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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 |
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 [
Keywords:
Modulus of continuity,
Ditzian-Totik modulus of smoothness,
Peetre's K-functional,
asymptotic formula,
local approximation.
Mathematics Subject Classification:
Primary: 41A25; Secondary: 41A36.
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
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References:
| [1] |
T. Acar, P. 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. Acar, A. Aral and I. Raşa,
Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41.
|
| [3] |
A. M. Acu, P. 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. Kwun, A.-M. Acu, A. Rafiq, V. A. Radu, F. 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. Chen, J. Tan, Z. 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. Acar, P. 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. Acar, A. Aral and I. Raşa,
Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41.
|
| [3] |
A. M. Acu, P. 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. Kwun, A.-M. Acu, A. Rafiq, V. A. Radu, F. 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. Chen, J. Tan, Z. 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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