On approximation properties of generalized Lupaş type operators based on Polya distribution with Pochhammer $k$-symbol

Year 2022, , 338 - 361, 01.04.2022

Övgü Gürel Yılmaz , Rabia Aktaş , Fatma Taşdelen Yeşildal , Ali Olgun

https://doi.org/10.15672/hujms.911716

Abstract

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 $k$.

Keywords

Bernstein operators, Stancu operators, Lupas operators, Kantorovich operators, Polya distribution, modulus of continuity, Lipschitz class, Voronovskaja type theorem, Pochhammer k-symbol

Supporting Institution

Scientific Research Projects Coordination Unit of Kırıkkale University

Project Number

Project number 2020/045

References

• [1] A.M. Acu and H. Gonska, Perturbed Bernstein-type operators, Anal. Math. Phys. 10 (4), 1-26, 2020.
• [2] A.M. Acu, N. Manav and D.F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl. 2018, 1-12, 2018.
• [3] P.N. Agrawal and P. Gupta, q-Lupaş Kantorovich operators based on Pólya distribu- tion, Ann. Univ. Ferrara 64, 1-23, 2018.
• [4] 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.
• [5] 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.
• [6] 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.
• [7] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica- tions, de Gruyter Studies in Mathematics 17, Walter de Gruyter, Berlin, 1994.
• [8] D. Barbosu, Kantorovich Stancu type operators, J. Inequal.Pure Appl. Math. 5 (3), 2004.
• [9] 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.
• [10] D. Cárdenas-Morales and V. Gupta, Two families of Bernstein-Durrmeyer type op- erators, Appl. Math. Comput. 248, 342-353, 2014.
• [11] N. Deo, M. Dhamija and D. Miclăuş, Stancu-Kantorovich operators based on inverse Pólya–Eggenberger distribution, Appl. Math. Comput. 273, 281-289, 2016.
• [12] R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat. 15 (2), 179-192, 2007.
• [13] V. Gupta and T. Rassias, Lupaş-Durrmeyer operators based on Pólya distribution, Banach J. Math. Anal. 8 (2), 146-155, 2014.
• [14] A. Kajla and D. Miclăuş, Some smoothness properties of the Lupaş-Kantorovich type operators based on Pólya distribution, Filomat 32 (11), 3867-3880, 2018.
• [15] A. Kajla and D. Miclăuş, Approximation by Stancu-Durrmeyer type operators based on Pólya-Eggenberger distribution, Filomat 32 (12), 4249-4261, 2018.
• [16] C.G. Kokologiannaki, Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences 5 (14), 653-660, 2010.
• [17] V. Krasniqi, A limit for the k-Gamma and k-Beta Function, Int. Math. Forum 5 (33), 2010.
• [18] S.F. Li and Y. Dong, k-Hypergeometric series solutions to one type of non- homogeneous k-hypergeometric equations, Symmetry 11, 262, 2019.
• [19] L. Lupaş and A. Lupaş, Polynomials of binomial type and approximation operators, Stud. Univ. Babes-Bolyai Math. 32 (4), 61-69, 1987.
• [20] D. Miclăuş, The revision of some results for Bernstein Stancu type operators, Carpathian J. Math 28 (2), 289-300, 2012.
• [21] D. Miclăuş, On the monotonicity property for the sequence of Stancu type polynomials, An. Ştiint. Univ. Al. I. Cuza Iaşi, Mat. (N.S) 62 (1), 141-149, 2016.
• [22] D. Miclăuş, On the Stancu type bivariate approximation formula, Carpathian J. Math. 32 (1), 103-111, 2016.
• [23] S.A. Mohiuddine and F. Özger, Approximation of functions by Stancu variant of Bernstein Kantorovich operators based on shape parameter , Rev. R. Acad. Cienc. Exactas. Fis. Nat. Ser. A Math. RACSAM 114 (70), 2020.
• [24] S. Mubeen, k-Analogue of Kummer’s first formula, J. Inequal. Spec. Funct. 3 (3), 41-44, 2012.
• [25] S. Mubeen and A. Rehman, A note on k-gamma function and pochhammer k-symbol, J. Inform. Math. Sci. 6 (2), 93-107, 2014.
• [26] T. Neer and P.N. Agrawal, A genuine family of Bernstein-Durrmeyer type operators based on Pólya basis functions, Filomat, 31 (9), 2611-2623, 2017.
• [27] G. Nowak, Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl. 350, 50-55, 2009.
• [28] A.A. Opriş, Approximation by modified Kantorovich Stancu operators, J. Inequal. Appl. 2018 (346), 2018.
• [29] S. Ostrovska and M. Turan, The distance between two limit q-Bernstein operators, Rocky Mountain J. Math. 50 (3), 1085-1096, 2020.
• [30] M.A. Özarslan and O. Duman, Smoothness properties of modified Bernstein- Kantorovich operators, Numer. Funct. Anal. Optim. 37, 92-105, 2016.
• [31] S. Rahman, M. Mursaleen and A. Khan, A Kantorovich variant of Lupas-Stancu operators based on Pólya distribution with error estimation, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114 (75), 2020.
• [32] Q. Razi, Approximation of functions by Bernstein type operators, Master Thesis, Aligarh Muslim University, Aligarh, India, 1983.
• [33] Q. Razi, Approximation of a function by Kantorovich type operators, Mat. Vesnik 41, 183-192, 1989.
• [34] O. Shisha and B. Mond, The degree of convergence of sequences of linear positive operators, Proc. Natl. Acad. Sci. USA, 60, 1196-1200, 1968.
• [35] D.D. Stancu, Approximation of functions by a new class of linear polynomial opera- tors, Rev. Roumaine Math. Pures Appl. 13 (8), 1173-1194, 1968.
• [36] V.I. Volkov, On the convergence of sequences of linear positive operators in the space of two variables, Dokl. Akad. Nauk. SSSR (N.S.), 115, 17-19, 1957.