• Previous Article
    New proofs of Khinchin's law of large numbers and Lindeberg's central limit theorem –PDE's approach
  • MFC Home
  • This Issue
  • Next Article
    Adaptive attitude determination of bionic polarization integrated navigation system based on reinforcement learning strategy
doi: 10.3934/mfc.2022008
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, 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: 

Metrics

  • PDF downloads (243)
  • HTML views (116)
  • Cited by (0)

Other articles
by authors

[Back to Top]