We observe that the approximation of our operators to the function $f$ is better than that of some other operators in a certain range of values of $k$.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:The purpose of this paper is to introduce a Kantorovich variant of Lupa\c{s}-Stancu operators based on Polya distribution with Pochhammer $k$-symbol.
📊
Key Findings & Data:We obtain rates of convergence for these operators by means of the classical modulus of continuity.
💡
Conclusion & Impact:We observe that the approximation of our operators to the function $f$ is better than that of some other operators in a certain range of values of $k$.
The purpose of this paper is to introduce a Kantorovich variant of Lupa\c{s}-Stancu operators based on Polya distribution with Pochhammer $k$-symbol. We obtain rates of convergence for these operators by means of the classical modulus of continuity. Also, we give a Voronovskaja type theorem for the pointwise approximation. Furthermore, we construct a bivariate generalization of these operators and we discuss some convergence properties of them. Finally, we present some figures to compare approximation properties of our new operators with those of other operators which are mentioned in this paper. We observe that the approximation of our operators to the function $f$ is better than that of some other operators in a certain range of values of $k$.
YILMAZ, Ö. G., AKTAŞ, R., YEŞİLDAL, F. T., OLGUN, A. (2022). On approximation properties of generalized Lupaş type operators based on Polya distribution with Pochhammer $k$-symbol. Hacettepe Journal of Mathematics and Statistics, Cilt: 51 - Sayı: 2, 338–361. https://ajindex.com/articles/669211
A.M. Acu and H. Gonska, Perturbed Bernstein-type operators, Anal. Math. Phys. 10 (4), 1-26, 2020
A.M. Acu, N. Manav and D.F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl. 2018, 1-12, 2018
P.N. Agrawal and P. Gupta, q-Lupaş Kantorovich operators based on Pólya distribu- tion, Ann. Univ. Ferrara 64, 1-23, 2018
P.N. Agrawal, N. Ispir and A. Kajla, Approximation properties of Bézier-summation integral type operators based on Pólya-Bernstein functions, Appl. Math. Comput. 259, 533-539, 2015
P.N. Agrawal, N. Ispir and A. Kajla, GBS Operators of Lupaş-Durrmeyer type based on Pólya Distribution, Results Math. 69 (3-4), 397-418, 2016
P.N. Agrawal, N. Ispir and A. Kajla, Approximation properties of Lupas-Kantorovich operators based on Polya distribution, Rend. Circ. Mat. Palermo 65, 185-208, 2016
F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica- tions, de Gruyter Studies in Mathematics 17, Walter de Gruyter, Berlin, 1994
D. Barbosu, Kantorovich Stancu type operators, J. Inequal.Pure Appl. Math. 5 (3), 2004
S.N. Bernstein, Demonstration du theoreme de Weierstrass Fondee sur le calcul des probabilites, Comp. Comm. Soc. Mat. Charkow Ser. 13 (2), 1-2, 1912
D. Cárdenas-Morales and V. Gupta, Two families of Bernstein-Durrmeyer type op- erators, Appl. Math. Comput. 248, 342-353, 2014
N. Deo, M. Dhamija and D. Miclăuş, Stancu-Kantorovich operators based on inverse Pólya–Eggenberger distribution, Appl. Math. Comput. 273, 281-289, 2016
R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat. 15 (2), 179-192, 2007
V. Gupta and T. Rassias, Lupaş-Durrmeyer operators based on Pólya distribution, Banach J. Math. Anal. 8 (2), 146-155, 2014
A. Kajla and D. Miclăuş, Some smoothness properties of the Lupaş-Kantorovich type operators based on Pólya distribution, Filomat 32 (11), 3867-3880, 2018
A. Kajla and D. Miclăuş, Approximation by Stancu-Durrmeyer type operators based on Pólya-Eggenberger distribution, Filomat 32 (12), 4249-4261, 2018
C.G. Kokologiannaki, Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences 5 (14), 653-660, 2010
V. Krasniqi, A limit for the k-Gamma and k-Beta Function, Int. Math. Forum 5 (33), 2010
S.F. Li and Y. Dong, k-Hypergeometric series solutions to one type of non- homogeneous k-hypergeometric equation