November  2022, 5(4): 331-342. doi: 10.3934/mfc.2022008

On approximation of Bernstein-Durrmeyer operators in movable interval

Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

*Corresponding author: Dansheng Yu

Received  December 2021 Revised  February 2022 Published  November 2022 Early access  March 2022

In the present paper, we introduce a new type of Bernstein-Durrmeyer operators preserving linear functions in movable interval. The approximation rate of the new operators for continuous functions and Voronovskaja's asymptotic estimate are obtained.

Citation: Fengfeng Wang, Dansheng Yu, Bin Zhang. On approximation of Bernstein-Durrmeyer operators in movable interval. Mathematical Foundations of Computing, 2022, 5 (4) : 331-342. doi: 10.3934/mfc.2022008
References:
[1]

T. AcarA. Aral and V. Gupta, On approximation properties of a new type of Bernstein-Durrmeyer operators, Math. Slovaca, 65 (2015), 1107-1122.  doi: 10.1515/ms-2015-0076.

[2]

A. M. Acu and P. N. Agrawal, Better approximation of functions by genuine Bernstein-Durrmeyer type operators, J. Math. Inequal., 12 (2018), 975-987. 

[3]

Q. B. CaiÜ. D. Kantar and B. Çekim, Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators, Appl. Math. J. Chinese Univ. Ser. B, 35 (2020), 468-478.  doi: 10.1007/s11766-020-3918-y.

[4]

Q. B. CaiB. Y. Lian and G. Zhou, Approximation properties of $\lambda$-Bernstein operators, J. Inequal. Appl., 2018 (2018), 1-11.  doi: 10.1186/s13660-018-1653-7.

[5]

W. Chen, On the Modified Durrmeyer-Bernstein Operator, Report of the Fifth Chinese Conference on Approximation Theory, Zhenzhou, China, 1987.

[6]

F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge Monographs on Applied and Computational Mathematics, 24. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618796.

[7]

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

[8]

L. X. Dong and D. S. Yu, Pointwise approximation by a Durrmeyer variant of Bernstein-Stancu operators, J. Inequal. Appl., 2017 (2017), Paper No. 28, 13 pp. doi: 10.1186/s13660-016-1291-x.

[9]

A. D. Gadjiev and A. M. Ghorbanalizaeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput., 216 (2010), 890-901.  doi: 10.1016/j.amc.2010.01.099.

[10]

H. GonskaD. Kacsó and I. Raşa, On genuine Bernstein-Durrmeyer operators, Result. Math., 50 (2007), 213-225.  doi: 10.1007/s00025-007-0242-8.

[11]

H. GonskaD. Kacsó and I. Raşa, The genuine Bernstein-Durrmeyer operators revisited, Result. Math., 62 (2012), 295-310.  doi: 10.1007/s00025-012-0287-1.

[12]

H. Gonska and R. Păltănea, Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Cze. Math. J., 60 (2010), 783-799.  doi: 10.1007/s10587-010-0049-8.

[13]

H. GonskaI. Raşa and E. D. Stănilă, The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, Medit. J. Math., 11 (2014), 561-576.  doi: 10.1007/s00009-013-0347-0.

[14]

T. N. T. Goodman and A. Sharma, A modified Bernstein-Schoenberg operators, In Bl. Sendov et al. (eds), Constructive Theory of Functions Verna 1987, Publ. House Bulgar. Acad. Sci., Sofia, (1988), 166–173.

[15]

T. N. T. Goodman and A. Sharma, A Bernstein type operators on the simplex, Math. Balkanica (N. S.), 5 (1991), 129-145. 

[16]

X. GuoL. X. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Mathematical Foundations of Computing., 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[17]

Z. C. GuoD. H. XiangX. Guo and D. X. Zhou, Thresholded spectral algorithms for sparse approximations, Anal. Appl., 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[18]

V. Gupta and P. Maheshwari, Bezier variant of a new Durrmeyer type operators, Riv. Mat. Univ. Parma, 7 (2003), 9-21. 

[19]

V. Gupta and H. M. Srivastava, A general family of the Srivastava-Gupta operators preserving linear functions, Eur. J. Pure Appl. Math., 11 (2018), 575-579.  doi: 10.29020/nybg.ejpam.v11i3.3314.

[20]

G. İçöz, A kantorovich variant of a new type Bernstein-Stancu polynomails, Appl. Math. Comput., 218 (2012), 8552-8560.  doi: 10.1016/j.amc.2012.02.017.

[21]

B. Jiang and D. S. Yu, On Approximation by Bernstein-Stancu polynomials in movable compact disks, Result. Math., 72 (2017), 1535-1543.  doi: 10.1007/s00025-017-0669-5.

[22]

B. Jiang and D. S. Yu, Approximation by Durrmeyer type Bernstein-Stancu polynomials in movable compact disks, Result. Math., 74 (2019), Paper No. 28, 12 pp. doi: 10.1007/s00025-018-0952-0.

[23]

B. Jiang and D. S. Yu, On approximation by Stancu type Bernstein-Schurer polynomials in compact disks, Result. Math., 72 (2017), 1623-1638.  doi: 10.1007/s00025-017-0740-2.

[24]

H. S. JungN. Deo and M. Dhamija, Pointwise approximation by Bernstein type operators in mobile interval, Appl. Math. Comput., 244 (2014), 683-694.  doi: 10.1016/j.amc.2014.07.034.

[25]

U. D. Kantar and G. Ergelen, A Voronovskaja-Type Theorem for a Kind of Durrmeyer-Bernstein-Stancu Operators, Gazi. Univ. J. Sci., 32 (2019), 1228-1236. 

[26]

B. Z. Li and D. X. Zhou, Analysis of approximation by linear operators on variable $L^{P(\cdot)}_{\rho}$ spaces and applications in learning theory, Abstr. Appl. Anal., 2014 (2014), Art. ID 454375, 10 pp. doi: 10.1155/2014/454375.

[27]

N. I. Mahmudova and V. Gupta, Approximation by genuine Durrmeyer-Stancu polynomials in compact disks, Math. Comput. Modelling, 55 (2012), 278-285.  doi: 10.1016/j.mcm.2011.06.018.

[28]

P. E. Parvanov and B. D. Popov, The limit case of Bernstein's operators with Jacobi weights, Math. Balk. (N. S.), 8 (1994), 165-177. 

[29]

T. Sauer, The genuine Bernstein-Durrmeyer operator on a simplex, Result. Math., 26 (1994), 99-130.  doi: 10.1007/BF03322291.

[30]

H. M. SrivastavaZ. Finta and V. Gupta, Direct results for a certain family of summation-integral type operators, Appl. Math. Comput., 190 (2007), 449-457.  doi: 10.1016/j.amc.2007.01.039.

[31]

H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling, 37 (2003), 1307-1315.  doi: 10.1016/S0895-7177(03)90042-2.

[32]

H. M. Srivastava and V. Gupta, Rate of convergence for Bézier variant of the Bleimann-Butzer-Hahn operators, Appl. Math. Letter, 18 (2005), 849-857.  doi: 10.1016/j.aml.2004.08.014.

[33]

H. M. Srivastava, G. Îçöz and B. Çekim, Approximation properties of an extended family of the Szász-Mirakjan Beta-type operators, Axiom, 8 (2019), Article ID 111, 1–13. doi: 10.3390/axioms8040111.

[34]

H. M. Srivastava, F. Özarslan and S. A. Mohiuddine, Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter $\lambda$, Symmetry, 11 (2019), Article ID 316, 1–22.

[35]

H. M. Srivastava and X. M. Zeng, Approximation by means of the Szász-Bézier integral operators, Int. J. Pure Appl. Math., 14 (2004), 283-294. 

[36]

F. TaşdelenG. Başcanbaz-Tunca and A. Erençin, On a new type Bernstein-Stancu operators, Fasci. Math., 48 (2012), 119-128. 

[37]

F. F. Wang and D. S. Yu, On approximation of Bernstein-Durrmeyer-Type operators in movable interval, Filomat, 35 (2021), 1191-1203.  doi: 10.2298/FIL2104191W.

[38]

M. L. WangD. S. Yu and P. Zhou, On the approximation by operators of Bernstein-Stancu type, Appl. Math. Comput., 246 (2014), 79-87.  doi: 10.1016/j.amc.2014.08.015.

[39]

S. Waldron, A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122 (2003), 141-150.  doi: 10.1016/S0021-9045(03)00041-8.

[40]

D. X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl., 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

[41]

D. X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344.  doi: 10.1007/s10444-004-7206-2.

show all references

References:
[1]

T. AcarA. Aral and V. Gupta, On approximation properties of a new type of Bernstein-Durrmeyer operators, Math. Slovaca, 65 (2015), 1107-1122.  doi: 10.1515/ms-2015-0076.

[2]

A. M. Acu and P. N. Agrawal, Better approximation of functions by genuine Bernstein-Durrmeyer type operators, J. Math. Inequal., 12 (2018), 975-987. 

[3]

Q. B. CaiÜ. D. Kantar and B. Çekim, Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators, Appl. Math. J. Chinese Univ. Ser. B, 35 (2020), 468-478.  doi: 10.1007/s11766-020-3918-y.

[4]

Q. B. CaiB. Y. Lian and G. Zhou, Approximation properties of $\lambda$-Bernstein operators, J. Inequal. Appl., 2018 (2018), 1-11.  doi: 10.1186/s13660-018-1653-7.

[5]

W. Chen, On the Modified Durrmeyer-Bernstein Operator, Report of the Fifth Chinese Conference on Approximation Theory, Zhenzhou, China, 1987.

[6]

F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge Monographs on Applied and Computational Mathematics, 24. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618796.

[7]

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

[8]

L. X. Dong and D. S. Yu, Pointwise approximation by a Durrmeyer variant of Bernstein-Stancu operators, J. Inequal. Appl., 2017 (2017), Paper No. 28, 13 pp. doi: 10.1186/s13660-016-1291-x.

[9]

A. D. Gadjiev and A. M. Ghorbanalizaeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput., 216 (2010), 890-901.  doi: 10.1016/j.amc.2010.01.099.

[10]

H. GonskaD. Kacsó and I. Raşa, On genuine Bernstein-Durrmeyer operators, Result. Math., 50 (2007), 213-225.  doi: 10.1007/s00025-007-0242-8.

[11]

H. GonskaD. Kacsó and I. Raşa, The genuine Bernstein-Durrmeyer operators revisited, Result. Math., 62 (2012), 295-310.  doi: 10.1007/s00025-012-0287-1.

[12]

H. Gonska and R. Păltănea, Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Cze. Math. J., 60 (2010), 783-799.  doi: 10.1007/s10587-010-0049-8.

[13]

H. GonskaI. Raşa and E. D. Stănilă, The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, Medit. J. Math., 11 (2014), 561-576.  doi: 10.1007/s00009-013-0347-0.

[14]

T. N. T. Goodman and A. Sharma, A modified Bernstein-Schoenberg operators, In Bl. Sendov et al. (eds), Constructive Theory of Functions Verna 1987, Publ. House Bulgar. Acad. Sci., Sofia, (1988), 166–173.

[15]

T. N. T. Goodman and A. Sharma, A Bernstein type operators on the simplex, Math. Balkanica (N. S.), 5 (1991), 129-145. 

[16]

X. GuoL. X. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Mathematical Foundations of Computing., 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[17]

Z. C. GuoD. H. XiangX. Guo and D. X. Zhou, Thresholded spectral algorithms for sparse approximations, Anal. Appl., 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[18]

V. Gupta and P. Maheshwari, Bezier variant of a new Durrmeyer type operators, Riv. Mat. Univ. Parma, 7 (2003), 9-21. 

[19]

V. Gupta and H. M. Srivastava, A general family of the Srivastava-Gupta operators preserving linear functions, Eur. J. Pure Appl. Math., 11 (2018), 575-579.  doi: 10.29020/nybg.ejpam.v11i3.3314.

[20]

G. İçöz, A kantorovich variant of a new type Bernstein-Stancu polynomails, Appl. Math. Comput., 218 (2012), 8552-8560.  doi: 10.1016/j.amc.2012.02.017.

[21]

B. Jiang and D. S. Yu, On Approximation by Bernstein-Stancu polynomials in movable compact disks, Result. Math., 72 (2017), 1535-1543.  doi: 10.1007/s00025-017-0669-5.

[22]

B. Jiang and D. S. Yu, Approximation by Durrmeyer type Bernstein-Stancu polynomials in movable compact disks, Result. Math., 74 (2019), Paper No. 28, 12 pp. doi: 10.1007/s00025-018-0952-0.

[23]

B. Jiang and D. S. Yu, On approximation by Stancu type Bernstein-Schurer polynomials in compact disks, Result. Math., 72 (2017), 1623-1638.  doi: 10.1007/s00025-017-0740-2.

[24]

H. S. JungN. Deo and M. Dhamija, Pointwise approximation by Bernstein type operators in mobile interval, Appl. Math. Comput., 244 (2014), 683-694.  doi: 10.1016/j.amc.2014.07.034.

[25]

U. D. Kantar and G. Ergelen, A Voronovskaja-Type Theorem for a Kind of Durrmeyer-Bernstein-Stancu Operators, Gazi. Univ. J. Sci., 32 (2019), 1228-1236. 

[26]

B. Z. Li and D. X. Zhou, Analysis of approximation by linear operators on variable $L^{P(\cdot)}_{\rho}$ spaces and applications in learning theory, Abstr. Appl. Anal., 2014 (2014), Art. ID 454375, 10 pp. doi: 10.1155/2014/454375.

[27]

N. I. Mahmudova and V. Gupta, Approximation by genuine Durrmeyer-Stancu polynomials in compact disks, Math. Comput. Modelling, 55 (2012), 278-285.  doi: 10.1016/j.mcm.2011.06.018.

[28]

P. E. Parvanov and B. D. Popov, The limit case of Bernstein's operators with Jacobi weights, Math. Balk. (N. S.), 8 (1994), 165-177. 

[29]

T. Sauer, The genuine Bernstein-Durrmeyer operator on a simplex, Result. Math., 26 (1994), 99-130.  doi: 10.1007/BF03322291.

[30]

H. M. SrivastavaZ. Finta and V. Gupta, Direct results for a certain family of summation-integral type operators, Appl. Math. Comput., 190 (2007), 449-457.  doi: 10.1016/j.amc.2007.01.039.

[31]

H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling, 37 (2003), 1307-1315.  doi: 10.1016/S0895-7177(03)90042-2.

[32]

H. M. Srivastava and V. Gupta, Rate of convergence for Bézier variant of the Bleimann-Butzer-Hahn operators, Appl. Math. Letter, 18 (2005), 849-857.  doi: 10.1016/j.aml.2004.08.014.

[33]

H. M. Srivastava, G. Îçöz and B. Çekim, Approximation properties of an extended family of the Szász-Mirakjan Beta-type operators, Axiom, 8 (2019), Article ID 111, 1–13. doi: 10.3390/axioms8040111.

[34]

H. M. Srivastava, F. Özarslan and S. A. Mohiuddine, Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter $\lambda$, Symmetry, 11 (2019), Article ID 316, 1–22.

[35]

H. M. Srivastava and X. M. Zeng, Approximation by means of the Szász-Bézier integral operators, Int. J. Pure Appl. Math., 14 (2004), 283-294. 

[36]

F. TaşdelenG. Başcanbaz-Tunca and A. Erençin, On a new type Bernstein-Stancu operators, Fasci. Math., 48 (2012), 119-128. 

[37]

F. F. Wang and D. S. Yu, On approximation of Bernstein-Durrmeyer-Type operators in movable interval, Filomat, 35 (2021), 1191-1203.  doi: 10.2298/FIL2104191W.

[38]

M. L. WangD. S. Yu and P. Zhou, On the approximation by operators of Bernstein-Stancu type, Appl. Math. Comput., 246 (2014), 79-87.  doi: 10.1016/j.amc.2014.08.015.

[39]

S. Waldron, A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122 (2003), 141-150.  doi: 10.1016/S0021-9045(03)00041-8.

[40]

D. X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl., 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

[41]

D. X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344.  doi: 10.1007/s10444-004-7206-2.

[1]

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

[2]

Danilo Costarelli, Gianluca Vinti. Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators. Mathematical Foundations of Computing, 2020, 3 (1) : 41-50. doi: 10.3934/mfc.2020004

[3]

Lucian Coroianu, Sorin G. Gal. New approximation properties of the Bernstein max-min operators and Bernstein max-product operators. Mathematical Foundations of Computing, 2022, 5 (3) : 259-268. doi: 10.3934/mfc.2021034

[4]

Purshottam Narain Agrawal, Jitendra Kumar Singh. Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2021040

[5]

Harun Karsli. On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators. Mathematical Foundations of Computing, 2021, 4 (1) : 15-30. doi: 10.3934/mfc.2020023

[6]

Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189

[7]

Nursel Çetin. On complex modified Bernstein-Stancu operators. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2021043

[8]

Uğur Kadak, Faruk Özger. A numerical comparative study of generalized Bernstein-Kantorovich operators. Mathematical Foundations of Computing, 2021, 4 (4) : 311-332. doi: 10.3934/mfc.2021021

[9]

Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51

[10]

Purshottam Narain Agrawal, Sompal Singh. Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022004

[11]

Mohd Qasim, Mohd Shanawaz Mansoori, Asif Khan, Zaheer Abbas, Mohammad Mursaleen. Convergence of modified Szász-Mirakyan-Durrmeyer operators depending on certain parameters. Mathematical Foundations of Computing, 2022, 5 (3) : 187-196. doi: 10.3934/mfc.2021027

[12]

Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041

[13]

Pablo Blanc, Juan J. Manfredi, Julio D. Rossi. Games for Pucci's maximal operators. Journal of Dynamics and Games, 2019, 6 (4) : 277-289. doi: 10.3934/jdg.2019019

[14]

Ling-Xiong Han, Wen-Hui Li, Feng Qi. Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces. Electronic Research Archive, 2020, 28 (2) : 721-738. doi: 10.3934/era.2020037

[15]

İsmail Aslan, Türkan Yeliz Gökçer. Approximation by pseudo-linear discrete operators. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021037

[16]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457

[17]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253

[18]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056

[19]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219

[20]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43

 Impact Factor: 

Article outline

[Back to Top]