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Approximation properties of univariate and bivariate new class \(\lambda \)-Bernstein–Kantorovich operators and its associated GBS operators

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Abstract

The main objective of this work is to derive some approximation properties of univariate and bivariate Kantorovich type new class Bernstein operators by means of shape parameter \(\lambda \in [-1,1]\). We obtained some basic results such as moments estimates and Korovkin type approximation theorem. Next, we estimated the degree of convergence in terms of the usual modulus of continuity, for an element of Lipschitz type continuous functions and Peetre’s K-functional, respectively. Moreover, we constructed the bivariate of newly introduced operators and computed the rate of approximation in connection with the partial and complete modulus of continuity, also for the elements of the Lipschitz type class. In addition, we proposed the associated Generalized Boolean Sum (GBS) type of bivariate operators and calculated the order of approximation with the help of mixed modulus of continuity and a class of the Lipschitz of Bögel type continuous functions. Finally, we presented some graphics and an error of estimation table to compare the convergence behavior of univariate, bivariate forms and its associated GBS type operators to certain functions.

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References

  • Acu AM, Manav N, Sofonea DF (2018) Approximation properties of \(\lambda \)-Kantorovich operators. J. Inequal. Appl. 2018:202

    Article  MathSciNet  MATH  Google Scholar 

  • Agrawal PN, Baxhaku B, Shukla R (2022) A new kind of bivariate \(\lambda \)-Bernstein-Kantorovich type operator with shifted knots and its associated GBS form. Math Found Comput 5(3):157

    Article  Google Scholar 

  • Aliaga E, Baxhaku B (2020) On the approximation properties of \(q\)-analogue bivariate \(\lambda \)-Bernstein type operators. J Funct Spaces 2020

  • Altomare F, Campiti M (2011) Korovkin-type approximation theory and its applications, volume 17, Walter de Gruyter

  • Aslan R (2021) Some approximation results on \(\lambda \)-Szász-Mirakjan-Kantorovich operators. Fundam J Math Appl 4:150–158

    Google Scholar 

  • Ansari KJ, Özger F, Ödemis Özger Z (2022) Numerical and theoretical approximation results for Schurer-Stancu operators with shape parameter \(\lambda \). Comp Appl Math 41:181

  • Aslan R, Mursaleen M (2022) Some approximation results on a class of new type \(\lambda -\)Bernstein polynomials. J Math Inequal 16(2):445–462

    Article  MathSciNet  MATH  Google Scholar 

  • Ayman Mursaleen M, Kilicman A, Nasiruzzaman Md (2022) Approximation by \(q\)-Bernstein-Stancu-Kantorovich operators with shifted knots of real parameters. Filomat 36(4):1179–1194

    Article  MathSciNet  Google Scholar 

  • Badea C (1995) K-functionals and moduli of smoothness of functions defined on compact metric spaces. Comput Math Appl 30:23–31

    Article  MathSciNet  MATH  Google Scholar 

  • Badea C, Badea I, Gonska HH (1988) Notes on degree of approximation on b-continuous and b-differentiable functions. Approx Theory Appl 4:95–108

    MathSciNet  MATH  Google Scholar 

  • Bernstein S (1912) Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comp Comm Soc Mat Charkow Ser 13:1–2

    MATH  Google Scholar 

  • Bögel K (1934) Über Mehrdimensionale differentiation von funktionen mehrerer reeller veränderlichen. J für die Reine und Angew Math 170:197–217

    Article  MathSciNet  MATH  Google Scholar 

  • Bögel K (1935) Über mehrdimensionale differentiation, integration und beschränkte variation. J für die Reine und Angew Math 173:5–30

    Article  MathSciNet  MATH  Google Scholar 

  • Bögel K (1962) Über die mehrdimensionale differentiation. Jahresber Deutsch Math Verein 65:45–71

    MATH  Google Scholar 

  • Braha NL, Mansour T, Mursaleen M, Acar T (2021) Convergence of \(\lambda \)-Bernstein operators via power series summability method. J Appl Math Comput 65(1):125–146

    Article  MathSciNet  MATH  Google Scholar 

  • Cai QB, Ansari KJ, Temizer Ersoy M, Özger F (2022) Statistical blending-type approximation by a class of operators that includes shape parameters \(\lambda \) and \(\alpha \). Mathematics 10(7):1149

  • Cai QB, Aslan R (2021) On a new construction of generalized \(q\)-Bernstein polynomials based on shape parameter \(\lambda \). Symmetry 13(10):1919

    Article  Google Scholar 

  • Cai QB, Aslan R (2022) Note on a new construction of Kantorovich form \(q\)-Bernstein operators related to shape parameter \(\lambda \). Comput Model Eng Sci 130(3):1479–1493

    Google Scholar 

  • Cai QB, Lian BY, Zhou G (2018) Approximation properties of \(\lambda \)-Bernstein operators. J. Inequal. Appl. 2018:61

    Article  MathSciNet  MATH  Google Scholar 

  • Cai QB, Zhou G, Li J (2019) Statistical approximation properties of \(\lambda \)-Bernstein operators based on \(q\)-integers. Open Math 17:487–498

    Article  MathSciNet  MATH  Google Scholar 

  • Deshwal S, İspir N, Agrawal PN (2017) Blending type approximation by bivariate Bernstein-Kantorovich operators. Appl Math Inform Sci 11:423–432

    Article  Google Scholar 

  • DeVore RA, Lorentz GG (1993) Constructive Approximation. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Dobrescu E, Matei I (1966) The approximation by Bernstein type polynomials of bidimensional continuous functions. An Univ Timisoara Ser Sti Mat-Fiz 4:85–90

    MathSciNet  MATH  Google Scholar 

  • Farin G (2014) Curves and surfaces for computer-aided geometric design: a practical guide. Elsevier

    MATH  Google Scholar 

  • Garg T, Acu AM, Agrawal PN (2019) Weighted approximation and GBS of Chlodowsky-Szász-Kantorovich type operators. Anal Math Phys 9:1429–1448

    Article  MathSciNet  MATH  Google Scholar 

  • Heshamuddin Md, Rao N, Lamichhane BP, Kılıçman A, Ayman Mursaleen M (2022) On one-and two-dimensional \(\alpha \)-Stancu-Schurer-Kantorovich operators and their approximation properties. Mathematics 10:3227

    Article  Google Scholar 

  • Izgi A (2012) Approximation by a class of new type Bernstein polynomials of one and two variables. Glob J Pure Appl Math 8:55–71

    MathSciNet  Google Scholar 

  • Kajla A, Miclăuş D (2018) Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type. Results Math 73:1

    Article  MathSciNet  MATH  Google Scholar 

  • Kantorovich LV (1930) Sur certain développements suivant les polynômes de la forme de s. Bernstein, I, II, CR Acad. URSS, 563–568

  • Khan K, Lobiyal DK (2017) Bézier curves based on Lupas \((p, q)\)-analogue of Bernstein functions in CAGD. J Comput Appl Math 317:458–477

    Article  MathSciNet  MATH  Google Scholar 

  • Khan K, Lobiyal DK, Kilicman A (2019) Bézier curves and surfaces based on modified Bernstein polynomials. Azerb J Math 9:3–21

    MathSciNet  MATH  Google Scholar 

  • Korovkin PP (1953) On convergence of linear positive operators in the space of continuous functions. Dokl Akad Nauk SSSR 90:961–964

    MathSciNet  Google Scholar 

  • Kumar A (2020) Approximation properties of generalized \(\lambda \)-Bernstein-Kantorovich type operators. Rend Circ Mat Palermo (2), 1–16

  • Mahmudov NI, Sabancigil P (2013) Approximation theorems for \(q-\)Bernstein-Kantorovich operators. Filomat 27(4):721–730

    Article  MathSciNet  MATH  Google Scholar 

  • Mursaleen M, Al-Abied AAH, Salman MA (2020) Chlodowsky type \((\lambda , q)\)-Bernstein-Stancu operators. Azerb J Math 10:75–101

    MathSciNet  MATH  Google Scholar 

  • Özger F (2019) Weighted statistical approximation properties of univariate and bivariate \(\lambda \)-Kantorovich operators. Filomat 33:3473–3486

    Article  MathSciNet  MATH  Google Scholar 

  • Özger F (2020) On new Bézier bases with Schurer polynomials and corresponding results in approximation theory. Commun Fac Sci Univ Ank Ser A1 Math Stat 69:376–393

    MathSciNet  MATH  Google Scholar 

  • Özger F, Aljimi E, Temizer Ersoy M (2022) Rate of weighted statistical convergence for generalized blending-type Bernstein-Kantorovich operators. Mathematics 10(12):2027

  • Özkan EY, Akpınar NN (2022) Some new inequalities and numerical results of bivariate Bernstein-type operator including Bézier basis and its GBS operator, Mathematical Foundations of Computing

  • Pop OT, Fărcas MD (2006) Approximation of b-continuous and b-differentiable functions by gbs operators of bernstein bivariate polynomials. J Inequal Pure Appl Math 7(9)

  • Radu VA, Agrawal PN, Singh JK (2021) Better numerical approximation by \(\lambda \)-Durrmeyer-Bernstein type operators. Filomat 35(4):1405–1419

    Article  MathSciNet  Google Scholar 

  • Rahman S, Mursaleen M, Acu AM (2019) Approximation properties of \(\lambda \)-Bernstein- Kantorovich operators with shifted knots. Math Method Appl Sci 42:4042–4053

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Özger F, Mohiuddine SA (2019) Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter \(\lambda \). Symmetry 11:316

    Article  MATH  Google Scholar 

  • Srivastava HM, Ansari KJ, Ödemis Özger Z (2021) A link between approximation theory and summability methods via four-dimensional infinite matrices. Mathematics 9(16):1895

  • Volkov VI (1957) On the convergence of sequences of linear positive operators in the space of continuous functions of two variable. Dokl Akad Nauk SSSR 17–19

  • Ye Z, Long X, Zeng X-M (2010) Adjustment algorithms for Bézier curve and surface, In: International Conference on Computer Science and Education. 1712–1716

  • Zeng X-M, Cheng F (2001) On the rates of approximation of Bernstein type operators. J Approx Theory 109:242–256

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Faculty of Sciences and Arts, Harran University, Şanlıurfa, 63100, Turkey

    Reşat Aslan

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  1. Reşat Aslan
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Aslan, R. Approximation properties of univariate and bivariate new class \(\lambda \)-Bernstein–Kantorovich operators and its associated GBS operators. Comp. Appl. Math. 42, 34 (2023). https://doi.org/10.1007/s40314-022-02182-w

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