## Abstract and Applied Analysis

### Approximation by the $q$-Szász-Mirakjan Operators

N. I. Mahmudov

#### Abstract

This paper deals with approximating properties of the q-generalization of the Szász-Mirakjan operators in the case $q>1$. Quantitative estimates of the convergence in the polynomial-weighted spaces and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by the q-Szász-Mirakjan operators ($q>1$ ) is of order ${q}^{-n}$ versus 1/n for the classical Szász-Mirakjan operators.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 754217, 16 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475992

Digital Object Identifier
doi:10.1155/2012/754217

Mathematical Reviews number (MathSciNet)
MR3004883

Zentralblatt MATH identifier
1258.41010

#### Citation

Mahmudov, N. I. Approximation by the $q$ -Szász-Mirakjan Operators. Abstr. Appl. Anal. 2012 (2012), Article ID 754217, 16 pages. doi:10.1155/2012/754217. https://projecteuclid.org/euclid.aaa/1364475992

#### References

• A. Lupaş, “A q-analogue of the Bernstein operator,” in Seminar on Numerical and Statistical Calculus, vol. 9, pp. 85–92, University of Cluj-Napoca, 1987.
• G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol. 4, no. 1–4, pp. 511–518, 1997.
• A. II'inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of Approximation Theory, vol. 116, no. 1, pp. 100–112, 2002.
• S. Ostrovska, “q-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123, no. 2, pp. 232–255, 2003.
• S. Ostrovska, “On the limit q-Bernstein operator,” Mathematica Balkanica. New Series, vol. 18, no. 1-2, pp. 165–172, 2004.
• S. Ostrovska, “On the Lupaş q-analogue of the Bernstein operator,” The Rocky Mountain Journal of Mathematics, vol. 36, no. 5, pp. 1615–1629, 2006.
• S. Ostrovska, “Positive linear operators generated by analytic functions,” Proceedings of the Indian Academy of Science, vol. 117, no. 4, pp. 485–493, 2007.
• T. N. T. Goodman, H. Oruç, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,” Proceedings of the Edinburgh Mathematical Society, vol. 42, no. 1, pp. 179–190, 1999.
• T. K. Boehme and R. E. Powell, “Positive linear operators generated by analytic functions,” SIAM Journal on Applied Mathematics, vol. 16, pp. 510–519, 1968.
• N. I. Mahmudov, “Korovkin-type theorems and applications,” Central European Journal of Mathematics, vol. 7, no. 2, pp. 348–356, 2009.
• V. S. Videnskii, “On some classes of q-parametric positive linear operators,” in Selected Topics in Complex Analysis, vol. 158 of Operator Theory: Advances and Applications, pp. 213–222, Birkhäuser, Basel, Switzerland, 2005.
• T. Trif, “Meyer-König and Zeller operators based on the q-integers,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 29, no. 2, pp. 221–229, 2000.
• O. Doğru and O. Duman, “Statistical approximation of Meyer-König and Zeller operators based on q-integers,” Publicationes Mathematicae Debrecen, vol. 68, no. 1-2, pp. 199–214, 2006.
• W. Heping, “Properties of convergence for the q-Meyer-König and Zeller operators,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1360–1373, 2007.
• A. Aral and O. Doğru, “Bleimann, Butzer, and Hahn operators based on the q-integers,” Journal of Inequalities and Applications, vol. 2007, Article ID 79410, 12 pages, 2007.
• N. I. Mahmudov and P. Sabanc\igil, “q-parametric Bleimann Butzer and Hahn operators,” Journal of Inequalities and Applications, vol. 2008, Article ID 816367, 15 pages, 2008.
• N. Mahmudov and P. Sabancigil, “A q-analogue of the Meyer-König and Zeller operators,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 1, pp. 39–51, 2012.
• A. Aral and V. Gupta, “The q-derivative and applications to q-Szász Mirakyan operators,” Calcolo, vol. 43, no. 3, pp. 151–170, 2006.
• A. Aral, “A generalization of Szász-Mirakyan operators based on q-integers,” Mathematical and Computer Modelling, vol. 47, no. 9-10, pp. 1052–1062, 2008.
• C. Radu, “On statistical approximation of a general class of positive linear operators extended in q-calculus,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2317–2325, 2009.
• N. I. Mahmudov, “On q-parametric Szász-Mirakjan operators,” Mediterranean Journal of Mathematics, vol. 7, no. 3, pp. 297–311, 2010.
• O. Agratini and C. Radu, “On q-Baskakov-Mastroianni operators,” Rocky Mountain Journal of Mathematics, vol. 42, no. 3, pp. 773–790, 2012.
• N. I. Mahmudov, “Statistical approximation of Baskakov and Baskakov-Kantorovich operators based on the q-integers,” Central European Journal of Mathematics, vol. 8, no. 4, pp. 816–826, 2010.
• V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.
• T. Ernst, “The history of q-calculus and a new method,” U.U.D.M. Report 16, Departament of Mathematics, Uppsala University, Uppsala, Sweden, 2000.
• M. Becker, “Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces,” Indiana University Mathematics Journal, vol. 27, no. 1, pp. 127–142, 1978.
• V. Totik, “Uniform approximation by Szász-Mirakjan type operators,” Acta Mathematica Hungarica, vol. 41, no. 3-4, pp. 291–307, 1983.
• J. de la Cal and J. Cárcamo, “On uniform approximation by some classical Bernstein-type operators,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 625–638, 2003.