May  2022, 5(2): 75-92. doi: 10.3934/mfc.2021024

Better degree of approximation by modified Bernstein-Durrmeyer type operators

1. 

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

2. 

Gazi University, Faculty of Science, Department Of Mathematics, 06500, Ankara, Turkey

* Corresponding author: Ş. Y. Güngör

Received  July 2021 Revised  September 2021 Published  May 2022 Early access  October 2021

In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function $ \tau(x), $ where $ \tau $ is infinitely differentiable function on $ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $ and $ \tau^{\prime }(x)>0, \;\forall\;\; x\in[0, 1]. $ We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function $ \tau(x) $ leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].

Citation: Purshottam Narain Agrawal, Şule Yüksel Güngör, Abhishek Kumar. Better degree of approximation by modified Bernstein-Durrmeyer type operators. Mathematical Foundations of Computing, 2022, 5 (2) : 75-92. doi: 10.3934/mfc.2021024
References:
[1]

T. AcarP. N. Agrawal and T. Neer, Bezier variant of the Bernstein-Durrmeyer type operators, Results. Math., 72 (2017), 1341-1358.  doi: 10.1007/s00025-016-0639-3.

[2]

T. AcarA. Aral and I. Raşa, Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41. 

[3]

A. M. AcuP. N. Agrawal and T. Neer, Approximation properties of the modified Stancu operators, Numer. Funct. Anal. Optim., 38 (2017), 279-292.  doi: 10.1080/01630563.2016.1248564.

[4]

A. M. Acu and V. A. Radu, Approximation by certain operators linking the $\alpha$- Bernstein and the genuine $\alpha$- Bernstein-Durrmeyer operators, in Mathematical Analysis I: Approximation Theory, Springer Proceedings in Mathematics & Statistics, 306 (eds. N. Deo et al.), Springer, (2020), 77-88. doi: 10.1007/978-981-15-1153-0_7.

[5]

P. N. Agrawal, N. Bhardwaj and P. Bawa, Bézier variant of modified $\alpha$- Bernstein operators, Rend. Circ. Mat. Palermo, II Ser., (2021). doi: 10.1007/s12215-021-00613-x.

[6]

S. Bernstein, Démonstration du theoréme de Weierstrass fondée sur le calcul des probabilitiés, Comm. Kharkov Math. Soc., 13 (1912), 1-2. 

[7]

N. Çetin and V. A. Radu, Approximation by generalized Bernstein-Stancu operators, Turk. J. Math., 43 (2019), 2032-2048.  doi: 10.3906/mat-1903-109.

[8]

R. A. Devore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, 1993. doi: 10.1007/978-3-662-02888-9.

[9]

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Ser. Comput. Math., Springer-Verlag New York, 1987. doi: 10.1007/978-1-4612-4778-4.

[10]

A. Kajla and T. Acar, Modified $\alpha$- Bernstein operators with better approximation properties, Ann. Funct. Anal., 10 (2019), 570-582.  doi: 10.1215/20088752-2019-0015.

[11]

A. Kajla and T. Acar, Blending type approximation by generalized Bernstein-Durrmeyer type operators, Miskolc Math. Notes, 19 (2018), 319-336.  doi: 10.18514/MMN.2018.2216.

[12]

A. Kajla and D. Miclăuş, Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type, Results Math., 73 (2018), Paper No. 1, 21 pp. doi: 10.1007/s00025-018-0773-1.

[13]

Y. C. KwunA.-M. AcuA. RafiqV. A. RaduF. Ali and S. M. Kang, Bernstein- Stancu type operators which preserve polynomials, J. Comput. Anal. Appl., 23 (2017), 758-770. 

[14]

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

[15]

M. A. Özarslan and H. Aktuğlu, Local approximation properties for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.

[16]

X. ChenJ. TanZ. Liu and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl., 450 (2017), 244-261.  doi: 10.1016/j.jmaa.2016.12.075.

show all references

References:
[1]

T. AcarP. N. Agrawal and T. Neer, Bezier variant of the Bernstein-Durrmeyer type operators, Results. Math., 72 (2017), 1341-1358.  doi: 10.1007/s00025-016-0639-3.

[2]

T. AcarA. Aral and I. Raşa, Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41. 

[3]

A. M. AcuP. N. Agrawal and T. Neer, Approximation properties of the modified Stancu operators, Numer. Funct. Anal. Optim., 38 (2017), 279-292.  doi: 10.1080/01630563.2016.1248564.

[4]

A. M. Acu and V. A. Radu, Approximation by certain operators linking the $\alpha$- Bernstein and the genuine $\alpha$- Bernstein-Durrmeyer operators, in Mathematical Analysis I: Approximation Theory, Springer Proceedings in Mathematics & Statistics, 306 (eds. N. Deo et al.), Springer, (2020), 77-88. doi: 10.1007/978-981-15-1153-0_7.

[5]

P. N. Agrawal, N. Bhardwaj and P. Bawa, Bézier variant of modified $\alpha$- Bernstein operators, Rend. Circ. Mat. Palermo, II Ser., (2021). doi: 10.1007/s12215-021-00613-x.

[6]

S. Bernstein, Démonstration du theoréme de Weierstrass fondée sur le calcul des probabilitiés, Comm. Kharkov Math. Soc., 13 (1912), 1-2. 

[7]

N. Çetin and V. A. Radu, Approximation by generalized Bernstein-Stancu operators, Turk. J. Math., 43 (2019), 2032-2048.  doi: 10.3906/mat-1903-109.

[8]

R. A. Devore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, 1993. doi: 10.1007/978-3-662-02888-9.

[9]

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Ser. Comput. Math., Springer-Verlag New York, 1987. doi: 10.1007/978-1-4612-4778-4.

[10]

A. Kajla and T. Acar, Modified $\alpha$- Bernstein operators with better approximation properties, Ann. Funct. Anal., 10 (2019), 570-582.  doi: 10.1215/20088752-2019-0015.

[11]

A. Kajla and T. Acar, Blending type approximation by generalized Bernstein-Durrmeyer type operators, Miskolc Math. Notes, 19 (2018), 319-336.  doi: 10.18514/MMN.2018.2216.

[12]

A. Kajla and D. Miclăuş, Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type, Results Math., 73 (2018), Paper No. 1, 21 pp. doi: 10.1007/s00025-018-0773-1.

[13]

Y. C. KwunA.-M. AcuA. RafiqV. A. RaduF. Ali and S. M. Kang, Bernstein- Stancu type operators which preserve polynomials, J. Comput. Anal. Appl., 23 (2017), 758-770. 

[14]

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

[15]

M. A. Özarslan and H. Aktuğlu, Local approximation properties for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.

[16]

X. ChenJ. TanZ. Liu and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl., 450 (2017), 244-261.  doi: 10.1016/j.jmaa.2016.12.075.

Figure 2.  The convergence of $ M_{n}^{(\alpha )}(f;x) $ and $ M_{n, \tau }^{(\alpha )}(f;x) $ operators to the function $ f(x) $ for $ n = 10 $
Figure 3.  The convergence of $ M_{n}^{(\alpha )}(f;x) $ and $ M_{n, \tau }^{(\alpha )}(f;x) $ operators to the function $ f(x) $ for $ n = 20 $
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