×
Nisar, K. S.; Sharma, V.; Khan, A.

On King type modification of \((p,q)\)-Lupaş Bernstein operators with improved estimates. (English) Zbl 1517.41009

Carpathian Math. Publ. 15, No. 1, 20-30 (2023).
Summary: This paper aims to modify the \((p,q)\)-Lupaş Bernstein operators using King’s technique and to establish convergence results of these operators by using of modulus of continuity and Lipschitz class functions. Some approximation results for this new sequence of operators are obtained. It has been shown that the convergence rate of King type modification is better than the \((p,q)\)-Lupaş Bernstein operators. King type modification of operators also provide better error estimation within some subinterval of \([0,1]\) in comparison to \((p,q)\)-Lupaş Bernstein operators. In the last section, some graphs and tables provided for simulation purposes using MATLAB (R2015a).

MSC:

41A36 Approximation by positive operators
65D17 Computer-aided design (modeling of curves and surfaces)

Keywords:

post-quantum calculus; \((p, q)\)-Lupaş Bernstein operator; modulus of continuity; King type approximation; error estimate

Software:

Matlab
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Acar T. (p,q)(p,q)-Generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. 2016, 39 (16), 2685-2695. doi:10.1002/mma.3721 · Zbl 1342.41019
[2] Acar T., Aral A., Mohiuddine S.A. On Kantorovich modification of (p,q)(p, q)-Bernstein operators. Iran. J. Sci. Technol. Trans. A Sci. 2017, 42 (3), 1459-1464. doi:10.1007/s40995-017-0154-8 · Zbl 1397.41007
[3] Acar T., Aral A., Mursaleen M. Approximation by Baskakov-Durrmeyer operators based on (p,q)(p, q)-integers. Math. Slovaca 2018, 68 (4), 897-906. doi:10.1515/ms-2017-0153 · Zbl 1505.41011
[4] Acar T., Agrawal P.N., Kumar A.S. On a Modification of (p,q)(p,q)-Szász-Mirakyan Operators. Complex Anal. Oper. Theory 2018, 12, 155-167. doi:10.1007/s11785-016-0613-9 · Zbl 1381.41016
[5] Bernstein S.N. Démonstration du theorème de Weierstrass fondeé sur le calcul des probabilités. Commun. Kharkov Math. Soc. 1912, 13 (1), 1-2. · JFM 43.0301.03
[6] Belen C., Mohiuddine S.A. Generalized weighted statistical convergence and application. Appl. Math. Comput. 2013, 219 (18), 9821-9826. · Zbl 1308.40003
[7] Ilarslan H.G.I., Acar T. Approximation by bivariate (p,q)(p,q)-Baskakov-Kantorovich operators. Georgian Math. J. 2016, 25 (3), 397-407. doi:10.1515/gmj-2016-0057. · Zbl 1400.41011
[8] Cai Q.-B., Zhoub G. On (p,q)(p, q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators. Appl. Math. Comput. 2016, 276 (5), 12-20. doi:10.1016/j.amc.2015.12.006 · Zbl 1410.41029
[9] Cai Q.-B., Cheng W.-T. Convergence of λ\lambda-Bernstein operators based on (p,q)(p, q)-integers. J. Inequal. Appl. 2020, 2020 (35). doi:10.1186/s13660-020-2309-y · Zbl 1503.41016
[10] Edely H.H. Osama, Mohiuddine S.A., Noman K.A. Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 2010, 23 (11), 1382-1387. doi:10.1016/j.aml.2010.07.004 · Zbl 1206.40003
[11] Gadjiev A.D., Orhan C. Some approximation theorems via statistical convergence. Rocky Mountain J. Math. 2002, 32 (1), 129-138. doi:10.1216/rmjm/1030539612 · Zbl 1039.41018
[12] Lupaş A. A qq-analogue of the Bernstein operator. Seminar on Numerical and Statistical Calculus. University of Cluj-Napoca. 1987, 9, 85-92. · Zbl 0684.41014
[13] Dalmanoğlu Ö., Örkcü M. Approximation Properties of King Type (p,q)(p,q)-Bernstein Operators. Iran. J. Sci. Technol. Trans. A Sci. 2017, 43 (10), 249-254. doi:10.1007/s40995-017-0434-3
[14] Phillips G.M. Interpolation and Approximation by Polynomials. In: Dilcher K., Taylor K. (Ed.) CMS Books in Mathematics. Springer New York, NY, 2003. · Zbl 1023.41002
[15] Kadak U., Mishra V.N., Pandey S. Chlodowsky type generalization of (p,q)(p, q)-Szász operators involving Brenke type polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 2018, 112 (1), 1443-1462. doi:10.1007/s13398-017-0439-y · Zbl 1400.41018
[16] Khan A., Sharma V. Approximation by (p,q)(p,q)-Lupaş Stancu Operators. Iran. J. Math. Sci. Inform. 2019, 14 (2), 43-60. · Zbl 1455.41001
[17] King J.P. Positive linear operators which preserve x2x^2. Acta Math. Hungar. 2003, 99, 203-208. doi:10.1023/A:1024571126455 · Zbl 1027.41028
[18] Khan K., Lobiyal D.K. Bèzier curves based on Lupaş (p,q)(p,q)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math. 2017, 317, 458-477. doi:10.1016/j.cam.2016.12.016 · Zbl 1357.65021
[19] Mohiuddine S.A., Acar T., Alotaib A. Construction of a new family of Bernstein-Kantorovich operators. Math. Methods Appl. Sci. 2017, 40 (18), 7749-7759. doi:org/10.1002/mma.4559 · Zbl 1387.41008
[20] Mursaleen M., Khan A. Generalized qq-Bernstein-Schurer Operators and Some Approximation Theorems. J. Funct. Spaces 2013, article ID 719834. doi:10.1155/2013/719834 · Zbl 1291.41020
[21] Mursaleen M., Nasiruzzaman Md., Ansari K.J., Alotaibi A. Generalized (p,q)(p,q)-Bleimann-Butzer-Hahn operators and some approximation results. J. Inequal. Appl. 2017, 2017 (1), article ID 310. doi:10.1186/s13660-017-1582-x · Zbl 1386.41002
[22] Mursaleen M., Nasiruzzaman Md., Nurgali A., Abzhapbarov A. Higher order generalization of Bernstein type operators defined by (p,q)(p,q)-integers. J. Comput. Anal. Appl. 2018, 25 (5), 817-829.
[23] Mursaleen M., Ansari K.J., Khan A. On (p,q)(p,q)-analogue of Bernstein Operators. Appl. Math. Comput. 2015, 266, 874-882. doi:10.1016/j.amc.2015.04.090 · Zbl 1410.41004
[24] Mursaleen M., Khan F., Khan A. Approximation by (p,q)(p,q)-Lorentz polynomials on a compact disk. Complex Anal. Oper. Theory 2016, 10 (8), 1725-1740. doi:10.1007/s11785-016-0553-4 · Zbl 1360.41004
[25] Fast H. Sur la convergence statistique. Colloq. Math. 1951, 2, 241-244. · Zbl 0044.33605
[26] Rao N., Wafi A. (p,q)(p,q)-Bivariate-Bernstein-Chlowdosky Operators. Filomat 2018, 32 (2), 369-378. · Zbl 1488.41063
[27] Weierstrass K. Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsber. Akad. Berlin 1885, 633-639. · JFM 17.0384.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.