On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers


  • Purshottam Agrawal Indian Institute of Technology Roorkee, India
  • Dharmendra Kumar Indian Institute of Technology Roorkee, India
  • Behar Baxhaku University of Prishtina "Hasan Prishtina", Albania




Steklov mean, Peetre's K-functional, modulus of continuity, Lipschitz class, Voronovskaja type theorem, Gruss-Voronovskaja type theorem, mixed modulus of smoothness
Abstract views: 334


In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.


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P.N. Agrawal, B. Baxhaku, R. Chauhan, Quantitative Voronovskaya and Gruss-Voronovskaya type theorems by the blending of Szasz operators including Brenke type polynomials, Turk. J. Math., 42 (2018), 1610-1629. https://doi.org/10.3906/mat-1708-1 DOI: https://doi.org/10.3906/mat-1708-1

P.N. Agrawal , V. Gupta, A.S. Kumar, On q-analogue of Bernstein-Schurer-Stancu operators, Appl. Math. Comput., 219 (14) (2013), 7754-7764. http://dx.doi.org/10.1016/j.amc.2013.01.063 DOI: https://doi.org/10.1016/j.amc.2013.01.063

P.N. Agrawal, N. Ispir, A. Kajla, GBS operators of Lupas-Durrmeyer type based on Polya distribution, Results Math., 69 (3-4) (2016), 397-418. https://doi.org/10.1007/s00025-015-0507-6 DOI: https://doi.org/10.1007/s00025-015-0507-6

E. Aliaga, B. Baxhaku, On the approximation properties of q-analogue bivariate λ-Bernstein type operators, J. Funct. Spaces, Volume 2020, art. ID 4589310, 2020, 11 pp. https://doi.org/10.1155/2020/45893102020 DOI: https://doi.org/10.1155/2020/4589310

A. Aral, V. Gupta, R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer (2013). DOI: https://doi.org/10.1007/978-1-4614-6946-9

C. Badea, I. Badea, H.H. Gonska, Notes on the degree of approximation of B-continuous and B-differentiable functions, J. Approx. Theory Appl., 4 (1988), 95-108.

C. Badea, I. Badea, H.H. Gonska, A test function theorem and approximation by pseudopolynomials, Bull. Austral. Math. Soc., 34 (1986), 53-64. http://dx.doi.org/10.1017/S0004972700004494 DOI: https://doi.org/10.1017/S0004972700004494

D. Barbosu, A.M. Acu and C.V. Muraru, On certain GBS-Durrmeyer operators based on q-integers, Turk. J. Math., 41(2) (2017), 368-380. http://dx.doi.org/10.3906/mat-1601-34 DOI: https://doi.org/10.3906/mat-1601-34

S.N. Bernstein, Demonstration du theoreme de Weierstrass fondee sur le calcul de probabilities, Commun. Soc. Math. Kharkow, 13 (2) 1-2, 1912-1913.

K. Bogel, Mehrdimensionale Differentiation von Funtionen mehrerer Ver ̈anderlicher, J. Reine Angew. Math., 170 (1934), 197-217. https://doi.org/10.1515/crll.1934.170.197 DOI: https://doi.org/10.1515/crll.1934.170.197

P.L. Butzer, H. Berens, Semi-groups of Operators and Approximation, vol. 145, Springer Science and Business Media, 2013.

Q.B. Cai, X.W. Xu, Shape-preserving properties of a new family of generalized Bernstein operators, J. Inequal. Appl., 241 (2018). https://doi.org/10.1186/s13660-018-1821-9 DOI: https://doi.org/10.1186/s13660-018-1821-9

X. Chen, J. Tan, Z. Liu, J. Xie, Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl., 450 (2017), 244-261. https://doi.org/10.1016/j.jmaa.2016.12.075 DOI: https://doi.org/10.1016/j.jmaa.2016.12.075

R.A. Devore and G.G. Lorentz, Constructive Approximation, volume 303, Springer Science and Business Media, 1993. DOI: https://doi.org/10.1007/978-3-662-02888-9_10

C. Disibyuk, H. Oruc, A generalization of rational Bernstein-Bezier curves, BIT Numer. Math., 47 (2007), 313-323, https://doi.org/10.1007/s10543-006-0111-y DOI: https://doi.org/10.1007/s10543-006-0111-y

C. Disibyuk, H. Oruc, Tensor product q-Bernstein polynomials, BIT Numer. Math., 48 (2008), 689–700. https://doi.org/10.1007/S10543-008-0192-XCorpus DOI: https://doi.org/10.1007/s10543-008-0192-x

E. Dobrescu, I. Matei, The approximation by Bernstein polynomials of bidimensionally continuous functions, Univ. Timisoara Ser. St. Mat. Fiz., 4 (1966), 85-90.

S.G. Gal and H. Gonska, Gr ̈uss and Gr ̈uss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables, Jaen J. Approx., 7 (1) (2015), 97-122. https://doi.org/10.48550/arXiv.1401.6824

V. Gupta, T.M. Rassias, P.N. Agrawal, A.M. Acu, Bivariate operators of discrete and integral type, Recent Advances in Constructive Approximation Theory. Springer Optimization and Its Applications, vol. 138, Springer, Cham, 2018. DOI: https://doi.org/10.1007/978-3-319-92165-5

G. Gruss, ̈uber, das maximum des absolutew betrages von G. Gruss, uber, das maximum des absolutew betrages von (1/(b-a)) ∫_{a}^{b}f(x)g(x)dx-(1/((b a)²))∫_{a}^{b}f(x)dx.∫_{a}^{b}g(x)dx, , Math. Z., 39 (1935), 215-226. https://doi.org/10.1007/BF01201355 DOI: https://doi.org/10.1007/BF01201355

A.D. Gadjiev, Theorems of Korovkin type, Mat. Zametki 20 (5) (1976), 781-786. ((in Russian), Math. Notes 20 (1976) (5-6) 995-998 (Engl. Trans.)). https://doi.org/10.1007/BF01146928 DOI: https://doi.org/10.1007/BF01146928

V. Gupta and R.P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Berlin (2014). DOI: https://doi.org/10.1007/978-3-319-02765-4

V. Gupta, G. Tachev, A.M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algor., 81 (1) (2019), 125-149. https://doi.org/10.1007/s11075-018-0538-7 DOI: https://doi.org/10.1007/s11075-018-0538-7

V. Gupta, C. Radu, Statistical approximation properties of q-Baskakov-Kantorovich operators, Cent. Eur. J. Math., 7 (4) (2009), 809-818. 809-818 https://doi.org/10.2478/s11533-009-0055-y DOI: https://doi.org/10.2478/s11533-009-0055-y

A. Kajla and D. Miclaus, Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type, Results Math., 73(1) (2018), 1-21. https://doi.org/10.1007/s00025-018-0773-1 DOI: https://doi.org/10.1007/s00025-018-0773-1

A. Kajla, T. Acar, Modified α-Bernstein operators with better approximation properties, Ann. Funct. Anal., 10 (4) (2019), 570-582. https://doi.org/10.1215/20088752-2019-0015 DOI: https://doi.org/10.1215/20088752-2019-0015

H. Khosravian-Arab, M. Dehghan, M.R. Eslahchi, A new approach to improve the order of approximation of the Bernstein operators: Theory and application, Numer. Algor., 77(1) (2018), 111-150. https://doi.org/10.1007/s11075-017-0307-z DOI: https://doi.org/10.1007/s11075-017-0307-z

S.A. Mohiuddine, T. Acar, M.A. Alghamdi, Genuine modified Bernstein-Durrmeyer operators, J. Inequal. Appl., 104 (2018). https://doi.org/10.1186/s13660-018-1693-z DOI: https://doi.org/10.1186/s13660-018-1693-z

A. Lupas, A q-analogue of the Bernstein operators. Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 9 (1987), 85-98.

N.I. Mahmudov, P. Sabancigil, Approximation Theorems for q-Bernstein-Kantorovich Operator, Filomat, 27 (4) (2013), 721-730. https://doi.org/10.2298/FIL1304721M DOI: https://doi.org/10.2298/FIL1304721M

C.V. Muraru, Note on q-Bernstein-Schurer operators, Studia Univ. Babes-Bolyai Math., 56 (2011) no. 2, 1-11.

M. Mursaleen, A. Khan, Generalized q-Bernstein-Schurer Operators and Some Approximation Theorems, J. Funct. Spaces, vol. 2013, Article ID 719834, 7 pages, 2013, https://doi.org/10.1155/2013/719834. DOI: https://doi.org/10.1155/2013/719834

M.A. Ozarslan, O. Duman, Smoothnesss properties of modified Bernstein-Kantorovich operators, Numer. Funct. Anal. Opt., 37 (1) (2016), 92-105. https://doi.org/10.1080/01630563.2015.1079219 DOI: https://doi.org/10.1080/01630563.2015.1079219

H. Oruc, G.M. Phillips, q-Bernstein polynomials and Bezier curves, J. Comput. Appl. Math., 151, (2003), 1-12. https://doi.org/10.1016/S0377-0427(02)00733-1 DOI: https://doi.org/10.1016/S0377-0427(02)00733-1

G.M. Phillips, Bernstein polynomials based on q-integers, Ann. Numer. Math., 4 (1997), 511-518. https://doi.org/10.1002/mma.4771

R. Ruchi, B. Baxhaku, P.N. Agrawal, GBS operators of bivariate Bernstein-Durrmeyer-type on a triangle, Math. Methods Appl. Sci., 41 (7) (2018), 2673-2683. https://doi.org/10.1002/mma.4771 DOI: https://doi.org/10.1002/mma.4771

V.I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dolk. Akad. Nauk, SSRR, 115 (1) (1957), 17-19.




How to Cite

Agrawal, P., Kumar, D., & Baxhaku, B. (2022). On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers. J. Numer. Anal. Approx. Theory, 51(1), 3–36. https://doi.org/10.33993/jnaat511-1244