Shape preserving properties of $ (\mathfrak{p}, \mathfrak{q}) $ Bernstein Bèzier curves and corresponding results over $ [a,b] $

Vinita Sharma; Asif Khan; Mohammad Mursaleen

doi:10.3934/mfc.2022041

Article Contents

2023, Volume 6, Issue 4: 691-703. Doi: 10.3934/mfc.2022041

This issue Previous Article Stability analysis of fractional order modelling of social media addiction Next Article Dynamically consistent nonstandard numerical schemes for solving some computer virus and malware propagation models

Shape preserving properties of $ (\mathfrak{p}, \mathfrak{q}) $ Bernstein Bèzier curves and corresponding results over $ [a,b] $

1.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2.
Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

^*Corresponding author: Mohammad Mursaleen
^*Corresponding author: Mohammad Mursaleen

Received: April 2022

Revised: September 2022

Early access: October 18, 2022

Published online: October 18, 2022

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This article deals with shape preserving and local approximation properties of post-quantum Bernstein bases and operators over arbitrary interval $ [a, b] $ defined by Khan and Sharma (Iran J Sci Technol Trans Sci (2021)). The properties for $ (\mathfrak{p}, \mathfrak{q}) $-Bernstein bases and Bézier curves over $ [a, b] $ have been given. A de Casteljau algorithm has been discussed. Further we obtain the rate of convergence for $ (\mathfrak{p}, \mathfrak{q}) $-Bernstein operators over $ [a, b] $ in terms of Lipschitz type space having two parameters and Lipschitz maximal functions.

Keywords:

$ (\mathfrak{p}, \mathfrak{q}) $-Bernstein Bèzier curves,
de Casteljau algorithm,
Lipschitz maximal function.

Mathematics Subject Classification: Primary: 65D17; Secondary: 41A10, 41A25, 41A36.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	T. Acar and A. Aral, Approximation properties of two dimensional Bernstein-Stancu-Chlodowsky operators, Matematiche (Catania), 68 (2013), 15-31.
[2]	T. Acar, A. Aral and S. A. Mohiuddine, Approximation by bivariate $(p, q)$-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 655-662. doi: 10.1007/s40995-016-0045-4.
[3]	R. Aslan and M. Mursaleen, Some approximation results on a class of new type $\lambda$-Bernstein polynomials, J. Math. Inequal., 16 (2022), 445-462. doi: 10.7153/jmi-2022-16-32.
[4]	M. Ayman-Mursaleen, A. Kilicman and M. Nasiruzzaman, Approximation by $q$-Bernstein-Stancu-Kantorovich operators with shifted knots of real parameters, Filomat, 36 (2022), 1179-1194.
[5]	M. Ayman-Mursaleen and S. Serra-Capizzano, Statistical convergence via $q$-calculus and a korovkin's type approximation theorem, Axioms, 11 (2022), 70.
[6]	S. N. Bernstein, Constructive proof of Weierstrass approximation theorem, Comm. Kharkov Math. Soc, 1912.
[7]	P. E. Bèzier, Numerical Control-Mathematics and applications, John Wiley and Sons, London, 1972.
[8]	P. Blaga, T. Cătinaş and G. Coman, Bernstein-type operators on a triangle with one curved side, Mediterr. J. Math., 9 (2011), 1-13. doi: 10.1007/s00009-011-0156-2.
[9]	N. Braha, T. Mansour, M. Mursaleen and T. Acar, Convergence of $\lambda$-Bernstein operators via power series summability method, J. Appl. Math. Comput., 65 (2021), 125-146. doi: 10.1007/s12190-020-01384-x.
[10]	Q.-B. Cai and R. Aslan, On a new construction of generalized q-bernstein polynomials based on shape parameter$\lambda$, Symmetry, 13 (2021), 1919.
[11]	Q.-B. Cai and W.-T. Cheng, Convergence of $\lambda$-Bernstein operators based on $(p, q)$-integers, J. Ineq. App., 2002 (2020), 35. doi: 10.1186/s13660-020-2309-y.
[12]	Q.-B. Cai, A. Kilicman and M. Ayman-Mursaleen, Approximation properties and $q$-statistical convergence of stancu-type generalized baskakov-szász operators, J. Funct. Spaces, 2022 (2022), 2286500. doi: 10.1155/2022/2286500.
[13]	Q.-B. Cai, B.-Y. Lian and G. Zhou, Approximation properties of $\lambda$-Bernstein operators, J. Ineq. App., 2018 (2018), Paper No. 61, 11 pp. doi: 10.1186/s13660-018-1653-7.
[14]	Feng, et. al., CNN models for readability of Chinese texts, Mathematical Foundations of Computing, 5 (2022), 351-362.
[15]	L.-W. Han, Y. Chu and Z.-Y. Qiu, Generalized Bèzier curves and surfaces based on Lupaş$q$-analogue of Bernstein operator, J. Comput. Appl. Math., 261 (2014), 352-363. doi: 10.1016/j.cam.2013.11.016.
[16]	M. N. Hounkonnou and J. D. B. Kyemba, $\mathcal{R}(\mathfrak{p}, \mathfrak{q})$-calculus:differentiation and integration, SUT J. Math., 49 (2013), 145-167.
[17]	S. Huang, et. al., Learning theory of minimum errorentropy under weak moment conditions, Anal. Appl., 20 (2022), 121-139. doi: 10.1142/S0219530521500044.
[18]	V. Kac and P. Cheung, Quantum Calculus, Universitext. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.
[19]	A. Khan, M. S. Mansoori, K. Khan and M. Mursaleen, Phillips-type $q$-bernstein on triangles, J. Funct. Spaces, 2021 (2021), Article ID 6637893, 13 pp. doi: 10.1155/2021/6637893.
[20]	A. Khan, M. S. Mansoori, K. Khan and M. Mursaleen, Lupaş type Bernstein operators on triangles based on quantum analogue, Alexandria Engineering Journal, 60 (2021), 5909-5919.
[21]	A. Khan, V. Sharma and K. Khan, $(\mathfrak{p}, \mathfrak{q})$-Bernstein bases and operators over arbitrary intervals, Iran. J. Sci. Technol. Trans. A Sci., 45 (2021), 1447-1456. doi: 10.1007/s40995-021-01118-z.
[22]	K. Khan, D. K. Lobiyal and A. Kilicman, A de casteljau algorithm for bernstein type polynomials based on $(\mathfrak{p}, \mathfrak{q})$-integers, Appl. Appl. Math., 13 (2018), 997-1017.
[23]	K. Khan, D. K. Lobiyal and A. Kilicman, Bèzier curves and surfaces based on modified Bernstein polynomials, Azerb. J. Math., 9 (2019), 3-21.
[24]	P. P. Korovkin, Linear operators and approximation theory, Hindustan Publishing Corporation, Delhi, 1960.
[25]	B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces, Nederl. Akad. Wetensch. Indag. Math., 91 (1988), 53-63.
[26]	B. Lenze, Bernstein-Baskakov-Kantorovich operators and Lipschitz-type maximal functions, Approximation Theory, Kecskemet, Colloq. Math. Soc. Janos Bolyai, North-Holland, Amsterdam, 58 (1991), 469-496.
[27]	A. Lupaş, A $q$-analogue of the bernstein operator, seminar on numerical and statistical calculus, University of Cluj-Napoca, 9 (1987), 85-92.
[28]	M. Mursaleen, A. A. H. Al-Abied and M. A. Salman, Chlodowsky type $(\lambda, q)$-Bernstein-Stancu operators, Azerb. J. Math., 10 (2020), 75-101.
[29]	M. Mursaleen, K. J. Ansari and A. Khan, On $(\mathfrak{p}, \mathfrak{q})$-analogue of bernstein operators, Appl. Math. Comput., 266 (2015), 874-882. doi: 10.1016/j.amc.2015.04.090.
[30]	M. Mursaleen, M. Nasiruzzaman, A. Khan and K. J. Ansari, Some approximation results on Bleimann-Butzer-Hahn operators defined by $(\mathfrak{p}, mathfrak{q})$-integers, Filomat, 30 (2016), 639-648. doi: 10.2298/FIL1603639M.
[31]	H. Oruç and G. M. Phillips, $q$-Bernstein polynomials and Bèzier curves, J. Comput. Appl. Math., 151 (2003), 1-12. doi: 10.1016/S0377-0427(02)00733-1.
[32]	S. Ostrovska, On the lupaş, $q$-analogue of the bernstein operator, Rocky Mountain J. Math., 36 (2006), 1615-1629. doi: 10.1216/rmjm/1181069386.
[33]	M. A. Özarslan and H. Aktuğlu, Local approximation properties for certain king type operators, Filomat, 27 (2013), 173-181. doi: 10.2298/FIL1301173O.
[34]	G. M. Phillips, A de Casteljau algorithm for generalized Bernstein polynomials, BIT, 36 (1996), 232-236. doi: 10.1007/BF02510184.
[35]	G. M. Phillips, Bernstein polynomials based on the $q$-integers, The heritage of P. L. Chebyshev, Ann. Numer. Math., 4 (1997), 511-518.
[36]	G. M. Phillips, A generalization of the Bernstein polynomials based on the $q$-integers, ANZIAM J., 42 (2000), 79-86. doi: 10.1017/S1446181100011615.
[37]	G. M. Phillips, Interpolation and Approximation by Polynomials, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York, 2003. doi: 10.1007/b97417.
[38]	A. Rababah and S. Manna, Iterative process for G2-multi degree reduction of B$\acute{e}$zier curves, Appl. Math. Comput., 217 (2011), 8126-8133. doi: 10.1016/j.amc.2011.03.016.
[39]	S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of $\lambda$-Bernstein-Kantorovich operators with shifted knots, Math. Meth. Appl. Sci., 42 (2019), 4042-4053. doi: 10.1002/mma.5632.
[40]	T. W. Sederberg, Computer aided geometric design course notes, Department of Computer Science Brigham Young University, 9 (2014).
[41]	P. Simeonova, V. Zafirisa and R. Goldman, $q$-Blossoming: A new approach to algorithms and identities for $q$-Bernstein bases and $q$-B$\acute{e}$zier curves, J. Approx. Theory, 164 (2012), 77-104. doi: 10.1016/j.jat.2011.09.006.
[42]	F. Wang, et. al., On approximation of Bernstein-Durrmeyer-type operators in movable interval, Filomat, 5 (2022), 331-342. doi: 10.2298/fil2104191w.