Symmetric Properties of λ-Szász Operators Coupled with Generalized Beta Functions and Approximation Theory
Abstract
:1. Introduction
2. Some Estimates and Approximation Results
3. Graphical and Numerical Analysis
4. Local Approximation
5. Bivariate Extension of Generalized Beta Type -Szász–Mirakjan Operators
6. Bivariate Graphical Analysis
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Correction Statement
References
- Szász, O. Generalization of S. Bernstein’s polynomials to the infinite interval. Res. Nat. Bur. Stand. 1950, 45, 239–245. [Google Scholar] [CrossRef]
- Bernšteın, S. Demonstration du théoreme de Weierstrass fondée sur le calcul des probabilities. Commum. Soc. Math. Kharkov. 1912, 13, 1–2. [Google Scholar]
- Izadbakhsh, A.; Kalat, A.A.; Khorashadizadeh, S. Observer-based adaptive control for HIV infection therapy using the Baskakov operator. Biomed. Signal Process. Control 2021, 65, 102343. [Google Scholar] [CrossRef]
- Uyan, H.; Aslan, A.O.; Karateke, S.; Buyukyazıcı, I. Interpolation for neural network operators activated with a generalized logistic-type function. J. Inequal. Appl. 2024, 125, 31. [Google Scholar] [CrossRef]
- Zhang, Q.; Mu, M.; Wang, X. A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator. Micromachines 2022, 14, 44. [Google Scholar] [CrossRef] [PubMed]
- Khan, K.; Lobiyal, D.K. Bezier curves based on Lupas (p,q)-analogue of Bernstein functions in CAGD. Comput. Appl. Math. 2017, 317, 458–477. [Google Scholar] [CrossRef]
- Rao, N.; Farid, M.; Ali, R. A Study of Szász–Durremeyer-Type Operators Involving Adjoint Bernoulli Polynomials. Mathematics 2024, 12, 3645. [Google Scholar] [CrossRef]
- Braha, N.L.; Mansour, T.; Mursaleen, M. Some Approximation Properties of Parametric Baskakov–Schurer–Szász Operators Through a Power Series Summability Method. Complex Anal. Oper. Theory 2024, 18, 71. [Google Scholar] [CrossRef]
- Özger, F.; Demiric, K. Approximation by Kantorovich Variant of λ—Schurer Operators and Related Numerical Results. In Topics in Contemporary Mathematics, Analysis and Applications; CRC Press: Boca Raton, FL, USA, 2020; pp. 77–94. [Google Scholar]
- Ansari, K.J.; Özger, F.; Ödemiş, Ö.Z. Numerical and theoretical approximation results for Schurer–Stancu operators with shape parameter λ. Comput. Appl. Math. 2022, 41, 181. [Google Scholar] [CrossRef]
- Khan, A.; Iliyas, M.; Khan, K.; Mursaleen, M. Approximation of conic sections by weighted Lupaş post-quantum Bézier curves. Demonstr. Math. 2022, 55, 328–342. [Google Scholar] [CrossRef]
- Acar, T.; Mursaleen, M.; Mohiuddine, S.A. Stancu type (p, q)-Szász-Mirakyan-Baskakov operators. Commun. Fac. Sci. Univ. Ank. Ser. Math. Stat. 2018, 67, 116–128. [Google Scholar]
- Alotaibi, A. Approximation of GBS type q-Jakimovski-Leviatan-Beta integral operators in Bögel space. Mathematics 2022, 10, 675. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Singh, K.K.; Alotaibi, A. On the order of approximation by modified summation-integral-type operators based on two parameters. Demonstr. Math. 2023, 8, 20220182. [Google Scholar] [CrossRef]
- Nasiruzzaman, M.; Rao, N.; Kumar, M.; Kumar, R. Approximation on bivariate parametric extension of Baskakov-Durrmeyer-opeator. Filomat 2021, 35, 2783–2800. [Google Scholar] [CrossRef]
- Çiçek, H.; İzgi, A. Approximation by modified bivariate Bernstein-Durrmeyer and GBS bivariate Bernstein-Durrmeyer operators on a triangular region. Fund. J. Math. Appl. 2022, 5, 135–144. [Google Scholar] [CrossRef]
- Cai, Q.-B.; Aslan, R.; Özger, F.; Srivastava, H.M. Approximation by a new Stancu variant of generalized (λ, μ)-Bernstein operators. Alex. Eng. J. 2024, 107, 205–214. [Google Scholar] [CrossRef]
- Aslan, R. Rate of approximation of blending type modified univariate and bivariate λ-Schurer-Kantorovich operators. Kuwait J. Sci. 2024, 51, 100168. [Google Scholar] [CrossRef]
- Rao, N.; Ayman-Mursaleen, M.; Aslan, R. A note on a general sequence of λ-Szász Kantorovich type operators. Comput. Appl. Math. 2024, 43, 428. [Google Scholar] [CrossRef]
- Izgi, A.; Serenbay, S.K. Approximation by complex Chlodowsky-Szász-Durrmeyer operators in compact disks. Creat. Math. Inform. 2020, 29, 37–44. [Google Scholar] [CrossRef]
- Qi, Q.; Guo, D.; Yang, G. Approximation properties of λ-Szász-Mirakian operators. Int. J. Eng. Res. 2019, 12, 662–669. [Google Scholar]
- Păltănea, R. A class of Durrmeyer type operators preserving linear functions. Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. 2007, 5, 109–117. [Google Scholar]
- Rao, N.; Raiz, M.; Mursaleen, M.A.; Mishra, V.N. Approximation properties of Generalized beta-type Szász–Mirakjan operators. Iran. J. Sci. 2023, 47, 1771–1781. [Google Scholar] [CrossRef]
- DeVore, R.A.; Lorentz, G.G. Constructive Approximation. In Grundlehren der Mathematischen Wissenschaften; Springer: Berlin/Heidelberg, Germany, 1993; p. 303. [Google Scholar]
- Altomare, F.; Campiti, M. Korovkin-Type Approximation Theory and Its Applications; Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff; de Gruyter Studies in Mathematics; Walter de Gruyter and Co.: Berlin, Germany, 1994. [Google Scholar]
- Özarslan, M.A.; Aktuglu, H. Local approximation for certain King type operators. Filomat 2013, 27, 173–181. [Google Scholar] [CrossRef]
- Lenze, B. On Lipschitz type maximal functions and their smoothness spaces. Nederl. Akad. Indag. Math. 1988, 50, 53–63. [Google Scholar] [CrossRef]
- Volkov, V.I. On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR (NS) 1957, 115, 17–19. (In Russian) [Google Scholar]
- Stancu, F. Apoximarea Funcțiilor de Două și mai Multe Variabile Prin Șiruri de Operatori Liniari și Pozitivi. Ph.D. Thesis, Cluj-Napoca, Romania, 1984. (In Romanian). [Google Scholar]
y | |||
---|---|---|---|
0.1 | 0.00855716 | 0.00868693 | 0.00888307 |
0.2 | 0.00331262 | 0.000933993 | 0.000881886 |
0.3 | 0.010011 | 0.00441485 | 0.0014342 |
0.4 | 0.0103584 | 0.00364727 | 0.00106068 |
0.5 | 0.00776931 | 0.00208241 | 0.000478021 |
0.6 | 0.00488011 | 0.000981387 | 0.000172209 |
0.7 | 0.00272745 | 0.00040907 | 0.000054154 |
0.8 | 0.00140172 | 0.000156346 | 0.0000155106 |
0.9 | 0.000676062 | 0.0000559933 | 4.14678 × |
y | |||
---|---|---|---|
0.1 | 0.0668425 | 0.05911693 | 0.0473892 |
0.2 | 0.0496822 | 0.00973307 | 0.0267445 |
0.3 | 0.032826 | 0.0654061 | 0.0.0675773 |
0.4 | 0.107994 | 0.0938944 | 0.0542833 |
0.5 | 0.146655 | 0.0958518 | 0.0478268 |
0.6 | 0.153884 | 0.0947532 | 0.000172209 |
0.7 | 0.13949 | 0.0908188 | 0.0706674 |
0.8 | 0.109585 | 0.0765646 | 0.0667936 |
0.9 | 0.0696738 | 0.0499619 | 0.0457327 |
0.1 0.1 | 0.000124468 | 2.3644 × | 1.46835 × |
0.2 0.2 | 0.000054595 | 0.0000251022 | 0.0000102038 |
0.3 0.3 | 0.000126461 | 0.0000319147 | 7.12176 × |
0.4 0.4 | 0.000114291 | 0.0000158371 | 1.94043 × |
0.5 0.5 | 0.0000616724 | 4.69277 × | 3.15716 × |
0.6 0.6 | 0.0000240142 | 1.00341 × | 3.70668 × |
0.7 0.7 | 7.46513 × | 1.71277 × | 3.47392 × |
0.8 0.8 | 1.96793 × | 2.47905 × | 2.76057 × |
0.9 0.9 | 4.574 × | 3.1634 × | 1.9339 × |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 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 (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Rao, N.; Farid, M.; Raiz, M. Symmetric Properties of λ-Szász Operators Coupled with Generalized Beta Functions and Approximation Theory. Symmetry 2024, 16, 1703. https://doi.org/10.3390/sym16121703
Rao N, Farid M, Raiz M. Symmetric Properties of λ-Szász Operators Coupled with Generalized Beta Functions and Approximation Theory. Symmetry. 2024; 16(12):1703. https://doi.org/10.3390/sym16121703
Chicago/Turabian StyleRao, Nadeem, Mohammad Farid, and Mohd Raiz. 2024. "Symmetric Properties of λ-Szász Operators Coupled with Generalized Beta Functions and Approximation Theory" Symmetry 16, no. 12: 1703. https://doi.org/10.3390/sym16121703
APA StyleRao, N., Farid, M., & Raiz, M. (2024). Symmetric Properties of λ-Szász Operators Coupled with Generalized Beta Functions and Approximation Theory. Symmetry, 16(12), 1703. https://doi.org/10.3390/sym16121703