On q-Szász-Durrmeyer operators

• Autorzy: Nazim Mahmudov , Havva Kaffaoğlu
• Języki publikacji: EN

Abstrakty

EN

In the present paper, we introduce the q-Szász-Durrmeyer operators and justify a local approximation result for continuous functions in terms of moduli of continuity. We also discuss a Voronovskaya type result for the q-Szász-Durrmeyer operators.

Słowa kluczowe

EN

q-Szász-Durrmeyer operators; K-functional; Modulus of continuity; q-calculus

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 2
• Strony: 399-409

Daty

wydano

2010-04-01

online

2010-04-14

Twórcy

autor

Nazim Mahmudov

• Eastern Mediterranean University,

autor

Havva Kaffaoğlu

• Eastern Mediterranean University,

Bibliografia

• [1] DeVore R.A., Lorentz G.G., Constructive Approximation, Springer Verlag, Berlin, 1993
• [2] Aral A., Gupta V., The q-derivative and applications to q-Szász Mirakyan operators, Calcolo 2006, 43, 151–170 http://dx.doi.org/10.1007/s10092-006-0119-3
• [3] Aral A., A generalization of Szász-Mirakyan operators based on q-integers, Mathematical and Computer Modelling 47, 2008, 9–10, 1052–1062 http://dx.doi.org/10.1016/j.mcm.2007.06.018
• [4] Aral A., Doğru O., Bleimann, Butzer, and Hahn operators based on the q-integers, J. Inequal. Appl., 2007, Art. ID 79410
• [5] Derriennic M.-M., Modifed Bernstein polynomials and Jacobi polynomials in q-calculus, Rendiconti Del Circolo Matematico Di Palermo, Serie II, Suppl. 2005, 76, 269–290
• [6] De Sole A., Kac V., On integral representations of q-gamma and q-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 2005, 16, 11–29
• [7] II’inskii A., Ostrovska S., Convergence of generalized Bernstein polynomials, J. Approx. Theory, 2002, 116, 100–112 http://dx.doi.org/10.1006/jath.2001.3657
• [8] Gupta V., Some approximation properties of q-Durrmeyer operators, Appl. Math. Comput., 2008, 197, 172–178 http://dx.doi.org/10.1016/j.amc.2007.07.056
• [9] Finta Z., Gupta V., Approximation by q-Durrmeyer operators, J. Appl. Math. Comput., 2009, 29, 401–415 http://dx.doi.org/10.1007/s12190-008-0141-5
• [10] Gupta V., Wang H., The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Math. Methods Appl. Sci., 2008, 31, 1946–1955 http://dx.doi.org/10.1002/mma.1012
• [11] Mahmudov N.I., Korovkin-type Theorems and Applications, Cent. Eur. J. Math., 2009, 7, 348–356 http://dx.doi.org/10.2478/s11533-009-0006-7
• [12] Mahmudov N.I., The moments for q-Bernstein operators in the case 0 < q < 1; Numer Algorithms, DOI 10.1007/s11075-009-9312-1
• [13] Mahmudov N.I., On q-parametric Szász-Mirakjan operators, preprint
• [14] Mahmudov N.I., Sabancıgil P., q-Parametric Bleimann Butzer and Hahn Operators, Journal of Inequalities and Applications, 2008, Article ID 816367
• [15] Kac V., Cheung P., Quantum Calculus, Universitext, Springer-Verlag, New York, 2002
• [16] Ostrovska S., q-Bernstein polynomials and their iterates, J. Approx. Theory, 2003, 123, 232–255 http://dx.doi.org/10.1016/S0021-9045(03)00104-7
• [17] Ostrovska S., On the Lupaş q-analogue of the Bernstein operator, Rocky Mountain Journal of Mathematics, 2006, 36, 1615–1629 http://dx.doi.org/10.1216/rmjm/1181069386
• [18] Özarslan M.A., Aktuğlu H., Local approximation properties of certain class of linear positive operators via I-convergence, Cent. Eur. J. Math., 2008, 6, 281–286 http://dx.doi.org/10.2478/s11533-008-0125-6
• [19] Phillips G.M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 1997, 4, 511–518
• [20] Trif T., Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numer. Theory Approx., 2000, 29, 221–229
• [21] Videnskii V.S., On some classes of q-parametric positive operators, Operator Theory: Advances and Applications, Birkhauser, Basel, 2005, 158, 213–222 http://dx.doi.org/10.1007/3-7643-7340-7_15
• [22] Wang H., Korovkin-type theorem and application, J. Approx. Theory, 2005, 132, 258–264 http://dx.doi.org/10.1016/j.jat.2004.12.010
• [23] Wang, H., Fanjun M., The rate of convergence of q-Bernstein polynomials for 0 < q < 1, J. Approx. Theory, 2005, 136, 151–158 http://dx.doi.org/10.1016/j.jat.2005.07.001
• [24] Wang H., Voronovskaya type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1, J. Approx. Theory, 2007, 145, 182–195 http://dx.doi.org/10.1016/j.jat.2006.08.005

Identyfikatory

DOI

10.2478/s11533-010-0016-5