# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021040
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## Better approximation by a Durrmeyer variant of $\alpha-$Baskakov operators

 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India-247667

*Corresponding author: Jitendra Kumar Singh

Received  October 2021 Revised  November 2021 Early access January 2022

Fund Project: The second author is supported by Council of Scientific and Industrial Research, New Delhi, India, grant no.- 09/143(0914)/2018-EMR-I

The aim of this paper is to study some approximation properties of the Durrmeyer variant of $\alpha$-Baskakov operators $M_{n,\alpha}$ proposed by Aral and Erbay [3]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr$\ddot{u}$ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions $e_0$ and $e_2$ and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators $M_{n,\alpha}$ and show the comparison of its rate of approximation vis-a-vis the modified operators.

Citation: Purshottam Narain Agrawal, Jitendra Kumar Singh. Better approximation by a Durrmeyer variant of $\alpha-$Baskakov operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021040
##### References:
 [1] O. Agratini, Linear operators that preserve some test functions, Int. J. Math. Math. Sci., 2006 (2006), Article ID 094136, 11 pp. doi: 10.1155/IJMMS/2006/94136.  Google Scholar [2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter, 2011. Google Scholar [3] A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Commun., 24 (2019), 119-131.   Google Scholar [4] A. Aral and V. Gupta, On q-Baskakov type operators, Demonstr. Math., 42 (2009), 109-122.   Google Scholar [5] V. Baskakov, An instance of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 113 (1957), 249-251.   Google Scholar [6] W. Z. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, 1989. Google Scholar [7] Z. Ditzian and V. Totik, Moduli of Smoothness, 9$^th$ edition, Springer Series in Computational Mathematics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.  Google Scholar [8] A. D Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki, 20 (1976), 781-786.   Google Scholar [9] V. Gupta, An estimate on the convergence of Baskakov-Bézier operators, J. Math. Anal. Appl., 312 (2005), 280-288.  doi: 10.1016/j.jmaa.2005.03.041.  Google Scholar [10] V. Gupta and G. C. Greubel, Moment estimations of new Szász-Mirakyan-Durrmeyer operators, Appl. Math. Comput., 271 (2015), 540-547.  doi: 10.1016/j.amc.2015.09.037.  Google Scholar [11] V. Gupta, G. Tachev and A. M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algorithms, 81 (2019), 125-149.  doi: 10.1007/s11075-018-0538-7.  Google Scholar [12] M. Heimann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl., 5 (1989), 105-127.   Google Scholar [13] N. Ispir, On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355-365.   Google Scholar [14] N. Ispir and . Atakut, Approximation by modified Szász-Mirakjan operators on weighted space, Proc. Indian. Acad. Sci. Math. Sci., 112 (2002), 571-578. doi: 10.1007/BF02829690.  Google Scholar [15] A. Kajla, Direct estimates of certain Miheşan-Durrmeyer type operators, Adv. Oper. Theory., 2 (2017), 162-178.  doi: 10.22034/aot.1612-1079.  Google Scholar [16] J. P. King, Positive linear operators which preserve $x^2$, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.  Google Scholar [17] A. Kumar and L. N. Mishra, Approximation by modified Jain-Baskakov-Stancu operators, Tbilisi Math. J., 10 (2017), 185-199.  doi: 10.1515/tmj-2017-0035.  Google Scholar [18] B. Lenze, On Lipschitz-type maximal functions and their smoothnes spaces, Nederl. Akad. Wetensch. Indag. Math., 50 (1988), 53-63.   Google Scholar [19] M. Nasiruzzaman, N. Rao, S. Wazir and R. Kumar, Approximation on parametric extension of Baskakov-Durrmeyer operators on weighted space, J. Inequal. Appl., 2019 (2019), 103, 11pp. doi: 10.1186/s13660-019-2055-1.  Google Scholar [20] A. Wafi and S. Khatoon, On the order of approximation of functions by generalized Baskakov operators, Indian J. Pure Appl. Math., 35 (2004), 347-358.   Google Scholar [21] I. Yuksel and N. Ispir, Weighted approximation by a certain family of summation integral-type operators, Comput. Math. Appl., 52 (2006), 1463-1470.  doi: 10.1016/j.camwa.2006.08.031.  Google Scholar [22] C. Zhang and Z. Zhu, Preservation properties of the Baskakov-Kantorovich operators, Comput. Math. Appl., 57 (2009), 1450-1455.  doi: 10.1016/j.camwa.2009.01.027.  Google Scholar

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##### References:
 [1] O. Agratini, Linear operators that preserve some test functions, Int. J. Math. Math. Sci., 2006 (2006), Article ID 094136, 11 pp. doi: 10.1155/IJMMS/2006/94136.  Google Scholar [2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter, 2011. Google Scholar [3] A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Commun., 24 (2019), 119-131.   Google Scholar [4] A. Aral and V. Gupta, On q-Baskakov type operators, Demonstr. Math., 42 (2009), 109-122.   Google Scholar [5] V. Baskakov, An instance of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 113 (1957), 249-251.   Google Scholar [6] W. Z. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, 1989. Google Scholar [7] Z. Ditzian and V. Totik, Moduli of Smoothness, 9$^th$ edition, Springer Series in Computational Mathematics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.  Google Scholar [8] A. D Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki, 20 (1976), 781-786.   Google Scholar [9] V. Gupta, An estimate on the convergence of Baskakov-Bézier operators, J. Math. Anal. Appl., 312 (2005), 280-288.  doi: 10.1016/j.jmaa.2005.03.041.  Google Scholar [10] V. Gupta and G. C. Greubel, Moment estimations of new Szász-Mirakyan-Durrmeyer operators, Appl. Math. Comput., 271 (2015), 540-547.  doi: 10.1016/j.amc.2015.09.037.  Google Scholar [11] V. Gupta, G. Tachev and A. M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algorithms, 81 (2019), 125-149.  doi: 10.1007/s11075-018-0538-7.  Google Scholar [12] M. Heimann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl., 5 (1989), 105-127.   Google Scholar [13] N. Ispir, On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355-365.   Google Scholar [14] N. Ispir and . Atakut, Approximation by modified Szász-Mirakjan operators on weighted space, Proc. Indian. Acad. Sci. Math. Sci., 112 (2002), 571-578. doi: 10.1007/BF02829690.  Google Scholar [15] A. Kajla, Direct estimates of certain Miheşan-Durrmeyer type operators, Adv. Oper. Theory., 2 (2017), 162-178.  doi: 10.22034/aot.1612-1079.  Google Scholar [16] J. P. King, Positive linear operators which preserve $x^2$, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.  Google Scholar [17] A. Kumar and L. N. Mishra, Approximation by modified Jain-Baskakov-Stancu operators, Tbilisi Math. J., 10 (2017), 185-199.  doi: 10.1515/tmj-2017-0035.  Google Scholar [18] B. Lenze, On Lipschitz-type maximal functions and their smoothnes spaces, Nederl. Akad. Wetensch. Indag. Math., 50 (1988), 53-63.   Google Scholar [19] M. Nasiruzzaman, N. Rao, S. Wazir and R. Kumar, Approximation on parametric extension of Baskakov-Durrmeyer operators on weighted space, J. Inequal. Appl., 2019 (2019), 103, 11pp. doi: 10.1186/s13660-019-2055-1.  Google Scholar [20] A. Wafi and S. Khatoon, On the order of approximation of functions by generalized Baskakov operators, Indian J. Pure Appl. Math., 35 (2004), 347-358.   Google Scholar [21] I. Yuksel and N. Ispir, Weighted approximation by a certain family of summation integral-type operators, Comput. Math. Appl., 52 (2006), 1463-1470.  doi: 10.1016/j.camwa.2006.08.031.  Google Scholar [22] C. Zhang and Z. Zhu, Preservation properties of the Baskakov-Kantorovich operators, Comput. Math. Appl., 57 (2009), 1450-1455.  doi: 10.1016/j.camwa.2009.01.027.  Google Scholar
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