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Convergence of λ-bernstein operators via power series summability method

JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2021年第1-2期65卷 125-146页

作者： Braha, Naim L. Mansour, Toufik Mursaleen, M. Acar, Tuncer Univ Prishtina Dept Math & Comp Sci Ave Mother Teresa 5 Pristina 10000 Kosovo Res Inst Ilirias Rr Janina 2 Ferizaj 70000 Kosovo Univ Haifa Dept Math IL-3498838 Haifa Israel China Med Univ Taiwan China Med Univ Hosp Dept Med Res Taichung Taiwan Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India Asia Univ Dept Comp Sci & Informat Engn Taichung Taiwan Selcuk Univ Dept Math TR-42003 Konya Turkey

In this paper we present uniform convergence of a sequence of lambda-bernstein operators viaA-statistical convergence and power summability method. A rate of convergence of the sequence of operators are also investigated by means of above mentioned summability methods. The last section is devoted to pointwise convergence (A-statistical convergence) of the sequence of operators in terms of Voronovskaya and Gruss-Voronovskaya type theorems.

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

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Approximation by a new Stancu variant of generalized (λ, μ)-bernstein operators

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ALEXANDRIA ENGINEERING JOURNAL 2024年 107卷 205-214页

作者： Cai, Qing-Bo Aslan, Resat Ozger, Faruk Srivastava, Hari Mohan Quanzhou Normal Univ Sch Math & Comp Sci Key Lab Intelligent Comp & Informat Proc Fujian Prov Key Lab Data Intens Comp Quanzhou 362000 Peoples R China Van Yuzuncu Yil Univ Dept Math TR-65080 Van Turkiye Igdir Univ Dept Comp Engn TR-76000 Igdir Turkiye Univ Victoria Dept Math & Stat Victoria BC V8W 3R4 Canada Azerbaijan Univ Dept Math & Informat 71 Jeyhun Hajibeyli St AZ-1007 Baku Azerbaijan Kyung Hee Univ Ctr Converging Humanities 26 Kyungheedae Ro Seoul 02447 South Korea China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan Chung Yuan Christian Univ Dept Appl Math Taoyuan 320314 Taiwan

The primary objective of this work is to explore various approximation properties of Stancu variant generalized (lambda, mu)-bernstein operators. Various moment estimates are analyzed, and several aspects of local direct approximation theorems are investigated. Additionally, further approximation features of newly defined operators are delved into, such as the Voronovskaya-type asymptotic theorem and pointwise estimates. By comparing the proposed operator graphically and numerically with some linear positive operators known in the literature, it is evident that much better approximation results are achieved in terms of convergence behavior, calculation efficiency, and consistency. Finally, the newly defined operators are used to obtain a numerical solution for a special case of the fractional Volterra integral equation of the second kind.

关键词： lambda-bernstein operators Computer graphics Fractional integral equations Error analysis Pointwise estimates

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Approximation properties of λ-bernstein-Kantorovich operators with shifted knots

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2019年第11期42卷 4042-4053页

作者： Rahman, Shagufta Mursaleen, Mohammad Acu, Ana Maria Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India Lucian Blaga Univ Sibiu Dept Math & Informat Str Dr I Ratiu 5-7 Sibiu 550012 Romania

In the present article, Kantorovich variant of lambda-bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r(th) order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.

关键词： lambda-bernstein operators Kantorovich operators rate of convergence modulus of continuity

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Applications of Generalized Weighted Statistical Convergence to Approximation Theorems for Functions of One and Two Variables

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2020年第16期41卷 1990-2006页

作者： Ozger, Faruk Izmir Katip Celebi Univ Dept Engn Sci TR-35620 Izmir Turkey

In this article, we apply weighted A-statistical convergence to obtain some general approximation results for lambda-bernstein operators. We prove a weighted A-statistical Voronovskaja-type approximation theorem. We support our theoretical parts about approximation properties of constructed bivariate lambda-bernstein operators with numerical results and graphics.

关键词： lambda-bernstein operators bivariate bernstein operators rate of weighted statistical convergence statistical approximation properties weighted statistical Voronovskajatype theorem

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λ-bernstein operators Based on Polya Distribution

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2023年第6期44卷 529-544页

作者： Lipi, Km Deo, Naokant Delhi Technol Univ Dept Appl Math Delhi India Delhi Technol Univ Dept Appl Math Bawana Rd Delhi 110042 India

In this manuscript, we propose a Polya distribution-based generalization of lambda-bernstein operators. We establish some fundamental results for convergence as well as order of approximation of the proposed operators. We present theoretical result and graph to demonstrate the proposed operator's intriguing ability to interpolate at the interval's end points. In order to illustrate the convergence of proposed operators as well as the effect of changing the parameter "mu," we provide a variety of results and graphs as our paper's conclusion.

关键词： lambda-bernstein operators Lipschitz continuous functions Polya-Eggenberger distribution

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Approximation by modified Un ^ρ operators

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2019年第3期113卷 2715-2729页

作者： Acu, Ana-Maria Acar, Tuncer Radu, Voichita Adriana Lucian Blaga Univ Sibiu Dept Math & Informat Str Dr I Ratiu 5-7 Sibiu 550012 Romania Selcuk Univ Dept Math Fac Sci TR-42003 Konya Turkey Babes Bolyai Univ FSEGA Dept Stat Forecasts Math Cluj Napoca Romania

The purpose of present paper is to extend the study of lambda-bernstein operators introduce by Cai et al. (J Inequal Appl 12: 1-11, 2018). In our paper we consider a generalization of the U-n (rho) operators introduced in 2007 by Radu Paltanea, using the new bernstein-Bezier bases { b n, k} with shape parameter.. Some approximation properties are given, including local approximation, error estimation in terms of moduli of continuity and Voronovskajatype asymptotic formulas. Finally, we give some numerical examples and graphs to put in evidence the convergence of U-n (rho) (f;x) to f (x).

关键词： lambda-bernstein operators Un operators Moduli of continuity Rate of convergence Voronovskaya type theorem

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Generalized blending type bernstein operators based on the shape parameter λ

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2022年第1期2022卷 1-19页

作者： Gezer, Halil Aktuglu, Huseyin Baytunc, Erdem Atamert, Mehmet Salih Cyprus Int Univ Fac Arts & Sci Dept Basic Sci & Humanities Mersin 10 Nicosia North Cyprus Turkey Eastern Mediterranean Univ Fac Arts & Sci Dept Math Mersin 10 Famagusta North Cyprus Turkey

In the present paper, we construct a new class of operators based on new type Bezier bases with a shape parameter lambda and positive parameter s. Our operators include some well-known operators, such as classical bernstein, alpha-bernstein, generalized blending type alpha-bernstein and lambda-bernstein operators as special case. In this paper, we prove some approximation theorems for these operators. Approximation properties of our operators are illustrated on graphs for variables s, alpha, lambda, and n. It should be mentioned that our operators for lambda = 1 have better approximation than bernstein and alpha-bernstein operators.

关键词： bernstein operators lambda-bernstein operators alpha-bernstein operators Modulus of continuity

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On the shape-preserving properties of λ-bernstein operators

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2022年第1期2022卷 1-11页

作者： Su, Lian-Ta Mutlu, Gokhan Cekim, Bayram Quanzhou Normal Univ Fujian Prov Key Lab Data Intens Comp Key Lab Intelligent Comp & Informat Proc Sch Math & Comp Sci Quanzhou Peoples R China Gazi Univ Fac Sci Dept Math Ankara Turkey

We investigate the shape-preserving properties of lambda-bernstein operators B-n,B-lambda(f;x) that were recently introduced bernstein-type operators defined by a new Bezier basis with shape parameter lambda is an element of [-1,1]. For this purpose, we express B-n,B-lambda(f;x) as a sum of a classical bernstein operator and a sum of first order divided differences of f. Using this new representation, we prove that B-n,B-lambda(f;x) preserves monotonic functions for all lambda is an element of [-1, 1]. However, we show by a counter example that B-n,B-lambda(f;x) does not preserve convex functions for some lambda is an element of [-1, 1]. We present a weaker result for the case lambda is an element of [0,1] for a special class of functions. Finally, we analyze the monotonicity of lambda-bernstein operators with n and show that B-n,B-lambda(f;x) is not monotonic with n for some lambda if 1/2 < lambda <= 1.

关键词： lambda-bernstein operators Shape-preserving properties Computer-aided design Computer graphics

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Convergence of λ-bernstein operators based on (p, q)-integers

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2020年第1期2020卷 1-17页

作者： Cai, Qing-Bo Cheng, Wen-Tao Quanzhou Normal Univ Sch Math & Comp Sci Quanzhou Peoples R China Anqing Normal Univ Sch Math & Computat Sci Anqing Peoples R China

In the present paper, we construct a new class of positive linear lambda-bernstein operators based on (p, q)-integers. We obtain a Korovkin type approximation theorem, study the rate of convergence of these operators by using the conception of K-functional and moduli of continuity, and also give a convergence theorem for the Lipschitz continuous functions.

关键词： lambda-bernstein operators (p, q)-integers Moduli of continuity Rate of convergence Lipschitz continuous functions

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Statistical approximation properties of Stancu type λ-bernstein operators 2

Statistical approximation properties of Stancu type λ-Berns...

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IEEE 2nd International Conference on Electronic Information and Communication Technology (ICEICT)

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

ISBN: (纸本)9781538692981

In this paper, we introduce a Stancu type lambda-bernstein operators, and obtain the moments and the limitation of central moments of these operators, study a statistical approximation theorem and a Voronovskaja type theorem by using the concept of statistical convergence. We also give an example to show the convergent effect of these operators to certain function f.

关键词： lambda-bernstein operators Statistical convergence basis function asymptotic formula

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