On Simultaneous Approximation and Combinations of Lupas Type Operators
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Authors: T. A. K. SINHA, K. K. SINGH AND A. K. SHARMA
DOI: 10.46793/KgJMat2404.619S
Abstract:
The purpose of the present paper is to study a sequence of linear and positive operators which was introduced by A. Lupas. First, we obtain estimate of moments of the operators and then prove a basic convergence theorem for simultaneous approximation. Further, we find error in approximation in terms of modulus of continuity of function. Finally, we establish a Voronovskaja asymptotic formula for linear combination of the above operators.
Keywords:
Lupas operators, simultaneous approximation, modulus of continuity, Voronovskaja asymptotic formula, linear combinations,
References:
[1] O. Agratini, On a sequence of linear and positive operators, Facta Univ. Ser. Math. Inform. 14 (1999), 41–48.
[2] P. L. Butzer, Linear combinations of Bernstein polynomials, Canadian J. Math. 5 (1953), 559–567.
[3] F. Özger, H. M. Srivastava and 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 114 (2020), Article ID 173. https://doi.org/10.1007/s13398-020-00903-6
[4] H. H. Gonska and X. L. Zhou, A global Inverse theorem on simultaneous approximation by Bernstein Durrmeyer operators, J. Approx. Theory 67 (1991), 284–302. https://doi.org/10.1016/0021-9045(91)90004-T
[5] V. Gupta, Diffrences of operators of Lupas type, Constructive Mathematical Analysis 1(1) (2018), 9–14. https://doi.org/10.33205/cma.452962
[6] V. Gupta and H. M. Srivastava, A general family of the Srivastava-Gupta operators preserving linear functions, Eur. J. Pure Appl. Math. 11(3) (2018), 575–579. https://doi.org/10.29020/nybg.ejpam.v11i3.3314
[7] V. Gupta, M. K. Gupta and V. Vasishtha, Simultaneous approximation by summation integral type operators, Journal of Nonlinear Functional Analysis 8(3) (2003), 399–412.
[8] M. Heilmann and M. W. Müller, On simultaneous approximation by the method of Baskakov-Durrmeyer operators, Numer. Funct. Anal. Optim. 10(1–2) (1989), 127–138. https://doi.org/10.1080/01630568908816295
[9] A. Kajla and T. Acar, Modified α-Bernstein operators with better approximation properties, Ann. Funct. Anal. 10(4) (2019), 570–582. https://doi.org/10.1215/20088752-2019-0015
[10] G. G. Lorentz, Benstein Polynomials, Chelsea Publishing Company, New York, 1986.
[11] A. Lupas, The approximation by some positive linear operators, in: M. W Müller et al. (Eds.), Proceedings of the International Dortmund Meeting on Approximation Theory, Akademie Verlag, Berlin, 1995, 201–229.
[12] A. Kajla, S. A. Mohiuddine and A. Alotaibi, Blending-type approximation by Lupaş-Durrmeyer-type operators involving Pólya distribution, Math. Methods Appl. Sci. 44 (2021), 9407–9418. https://doi.org/10.1002/mma.7368
[13] S. A. Mohiuddine and F. Özger, Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter α, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 114 (2020), Article ID 70. https://doi.org/10.1007/s13398-020-00802-w
[14] S. A. Mohiuddine, T. Acar and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Methods Appl. Sci. 40 (2017), 7749–7759. https://doi.org/10.1002/mma.4559
[15] S. A. Mohiuddine, N. Ahmad, F. Özger, A. Alotaibi and B. Hazarika, Approximation by the parametric generalization of Baskakov-Kantorovich operators linking with Stancu operators, Iran. J. Sci. Technol. Trans. A Sci. 45 (2021), 593–605. https://doi.org/10.1007/s40995-020-01024-w
[16] R. K. S. Rathore, Linear combination of linear positive operators and generating relations in special functions, Ph. D. Thesis, I. I. T. Delhi, India, 1973.
[17] K. K. Singh and P. N. Agrawal, Simultaneous approximation by a linear combination of Bernstein-Durrmeyer type polynomials, Bull. Math. Anal. Appl. 3(2) (2011), 70–82.
[18] R. P. Sinha, P. N. Agrawal and V. Gupta, On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. Ser. B. 43(2) (1991), 217–231.
[19] H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37 (2003), 1307–1315. https://doi.org/10.1016/S0895-7177(03)90042-2
[20] K. J. Thamer and A. I. Ibrahim, Simultaneous approximation with linear combination of integral Baskakov type operators, Revista De La Union Mathematica Argentina 46(1) (2005), 1–10.
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