smoothness Properties of Modified bernstein-kantorovich operators

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2016年第1期37卷 92-105页

作者： Ozarslan, Mehmet Ali Duman, Oktay Eastern Mediterranean Univ Dept Math Gazimagusa Mersin Turkey TOBB Econ & Technol Univ Dept Math TR-06530 Ankara Turkey

In this article, we consider modified bernstein-kantorovich operators and investigate their approximation properties. We show that the order of approximation to a function by these operators is at least as good as that of ones classically used. We obtain a simultaneous approximation result for our operators. Also, we prove two direct approximation results via the usual second-order modulus of smoothness and the second-order Ditzian-Totik modulus of smoothness, respectively. Finally, some graphical illustrations are provided.

关键词： bernstein-kantorovich operators Ditzian-Totik modulus of smoothness modulus of continuity simultaneous approximation

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Construction of a new family of bernstein-kantorovich operators

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2017年第18期40卷 7749-7759页

作者： Mohiuddine, S. A. Acar, Tuncer Alotaibi, Abdullah King Abdulaziz Univ Fac Sci Dept Math Operator Theory & Applicat Res Grp POB 80203 Jeddah 21589 Saudi Arabia Kirikkale Univ Fac Sci & Arts Dept Math TR-71450 Yahsihan Kirikkale Turkey

In the present paper, we construct a new sequence of bernstein-kantorovich operators depending on a parameter alpha. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first- and second-order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of bernstein-kantorovich operators and their approximation behaviors.

关键词： bernstein-kantorovich operators bivariate bernstein-kantorovich operators modulus of continuity rate of convergence

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Approximation by a new kind of(λ,μ)-bernstein-kantorovich operators

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第5期43卷 1-18页

作者： Cai, Qing-Bo Quanzhou Normal Univ Sch Math & Comp Sci Quanzhou 362000 Fujian Peoples R China

In this manuscript, a new family of (lambda,mu)-bernstein-kantorovich operators are introduced. A Kovovkin type approximation theorem is obtained, the rate of convergence and A-statistical convergence are investigated by using the modulus of smoothness, Steklov mean and Korovkin-type statistical approximation theorem, graphical representations and numerical examples are also presented to compare the newly defined ones with other forms. Finally, an asymptotic estimate on the rate of convergence for some absolutely continuous functions are derived by using the Bojainc-Cheng decomposition method and some analysis techniques.

关键词： bernstein operators bernstein-kantorovich operators Basis function Rate of convergence Steklov mean

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THE STRONG CONVERSE INEQUALITY FOR bernstein-kantorovich operators

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 1995年第3-6期30卷 103-128页

作者： GONSKA, HH ZHOU, XL University of Duisburg D - 47057 Duisburg Germany

We give a solution to a problem posed by Totik at the 1992 Texas conference concerning the strong converse inequality for approximation by bernstein-kantorovich operators. The approximation behaviour of these operators is characterized for 1 less than or equal to p less than or equal to infinity by using an appropriate K-functional which, for 1 < p less than or equal to infinity, is equivalent to a second order modulus and an extra term. Crucial in our approach are estimates for the derivatives of iterated kantorovich operators.

关键词： bernstein-kantorovich operators STRONG CONVERSE INEQUALITY DEGREE OF APPROXIMATION K-FUNCTIONAL

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APPROXIMATION PROPERTIES OF RIEMANN-LIOUVILLE TYPE FRACTIONAL bernstein-kantorovich operators OF ORDER

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MATHEMATICAL FOUNDATIONS OF COMPUTING 2024年第4期7卷 544-567页

作者： Baytunc, Erdem Aktuglu, Huseyin Mahmudov, Nazim I. Eastern Mediterranean Univ Dept Math 10 Mersin TR-99450 Famagusta TR Northern Cyp Turkiye

. In this paper, we construct a new sequence of Riemann-Liouville type fractional bernstein-kantorovich operators Kn & alpha;(f;x) depending on a parameter & alpha;. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first and second order modulus of continuity of these operators. Moreover, we introduce a new operator that preserves affine functions from Riemann-Liouville type fractional bernstein-kantorovich operators. Further, we define the bivariate case of Riemann-Liouville type fractional bernstein-kantorovich operators and investigate the order of convergence. Some numerical results are given to illustrate the convergence of these operators and its comparison with the classical case of these operators.

关键词： bernstein-kantorovich operators rate of convergence modulus of continuity bivariate bernstein-kantorovich operators affine functions positive linear operators

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Korovkin type theorem for bernstein-kantorovich operators via power summability method

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ANALYSIS AND MATHEMATICAL PHYSICS 2020年第4期10卷 62-62页

作者： Braha, Naim L. Univ Prishtina Dept Math & Comp Sci Ave Mother Teresa 5 Prishtine 10000 Kosovo

In this paper we will prove the Korovkin type theorem for bernstein-kantorovich type operators via A- statistical convergence and power summability method. Also we give the rate of the convergence related to the above summability methods and in the last sections we give a kind of Voronovskaya type theorem for A- statistical convergence and Gruss-Voronovskaya type theorem.

关键词： A-statistical convergence bernstein-kantorovich operators Power summability method Korovkin type theorem Voronovskaya type theorem Rate of convergence

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