A Voronovskaya type theorem for q-Szász-Mirakyan-Kantorovich operators

Gülen Başcanbaz-Tunca; Ayçegül Erençin

doi:10.33993/jnaat401-947

Authors

Gülen Başcanbaz-Tunca Ankara University, Turkey
Ayçegül Erençin Abant Izzet Baysal University, Turkey

DOI:

https://doi.org/10.33993/jnaat401-947

Keywords:

\(q\)-Szász-Mirakyan-Kantorovich operator, Riemann type \(q\)-integral, Voronovskaya type theorem

Abstract views: 306

Abstract

In this work, we consider a Kantorovich type generalization of \(q\)-Szász-Mirakyan operators via Riemann type \(q\)-integral and prove a Voronovskaya type theorem by using suitable machinery of \(q\)-calculus.

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References

Andrews, G.E., Askey, R. and Roy, R., Special Functions, Cambridge University Press, 1999. DOI: https://doi.org/10.1017/CBO9781107325937

Aral, A. and Gupta, V., The q-derivative and application to q-Szász-Mirakyan operators, Calcolo, 43 (3), pp. 151-170, 2006. https://doi.org/10.1007/s10092-006-0119-3 DOI: https://doi.org/10.1007/s10092-006-0119-3

Aral, A. and Doğru, O., Bleimann, Butzer and Hahn operators based on the q-integers, J. Inequal. Appl., Art. ID 79410, 12pp, 2007 DOI: https://doi.org/10.1155/2007/79410

Aral, A., A generalization of Szász-Mirakyan operators based on q-integers, Math. Comput. Modelling., 47, pp. 1052-1062, 2008. https://doi.org/10.1016/j.mcm.2007.06.018 DOI: https://doi.org/10.1016/j.mcm.2007.06.018

Butzer, P.L., On the extensions of Bernstein polynomials to the infinite interval, Proc. Amer. Soc., 5, pp. 547-553, 1954 https://doi.org/10.1090/s0002-9939-1954-0063483-7. DOI: https://doi.org/10.1090/S0002-9939-1954-0063483-7

Dalmanoğlu, Ö. and Doğru, O., Statistical approximation properties of Kantorovich type q-MKZ operators, Creat. Math. Inform., 19 (1), pp. 15-24, 2010.

Duman, O., Özarslan, M.A. and Della Vecchia, B., Modified q-Szász-Mirakjan-Kantorovich operators preserving linear functions, Turkish J. Math., 33 (2), pp. 151-158, 2009. DOI: https://doi.org/10.3906/mat-0801-2

Gupta, V., Vasishtha, V. and Gupta, M.K., Rate of convergence of the Szasz-Kantorovich-Bezier operators for bounded variation functions, Publ. Inst. Math. (Beograd) (N.S), 72 (86), pp. 137-143, 2002. DOI: https://doi.org/10.2298/PIM0272137G

Gupta, V. and Zeng, Xiao-Ming, Approximation by Bézier variant of the Szász-Kantorovich operators in case α<1, Georgian Math. J., 17(2), pp. 253-260, 2010. DOI: https://doi.org/10.1515/gmj.2010.017

Kac, V.G. and Cheung, P., Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. DOI: https://doi.org/10.1007/978-1-4613-0071-7

Karslı, H. and Gupta, V., Some approximation properties of q-Chlodowsky operators, Appl. Math. Comput., 195, pp. 220-229, 2008. https://doi.org/10.1016/j.amc.2007.04.085 DOI: https://doi.org/10.1016/j.amc.2007.04.085

Mahmudov, N.I., On q-parametric Szász-Mirakjan operators, Mediterr. J. Math., 7(3), pp. 297-311, 2010. https://doi.org/10.1007/s00009-010-0037-0 DOI: https://doi.org/10.1007/s00009-010-0037-0

Mahmudov, N.I. and Gupta, V., On certain q-analogue of Szász Kantorovich operators, Journal of Applied Mathematics and Computing, October 2011, Volume 37, Issue 1-2, pp 407–419 https://doi.org/10.1007/s12190-010-0441-4. DOI: https://doi.org/10.1007/s12190-010-0441-4

Marinković, S., Rajković, P. and Stanković, M., The inequalities for some types of q-integrals, Comput. Math. Appl., 56, pp. 2490-2498, 2008. https://doi.org/10.1016/j.camwa.2008.05.035 DOI: https://doi.org/10.1016/j.camwa.2008.05.035

Nowak, G. and Sikorska-Nowak, A., Some approximation properties for modified Szasz-Mirakyan-Kantorovich operators, Rev. Anal. Numér. Théor. Approx., 38(1), pp. 73-82, 2009, http://ictp.acad.ro/jnaat/journal/article/view/2009-vol38-no1-art7

Phillips, G.M., Bernstein polynomials based on the q-integers, Annals of Num. Math., 4, pp. 511-518, 1997.

Sikkema, P.C., On some linear positive operators, Indag. Math., 32, pp. 327-337, 1970. https://doi.org/10.1016/s1385-7258(70)80037-3 DOI: https://doi.org/10.1016/S1385-7258(70)80037-3

Stypinski, Z., Theorem of Voronovskaya for Szász-Chlodovsky operators, Funct. Approximatio Comment. Math., 1, pp 133-137, 1974.

Totik, V., Approximation by Szasz-Mirakjan-Kantorovich operators in L^{p}(p>1), Annal. Math., 9(2), pp. 147-167, 1983. https://doi.org/10.1007/bf01982010 DOI: https://doi.org/10.1007/BF01982010

Trif, T., Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numér. Théory Approx., 29(2), pp. 221-229, 2000, http://ictp.acad.ro/jnaat/journal/article/view/2000-vol29-no2-art13

Wang, H., Properties of convergence for the q-Meyer-König and Zeller operators, J. Math. Anal. Appl., 335, pp. 1360-1373, 2007. https://doi.org/10.1016/j.jmaa.2007.01.103 DOI: https://doi.org/10.1016/j.jmaa.2007.01.103