Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ
Abstract
:1. Introduction
- (i)
- (ii)
- (iii)
2. Some Auxiliary Lemmas and Approximation by Stancu-Type -Bernstein Operators
3. Voronovskaja-Type Theorems
4. The Bivariate Case of the Operators
Author Contributions
Funding
Conflicts of Interest
References
- Bernstein, S.N. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comm. Soc. Math. Kharkow 1913, 13, 1–2. [Google Scholar]
- Stancu, D.D. Asupra unei generalizari a polinoamelor lui Bernstein. Studia Univ. Babes-Bolyai Ser. Math.-Phys. 1969, 14, 31–45. [Google Scholar]
- Acar, T.; Mohiuddine, S.A.; Mursaleen, M. Approximation by (p,q)-Baskakov-Durrmeyer-Stancu operators. Complex Anal. Oper. Theory 2018, 12, 1453–1468. [Google Scholar] [CrossRef]
- Baxhaku, B.; Agrawal, P.N. Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators. Appl. Math. Comput. 2017, 306, 56–72. [Google Scholar] [CrossRef]
- Chauhan, R.; Ispir, N.; Agrawal, P.N. A new kind of Bernstein-Schurer-Stancu-Kantorovich-type operators based on q-integers. J. Inequal. Appl. 2017, 2017, 50. [Google Scholar] [CrossRef] [PubMed]
- Mursaleen, M.; Ansari, K.J.; Khan, A. Some approximation results by (p,q)-analogue of Bernstein-Stancu operators. Appl. Math. Comput. 2015, 264, 392–402. [Google Scholar] [CrossRef]
- Cai, Q.-B.; Lian, B.-Y.; Zhou, G. Approximation properties of λ-Bernstein operators. J. Inequal. Appl. 2018, 2018, 61. [Google Scholar] [CrossRef] [PubMed]
- Ye, Z.; Long, X.; Zeng, X.-M. Adjustment algorithms for Bézier curve and surface. In Proceedings of the International Conference on Computer Science and Education, Hefei, China, 24–27 August 2010; pp. 1712–1716. [Google Scholar]
- Cai, Q.-B. The Bézier variant of Kantorovich type λ-Bernstein operators. J. Inequal. Appl. 2018, 2018, 90. [Google Scholar] [CrossRef] [PubMed]
- Acu, A.M.; Manav, N.; Sofonea, D.F. Approximation properties of λ-Kantorovich operators. J. Inequal. Appl. 2018, 2018, 202. [Google Scholar] [CrossRef]
- Özger, F. Some general statistical approximation results for λ-Bernstein operators. arXiv, 2018; arXiv:1901.01099. [Google Scholar]
- Acar, T.; Aral, A. On pointwise convergence of q-Bernstein operators and their q-derivatives. Numer. Funct. Anal. Optim. 2015, 36, 287–304. [Google Scholar] [CrossRef]
- Acar, T.; Aral, A.; Mohiuddine, S.A. On Kantorovich modification of (p,q)-Bernstein operators. Iran. J. Sci. Technol. Trans. Sci. 2018, 42, 1459–1464. [Google Scholar] [CrossRef]
- Acar, T.; Aral, A.; Mohiuddine, S.A. Approximation by bivariate (p,q)-Bernstein-Kantorovich operators. Iran. J. Sci. Technol. Trans. Sci. 2018, 42, 655–662. [Google Scholar] [CrossRef]
- Acar, T.; Aral, A.; Mohiuddine, S.A. On Kantorovich modification of (p,q)-Baskakov operators. J. Inequal. Appl. 2016, 2016, 98. [Google Scholar] [CrossRef]
- Acu, A.M.; Muraru, C. Approximation properties of bivariate extension of q-Bernstein-Schurer-Kantorovich operators. Results Math. 2015, 67, 265–279. [Google Scholar] [CrossRef]
- Mishra, V.N.; Patel, P. On generalized integral Bernstein operators based on q-integers. Appl. Math. Comput. 2014, 242, 931–944. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Acar, T.; Alotaibi, A. Construction of a new family of Bernstein-Kantorovich operators. Math. Methods Appl. Sci. 2017, 40, 7749–7759. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Acar, T.; Alotaibi, A. Durrmeyer type (p,q)-Baskakov operators preserving linear functions. J. Math. Inequal. 2018, 12, 961–973. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Acar, T.; Alghamdi, M.A. Genuine modified Bernstein-Durrmeyer operators. J. Inequal. Appl. 2018, 2018, 104. [Google Scholar] [CrossRef] [PubMed]
- Mursaleen, M.; Ansari, K.J.; Khan, A. On (p,q)-analogue of Bernstein operators. Appl. Math. Comput. 2018, 266, 874–882, Erratum in Appl. Math. Comput. 2016, 278, 70–71. [Google Scholar] [CrossRef]
- Braha, N.L.; Srivastava, H.M.; Mohiuddine, S.A. A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl. Math. Comput. 2014, 228, 62–169. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Zeng, X.-M. Approximation by means of the Szász-Bézier integral operators. Int. J. Pure Appl. Math. 2004, 14, 283–294. [Google Scholar]
- Ditzian, Z.; Totik, V. Moduli of Smoothness; Springer: New York, NY, USA, 1987. [Google Scholar]
- DeVore, R.A.; Lorentz, G.G. Constructive Approximation; Springer: Berlin, Germany, 1993. [Google Scholar]
- Peetre, J. A Theory of Interpolation of Normed Spaces; Notas Mat.: Rio de Janeiro, Brazil, 1963. [Google Scholar]
- Ozarslan, M.A.; Aktuğlu, H. Local approximation for certain King type operators. Filomat 2013, 27, 173–181. [Google Scholar] [CrossRef]
- Büyükyazıcı, İ.; İbikli, E. The properties of generalized Bernstein polynomials of two variables. Appl. Math. Comput. 2004, 156, 367–380. [Google Scholar] [CrossRef]
- Martinez, F.L. Some properties of two-demansional Bernstein polynomials. J. Approx. Theory 1989, 59, 300–306. [Google Scholar] [CrossRef]
- Volkov, V.J. On the convergence of linear positive operators in the space of continuous functions of two variables. Doklakad Nauk SSSR 1957, 115, 17–19. [Google Scholar]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Srivastava, H.M.; Özger, F.; Mohiuddine, S.A. Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ. Symmetry 2019, 11, 316. https://doi.org/10.3390/sym11030316
Srivastava HM, Özger F, Mohiuddine SA. Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ. Symmetry. 2019; 11(3):316. https://doi.org/10.3390/sym11030316
Chicago/Turabian StyleSrivastava, Hari M., Faruk Özger, and S. A. Mohiuddine. 2019. "Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ" Symmetry 11, no. 3: 316. https://doi.org/10.3390/sym11030316
APA StyleSrivastava, H. M., Özger, F., & Mohiuddine, S. A. (2019). Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ. Symmetry, 11(3), 316. https://doi.org/10.3390/sym11030316