On Kantorovich form of generalized Sz\'{a}sz-type operators using Charlier polynomials

Abdul Wafi, Nadeem Rao, . Deepmala


The aim of this article is to introduce a new form of Kantorovich Sz\'{a}sz-type operators involving Charlier polynomials. In this manuscript, we discuss the rate of convergence, better error estimates. Further, we investigate order of approximation in the sense of local approximation results with the help of Ditzian-Totik modulus of smoothness, second order modulus of continuity, Peetre's K-functional and Lipschitz class.


Kantorovich Sz\'{a}sz operators, Charlier polynomials, Ditzian-Totik modulus of smoothness, Peetre's K-functional, Lipschitz class

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T.Acar, L.N.Mishra and V.N.Mishra, Simultaneous approximation for generalized Srivastava-Gupta operator, Journal of Function Spaces, 2015, Article ID 936308, (2015) 11 pages, doi:10.1155/2015/936308.

S.N. Bernstein, De ́monstration du the ́ore`me de Weierstrass, fondee ́ sur le calcul de ́s probabilite ́s, Commun. Soc. Math. Kharkow 2 (13) (1912-1913), 1–2.

R.A. DeVore and G.G. Lorentz, Constructive Approximation, Grudlehren der Mathematischen Wissenschaften [Fundamental principales of Mathematical Sci- ences], (Springer-Verlag, Berlin, 1993).

Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, 8. Springer-Verlag, New York, 1987.

R.B. Gandhi, Deepmala and V.N. Mishra, Local and global results for modified Sz ́asz - Mirakjan operators, Math. Method. Appl. Sci. (2016), DOI:10.1002/ mma.4171.

H.Gonska and I.Rasa, Asymptotic behaviour of differentiated Bernstein polynomials, Mat. Vesnik, 61 (2009), 53–60.

H.Gonska, M.Heilmann and I.Rasa, Kantorovich operators of order k, Numer. Funct. Anal. Optimiz. 32 (2011), 717–738.

M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in one Variable, Cambridge University Press, Cambridge, 2005.

L.V. Kantorovich, Sur certains de ́veloppements suivant les polynoˆmes la forme de ́ S. Bernstein, I, II, C. R. Acad URSS, (1930) 563–568, 595–600.

J.P. King, Positive linear opeartors which preserves x2, Acta Math. Hungar 99 (3) (2003), 203–208.

G.G. Lorentz, Mathematical Expositions, No. 8, Bernstein polynomials, Univer- sity of Toronto Press, Toronto 1953.

V.N. Mishra, K. Khatri and L.N. Mishra, On Simultaneous Approximation for Baskakov-Durrmeyer-Stancu type operators, Journal of Ultra Scientist of Phys- ical Sciences 24 (3) (2012), 567–577.

V.N. Mishra, K. Khatri, L.N.Mishra and Deepmala, Inverse result in simulta- neous approximation by Baskakov-Durrmeyer-Stancu operators, Journal of In- equalities and Applications 2013, 2013:586.doi:10.1186/1029-242X-2013-586.

V.N. Mishra, H.H. Khan, K. Khatri and L.N. Mishra, Hypergeometric Representation for Baskakov-Durrmeyer-Stancu Type Operators, Bulletin of Mathe- matical Analysis and Applications, 5 (3) (2013), 18–26.

V.N. Mishra, K. Khatri and L.N. Mishra, Some approximation properties of q- Baskakov-Beta-Stancu type operators, Journal of Calculus of Variations, Volume 2013, Article ID 814824, 8 pages. http://dx.doi.org/10.1155/2013/814824.

V.N. Mishra, K. Khatri and L.N. Mishra Statistical approximation by Kantorovich type Discrete q−Beta operators, Advances in Difference Equations 2013, 2013:345, DOI:10.1186/10.1186/1687-1847-2013-345.

V.N. Mishra, P. Sharma and L.N. Mishra, On statistical approximation properties of q−Baskakov-Sz ́asz-Stancu operators, Journal of Egyptian Mathematical Society 24 (3) (2016), 396–401. DOI:10.1016/j.joems.2015.07.005.

V.N. Mishra, R.B.Gandhi and F.Nasierh, Simultaneous approximation by Sz ́asz-Mirakjan-Durrmeyer-type operators, Bollettino dell’Unione Matematica Italiana 8 (4) (2016), 297–305.

V.N. Mishra and R.B. Gandhi, Simultaneous approximation by Sz ́asz-Mirakjan- Stancu-Durrmeyer type operators, Periodica Mathematica Hungarica 74 (1), (2017), 118–127. DOI:10.1007/s10998-016-0145-0.

R.N. Mohapatra and Z. Walczak, Remarks on a class of Szsz-Mirakyan type operators, East J. Approx., 15 (2) (2009), 197–206.

O. Sz ́asz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Research Nat. Bur. Standards Sci. 45 (3-4) (1950), 239–245.

V. Totik, Approximation by Sza ́sz-Mirakjan-Kantorovich operators in Lp(p > 1), Anal. Math. 9 (2) (1983), 147–167.

A. Wafi and N. Rao, Stancu-variant of generalized Baskakov operators, Filomat, (2015) (In Press).

A. Wafi, N.Rao and D. Rai, Approximation properties by generalized-Baskakov- Kantorovich-Stancu type operators, Appl. Math. Inf. Sci. Lett., 4 (3) (2016), 111–118.

A. Wafi and N. Rao, Sza ́sz-Durremeyer operators based on Dunkl analogue, Com- plex Anal. Oper. Theory, (2017) 1–18. doi:10.1007/s11785-017-0647-7.

A. Wafi and N. Rao, A generalization of Sz ́asz-type operators which preserves constant and quadratic test functions, Cogent Mathematics (2016), 3: 1227023.

S. Varma and F. Ta ̧sdelen, Sz ́asz type operators involving Charlier polynomials, Math. Comput. Modeling, 56 (5-6) (2012) 118–122.

DOI: http://dx.doi.org/10.11568/kjm.2017.25.1.99


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