Abstract
In this paper, we introduce Stancu type modification of generalized Baskakov–Durrmeyer operators and study their approximation properties. First, we derive the recurrence relation and central moments of these operators and then we study the local approximation, weighted approximation results for the new operators. The last section is devoted to A-statistical convergence behaviours of these operators by using the Korovkin type approximation of statistical convergence.
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Acar, T.: Rate of convergence for generalized Szasz operators with derivatives of bounded variation. Proc. Jangjeon Math. Soc. 16(1), 21–35 (2013)
Acar, T., Aral, A., Raşa, I.: Modi.ed Bernstein–Durrmeyer operators. Gen. Math. 22(1), 27–34 (2014)
Acar, T., Gupta, V., Aral, A.: Rate of convergence for generalized Szasz operators. Bull. Math. Sci. 1(1), 99–113 (2011)
Acar T., Ulusoy, G.: Approximation by modified Szasz-Durrmeyer operators. Period. Math. Hung. 72(1), 64–75 (2016)
Agratini, O.: Statistical convergence of intergal operators generated by a single kernal. Nonlinear Anal. 75, 3465–3469 (2012)
Agrawal, P.N., Gupta, V., Sathish Kumar, A.: Generalized Baskakov-Durrmeyer type operators. Rendi. Circ. Mat. Palermo. 63(2), 193–209 (2014)
Ali Özarslan, M., Duman, O., Srivastava, H.M.: Statistical approximation results for Kantorovich-type operators involving some special polynomials. Math. Comput. Modell. 48(3–4), 388–401 (2008)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Duman, O.: \(A\)-statistical convergence of sequences of convolution operators. Taiwanese J. Math. 12(2), 523–536 (2008)
Doğru, O., Örkcü, M.: Statistical approximation by a modification of q- Meyer-Kȯnig and Zeller operators. Appl. Math. Lett. 23(3), 261–266 (2010)
Erencin, A.: Durrmeyer type modification of generalized Baskakov operators. Appl. Math. Comput. 218(3), 4384–4390 (2011)
Erencin, A., Bascanbaz-Tunca, G.: Approximation properties of a class of linear positive operators in weighted spaces. C. R. Acad. Bulgare Sci. 63(10), 1397–1404 (2010)
Erkus, E., Duman, O., Srivastava, H.M.: Statistical approximation of certain positive linear operators constructed by means of the Chan–Chyan–Srivastava polynomials. Appl. Math. Comput. 182(1), 213–222 (2006)
Finta, Z., Govil, N.K., Gupta, V.: Some results on modified Szász–Mirakjan operators. J. Math. Anal. Appl. 327, 1284–1296 (2007)
Gadjiev, A.D.: On P.P. Korovkin type theorems. Math. Zametki 20(5), 781–786 (1976)
Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mountain J. Math. 32(1), 129–138 (2002)
Gupta, V., Radu, C.: Statistical approximation properties of q-Baskakov-Kantorovich operators. Cent. Eur. J. Math. 7(4), 809–818 (2009)
Gupta, V.: Error estimation for mixed summation-integral type operators. J. Math. Anal. Appl. 313, 632–641 (2006)
Karsli, H.: Rate of convergence of new gamma type operators for functions with derivatives of bounded variation. Math. Comput. Modell. 45, 617–624 (2007)
Lenze, B.: On Lipschitz-type maximal functions and their smoothness spaces. Nederl. Akad. Wetensch. Indag. Math. 50(1), 53–63 (1988)
Mihesan, V.: Uniform approximation with positive linear operators generated by generalized Baskakov method. Automat. Comput. Appl. Math. 7(1), 34–37 (1998)
Sahai, A., Prasad, G.: On simultaneous approximation by modified Lupas operators. J. Approx. Theory 45(12), 122–128 (1985)
Stancu, D.D.: Approximation of functions by a new class of linear polynomial operators. Rev. Rom. Math. Pures Appl. 13, 1173–1194 (1968)
Wafi, A., Khatoon, S.: Approximation by generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces. Thai. J. Math. 2(2), 203–216 (2004)
Wafi, A., Khatoon, S.: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cent. Eur. J. Math. 6(2), 325–334 (2008)
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The authors are extremely grateful to the reviewer for a careful reading of the manuscript and making valuable suggestions leading to a better presentation of the paper.
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Kumar, A.S., Acar, T. Approximation by generalized Baskakov–Durrmeyer–Stancu type operators. Rend. Circ. Mat. Palermo, II. Ser 65, 411–424 (2016). https://doi.org/10.1007/s12215-016-0242-1
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DOI: https://doi.org/10.1007/s12215-016-0242-1
Keywords
- Baskakov–Durrmeyer operators
- Rate of convergence
- Weighted approximation
- Modulus of continuity
- Statistical convergence