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Pointwise and weighted estimates for Bernstein-Kantorovich type operators including beta function

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Abstract

We establish a Kantorovich variant of Bernstein operators including the beta function inspired by the King-type operators which preserve certain functions. We provide some pointwise, weighted, and direct approximation theorems to examine the approximation properties of the introduced sequence of operators. Furthermore, by graphical and numerical examples, we demonstrate the convergence of defined operators with the help of the MATLAB program.

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References

  1. S. Rahman, M. Mursaleen, A. Khan, A Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution with error estimation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, (2020), 114:75.

  2. A. Kajla, S.A. Mohiuddine, A. Alotaibi, M. Goyal, K.K. Singh, Approximation by \(\upsilon \)-Baskakov-Durrmeyer-Type Hybrid Operators. Iran J Sci Technol Trans Sci, 44, (2020), 1111-1118. https://doi.org/10.1007/s40995-020-00914-3

    Article  MathSciNet  Google Scholar 

  3. A. Naaz, M. Mursaleen, Some Approximation Results on Compact Sets by \((p, q)\)-Bernstein-Faber Polynomials, \(q>p>1\). Iran J Sci Technol Trans Sci 43, (2019), 2585-2593.

    Article  MathSciNet  Google Scholar 

  4. F. Usta, On New Modification of Bernstein Operators: Theory and Applications. Iran J Sci Technol Trans Sci 44, (2020), 1119-1124. https://doi.org/10.1007/s40995-020-00919-y

    Article  MathSciNet  Google Scholar 

  5. A.D. Gadjiev, Theorems of Korovkin type, Mat. Zamekti, 20(5), (1976), 781-786.

    Google Scholar 

  6. F. Özger, On new Bézier bases with Schurer polynomials and corresponding results in approximation theory, Commun Fac Sci Univ Ank Ser Math Stat. 69(1), , 376-393. (2020)

    Article  MathSciNet  Google Scholar 

  7. S.A. Mohiuddine, F. Özger, Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter \(\alpha \), Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM, 114(70), (2020).

  8. F. Özger, Weighted statistical approximation properties of univariate and bivariate \( \lambda \)-Kantorovich operators, Filomat, 2019; 33(11):3473-3486.

    Article  MathSciNet  Google Scholar 

  9. F. Özger, H.M. Srivastava, S.A. Mohiuddine, Approximation of functions by a new class of generalized Bernstein-Schurer operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM (2020) 114:173.

  10. S.A. Mohiuddine, N. Ahmad, F. Özger, et al. Approximation by the parametric generalization of Baskakov-Kantorovich operators linking with Stancu operators, Iran J. Sci. Technol. Trans. Sci. 45, (2021), 593-605. https://doi.org/10.1007/s40995-020-01024-w

    Article  MathSciNet  Google Scholar 

  11. A. Alotaibi, F. Özger, S.A. Mohiuddine et al. Approximation of functions by a class of Durrmeyer-Stancu type operators which includes Euler’s beta function, Adv. Differ. Equ. 13, (2021), 2021. https://doi.org/10.1186/s13662-020-03164-0

    Article  MathSciNet  Google Scholar 

  12. F. Özger, Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numerical Functional Analysis and Optimization, 41(16), (2020), 1990-2006. https://doi.org/10.1080/01630563.2020.1868503

    Article  MathSciNet  Google Scholar 

  13. F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, de Gruyter Stud. Math., vol. 17, de Gruyter & Co., Berlin, 1994.

  14. P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corporation, Delhi, 1960.

    Google Scholar 

  15. J.P. King, Positive linear operators which preserve \(y^{2}\), Acta Math. Hungar. 99(3) , 203-208.(2003)

    Article  MathSciNet  Google Scholar 

  16. M. Mursaleen, K.J. Ansari, Approximation of \(q\)-Stancu-Beta operators which preserve \(\displaystyle x^2\), Bull. Malays. Math. Sci. Soc. 40, 1479-1491 (2017). https://doi.org/10.1007/s40840-015-0146-9

    Article  MathSciNet  Google Scholar 

  17. D.J. Bhatt, V.N. Mishra, R.K. Jana, On a new class of Bernstein type operators based on beta function, Khayyam J. Math. 6(1) (2020), 1-15.

    MathSciNet  Google Scholar 

  18. M. Mursaleen, S. Rahman, Dunkl generalization of \(q\)-Szász-Mirakjan operators which preserve \(y^2\), Filomat, 32(3), (2018), 733-747.

    Article  MathSciNet  Google Scholar 

  19. M. Mursaleen, A. Naaz, A. Khan, Improved approximation and error estimations by King type \((p,q)\)-Szász-Mirakjan Kantorovich operators, Appl. Math. Comp. 348, (2019) 175-185.

    Article  Google Scholar 

  20. R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.

    Book  Google Scholar 

  21. Z. Ditzian, V. Totik, Moduli of Smoothness, Springer Series in Computational Mathematics, 8. Springer-Verlag, New York, 1987.

    Google Scholar 

  22. B. Lenze, Bernstein-Baskakov-Kantorovic operators and Lipschitz-type maximal functions, in Approximation Theory (Kecskemet, 1990) 469-496, Colloq. Math. Soc. Janos Bolyai, 58 North-Holland, Amsterdam.

  23. H.M. Srivastava, K.J. Ansari, F. Özger, Z.Ö. Özger, A link between approximation theory and summability methods via four-dimensional infinite matrices, Mathematics, 9, (2021), 1895. https://doi.org/10.3390/math9161895

    Article  Google Scholar 

  24. Md. Nasiruzzaman, Adem Kilicman and Mohammad Ayman Mursaleen, Construction of \(q\)-Baskakov operators by wavelets and approximation properties, Iranian Journal of Science and Technology, Transactions A: Science, 46(5) (2022) 1495-1503.

    Article  MathSciNet  Google Scholar 

  25. Ming-Yu Chen, Md. Nasiruzzaman, M.A. Mursaleen, N. Rao, A. Kiliçman, On shape parameter \(\alpha \) based approximation properties and \(q\)-statistical convergence of Baskakov-Gamma operators, Journal of Mathematics, 2022 (2022) 4190732.

  26. K.J. Ansari, F. Özger, Z. Ödemiş Özger, Numerical and theoretical approximation results for Schurer-Stancu operators with shape parameter \(\lambda \). Comp. Appl. Math. 41, 181 (2022).

  27. Ansari, K. J., Karakılıç, S., and Özger, F. Bivariate Bernstein-Kantorovich operators with a summability method and related GBS operators. Filomat, 36(19),(2022) 6751-6765.

    Article  MathSciNet  Google Scholar 

  28. K.J. Ansari, M. Civelek, F. Usta, Jain’s Operator: A New Construction and Applications in Approximation Theory, Math. Meth. Appl. Sci., (2023), https://doi.org/10.1002/mma.9311

    Article  MathSciNet  Google Scholar 

  29. N.I. Mahmudov, P. Sabancıgil, Approximation Theorems for \(q\)-Bernstein Kantorovich Operators, Filomat, vol.27:4, pp. 721-730, 2013.

    Article  MathSciNet  Google Scholar 

  30. H. Hamal, P. Sabancıgil, Some Approximation Properties of new Kantorovich type \(q\)-analogue of Balazs Szabados Operators, Journal of Inequalities and Applications, 159, 2020.

  31. H. Hamal, P. Sabancıgil, Kantorovich Type Generalization of Berstein Type Rational Functions Based on \((p,q)\)-integers, Symmetry, 14(5), 1054, 2022.

    Article  Google Scholar 

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Acknowledgements

The first author extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding his work through the research groups program under Grant number RGP. 2/371/44.

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Correspondence to Khursheed J. Ansari.

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Communicated by NM Bujurke.

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Ansari, K.J., Özger, F. Pointwise and weighted estimates for Bernstein-Kantorovich type operators including beta function. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00587-3

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