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.

[2]

F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter, 2011.

[3]

A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Commun., 24 (2019), 119-131. 

[4]

A. Aral and V. Gupta, On q-Baskakov type operators, Demonstr. Math., 42 (2009), 109-122. 

[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. 

[6]

W. Z. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, 1989.

[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.

[8]

A. D Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki, 20 (1976), 781-786. 

[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.

[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.

[11]

V. GuptaG. 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.

[12]

M. Heimann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl., 5 (1989), 105-127. 

[13]

N. Ispir, On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355-365. 

[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.

[15]

A. Kajla, Direct estimates of certain Miheşan-Durrmeyer type operators, Adv. Oper. Theory., 2 (2017), 162-178.  doi: 10.22034/aot.1612-1079.

[16]

J. P. King, Positive linear operators which preserve $x^2$, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.

[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.

[18]

B. Lenze, On Lipschitz-type maximal functions and their smoothnes spaces, Nederl. Akad. Wetensch. Indag. Math., 50 (1988), 53-63. 

[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.

[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. 

[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.

[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.

show all references

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.

[2]

F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter, 2011.

[3]

A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Commun., 24 (2019), 119-131. 

[4]

A. Aral and V. Gupta, On q-Baskakov type operators, Demonstr. Math., 42 (2009), 109-122. 

[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. 

[6]

W. Z. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, 1989.

[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.

[8]

A. D Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki, 20 (1976), 781-786. 

[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.

[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.

[11]

V. GuptaG. 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.

[12]

M. Heimann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl., 5 (1989), 105-127. 

[13]

N. Ispir, On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355-365. 

[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.

[15]

A. Kajla, Direct estimates of certain Miheşan-Durrmeyer type operators, Adv. Oper. Theory., 2 (2017), 162-178.  doi: 10.22034/aot.1612-1079.

[16]

J. P. King, Positive linear operators which preserve $x^2$, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.

[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.

[18]

B. Lenze, On Lipschitz-type maximal functions and their smoothnes spaces, Nederl. Akad. Wetensch. Indag. Math., 50 (1988), 53-63. 

[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.

[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. 

[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.

[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.

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