On approximation properties of Stancu variant $\lambda$-Szász-Mirakjan-Durrmeyer operators
Main Article Content
Abstract
In the present paper, we aim to obtain several approximation properties of Stancu form Sz\'{a}sz-Mirakjan-Durrmeyer operators based on B\'{e}zier basis functions with shape parameter $\lambda \in\lbrack-1,1]$. We estimate some auxiliary results such as moments and central moments. Then, we obtain the order of convergence in terms of the Lipschitz-type class functions and Peetre's $K$-functional. Further, we prove weighted approximation theorem and also Voronovskaya-type asymptotic theorem. Finally, to see the accuracy and effectiveness of discussed operators, we present comparison of the convergence of constructed operators to certain functions with some graphical illustrations under certain parameters.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
References
[1] A. M. Acu, N. Manav and D. F. Sofonea, Approximation properties of λ-Kantorovich operators, J. Inequal. Appl., 2018 (2018), 202. Google Scholar
[2] A. Alotaibi, F. O ̈zger, S. A. Mohiuddine and M. A. Alghamdi, Approximation of functions by a class of Durrmeyer–Stancu type operators which includes Euler’s beta function, Adv. Differ. Equ., 2021 (2021), 1–14. Google Scholar
[3] F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, 17, Walter de Gruyter, 2011. Google Scholar
[4] K. J. Ansari, F. O ̈zger and Z. O ̈demi ̧s O ̈zger, Numerical and theoretical approximation results for Schurer–Stancu operators with shape parameter lambda, Comp. Appl. Math., 41 (2022), 1–18. Google Scholar
[5] R. Aslan, Some approximation results on λ-Szasz-Mirakjan-Kantorovich operators, FUJMA, 4 (2021), 150–158. Google Scholar
[6] R. Aslan, Approximation by Sz ́asz-Mirakjan-Durrmeyer operators based on shape parameter λ, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 71 (2022), pp, 407-421. Google Scholar
[7] M. Ayman Mursaleen, A. Kilicman and Md. Nasiruzzaman, Approximation by q-Bernstein- Stancu-Kantorovich operators with shifted knots of real parameters, Filomat, 36(4) (2022), 1179– 1194. Google Scholar
[8] M. Ayman Mursaleen and S. Serra-Capizzano, Statistical convergence via q-calculus and a ko- rovkin’s type Approximation theorem, Axioms, 11 (2022), 70. Google Scholar
[9] Q.B.Cai, K.J.Ansari, M.Temizer Ersoy and F.O ̈zger, Statistical blending-type approximation by a class of operators that includes shape parameters λ and α, Mathematics, 10(7) (2022), 1149. Google Scholar
[10] Q. B. Cai and R. Aslan, On a new construction of generalized q-Bernstein polynomials based on shape parameter λ, Symmetry, 13 (2021), 1919. Google Scholar
[11] Q. B. Cai and R. Aslan, Note on a new construction of Kantorovich form q-Bernstein operators related to shape parameter λ, Computer Modeling in Engineering & Sciences, 130 (2022), 1479– 1493. Google Scholar
[12] Q. B. Cai and W. T. Cheng, Convergence of λ-Bernstein operators based on (p,q)-integers, J. Inequal. Appl., 2020 (2020), 35. Google Scholar
[13] Q. B. Cai, A. Kilicman and M. Ayman Mursaleen, Approximation Properties and q-Statistical Convergence of Stancu-Type Generalized Baskakov-Sz ́asz Operators, J. Funct. Spaces, 2022 (2022). Google Scholar
[14] Q. B. Cai, B. Y. Lian and G. Zhou, Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 2018 (2018), 61. Google Scholar
[15] Q. B. Cai, G. Zhou and J. Li, Statistical approximation properties of λ-Bernstein operators based on q-integers, Open Math., 17 (2019), 487–498. Google Scholar
[16] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Heidelberg, 1993. Google Scholar
[17] G. Farin, Curves and surfaces for computer-aided geometric design: a practical guide, Elsevier, 2014. Google Scholar
[18] A. D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, Dokl. Akad. Nauk., 218 (1974), 1001–1004. Google Scholar
[19] V. Gupta, Simultaneous approximation by Sz ́asz-Durrmeyer operators, Math. Stud., 64 (1995), 27—36. Google Scholar
[20] M. K. Gupta, M. S. Beniwal and P. Goel, Rate of convergence for Sz ́asz–Mirakyan–Durrmeyer operators with derivatives of bounded variation, Appl. Math. comput., 199 (2008), 828–832. Google Scholar
[21] V. Gupta, M. A. Noor and M. S. Beniwal, Rate of convergence in simultaneous approximation for Sz ́asz–Mirakyan–Durrmeyer operators, J. Math. Anal. Appl., 322 (2006), 964–970. Google Scholar
[22] K. Khan, D. K. Lobiyal and A. Kilicman, B ́ezier curves and surfaces based on modified Bernstein polynomials, Azerb. J. Math., 9 (2019), 3–21. Google Scholar
[23] P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 90 (1953), 961–964. Google Scholar
[24] A. Kumar, Approximation properties of generalized λ-Bernstein-Kantorovich type operators, Rend. Circ. Mat. Palermo (2), 70 (2020), 505–520. Google Scholar
[25] S. Mazhar and V. Totik, Approximation by modified Sz ́asz operators, Acta Sci. Math., 49 (1985), 257–269. Google Scholar
[26] G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, In Dokl. Acad. Nauk SSSR, 31 (1941), 201–205. Google Scholar
[27] V. N. Mishra and R. B. Gandhi, A summation-integral type modification of Sz ́asz-Mirakjan operators, Math. Methods Appl. Sci., 40 (2017), 175–182. Google Scholar
[28] V. N. Mishra, R. B. Gandhi and R. N. Mohapatra, A summation-integral type modification of Szasz-Mirakjan-Stancu operators, J. Numer. Anal. Approx. Theory, 45 (2016), 27–36. Google Scholar
[29] V. N. Mishra, R. B. Gandhi and F. Nasaireh, Simultaneous approximation by Sz ́asz-Mirakjan- Durrmeyer-type operators, Bollettino dell’Unione Matematica Italiana, 8 (2016), 297–305. Google Scholar
[30] M. Mursaleen, A. A. H. Al-Abied and M. A. Salman, Chlodowsky type (λ, q)-Bernstein-Stancu operators, Azerb. J. Math., 10 (2020), 75–101. Google Scholar
[31] M. Mursaleen, A. Alotaibi and K. J. Ansari, On a Kantorovich variant of-Sz ́asz-Mirakjan operators, J. Funct. Spaces, 2016 (2016). Google Scholar
[32] H. Oruc ̧ and G. M. Phillips, q-Bernstein polynomials and B ́ezier curves, J. Comput. Appl. Math., 151 (2003), 1–12. Google Scholar
[33] F. O ̈zger, Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators, Filomat, 33 (2019), 3473–3486. Google Scholar
[34] F.O ̈zger, Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numer. Funct. Anal. Optim., 41 (2020), 1990-2006. Google Scholar
[35] F. O ̈zger, On new B ́ezier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 376–393. Google Scholar
[36] F.O ̈zger, E.Aljimi and M.Temizer Ersoy, Rate of weighted statistical convergence of generalized blending-type Bernstein-Kantorovich operators, Mathematics, 10(12) (2022), 2027. Google Scholar
[37] F. O ̈zger, K. Demirci and S. Yıldız, Approximation by Kantorovich variant of λ-Schurer op- erators and related numerical results, In: Topics in Contemporary Mathematical Analysis and Applications, pp. 77-94, CRC Press, Boca Raton, 2020. Google Scholar
[38] Q. Qi, D. Guo and G. Yang, Approximation properties of λ-Sz ́asz-Mirakian operators, Int. J. Eng. Res., 12 (2019), 662–669. Google Scholar
[39] S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of λ-Bernstein- Kantorovich operators with shifted knots, Math. Meth. Appl. Sci., 42 (2019), 4042–4053. Google Scholar
[40] T. W. Sederberg, Computer Aided Geometric Design Course Notes, Department of Computer Science Brigham Young University, October 9, 2014. Google Scholar
[41] H. M. Srivastava, K. J. Ansari, F. O ̈zger and Z. O ̈demi ̧s O ̈zger, A link between approximation theory and summability methods via four-dimensional infinite matrices, Mathematics, 9 (2021), 1895. Google Scholar
[42] H.M.Srivastava, F.O ̈zger and S.A.Mohiuddine, Construction of Stancu-type Bernstein operators based on B ́ezier bases with shape parameter λ, Symmetry, 11 (2019), 316. Google Scholar
[43] D. D. Stancu, Asupra unei generalizari a polinoamelor lui Bernstein, Studia Univ. Babes-Bolyai Ser. Math.-Phys., 14 (1969), 31–45. Google Scholar
[44] O. Sz ́asz, Generalization of the Bernstein polynomials to the infinite interval, J. Res. Nat. Bur. Stand., 45 (1950) 239–245. Google Scholar
[45] Z. Ye, X. Long and X. M. Zeng, Adjustment algorithms for B ́ezier curve and surface, In: Inter- national Conference on Computer Science and Education, pp, 1712–1716, 2010. Google Scholar