×

Note on approximation properties of generalized Durrmeyer operators. (English) Zbl 1264.41017

Summary: We study the rate of convergence of \(q\) analogue of Durrmeyer operator generalization proposed by N. Deo. We first estimate moments of \(q\)-Durrmeyer operators. We also study the rate of convergence our operators. We use Maple programming to draw the graphs for the approximation process for two operators. In all graphs, we observe that either classical operator has sharp convergence or both operators behave alike after a large number of iterations. We conclude that the modified operator does not improve the approximation process.

MSC:

41A25 Rate of convergence, degree of approximation
41A35 Approximation by operators (in particular, by integral operators)

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Derriennic MM: Sur ℓ approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies. J. Approx. Theory 1981, 31: 325-343. 10.1016/0021-9045(81)90101-5 · Zbl 0475.41025 · doi:10.1016/0021-9045(81)90101-5
[2] Deo, N.; Bhardwaj, N.; Singh, SP, Simultaneous approximation on generalized Bernstein Durrmeyer operators (2011)
[3] Ostrovska S: q-Bernstein polynomials and their iterates. J. Approx. Theory 2003, 123(2):232-255. 10.1016/S0021-9045(03)00104-7 · Zbl 1093.41013 · doi:10.1016/S0021-9045(03)00104-7
[4] Ostrovska S: The first decade of the q-Bernstein polynomials: results and perspectives. J. Math. Anal. and Approx. Theory 2007, 2(1):35-51. · Zbl 1159.41301
[5] Videnskii VS: On some classes of q-parametric positive operators. Operator Theory: Adv. Appl 2005, 158: 213-222. 10.1007/3-7643-7340-7_15 · Zbl 1088.41008 · doi:10.1007/3-7643-7340-7_15
[6] Wang H: Korovkin-type theorem and application. J. Approx. Theory 2005, 132(2):258-264. 10.1016/j.jat.2004.12.010 · Zbl 1118.41015 · doi:10.1016/j.jat.2004.12.010
[7] Wang H: Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 2007, 145: 182-195. 10.1016/j.jat.2006.08.005 · Zbl 1112.41016 · doi:10.1016/j.jat.2006.08.005
[8] Wang H: Properties of convergence for ω,q-Bernstein polynomials. J. Math. Anal. and Appl 2008, 340(2):1096-1108. 10.1016/j.jmaa.2007.09.004 · Zbl 1144.41004 · doi:10.1016/j.jmaa.2007.09.004
[9] Wang H, Meng F: The rate of convergence of q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 2005, 136(2):151-158. 10.1016/j.jat.2005.07.001 · Zbl 1082.41007 · doi:10.1016/j.jat.2005.07.001
[10] Gupta V: Some approximation properties of q-Durrmeyer operators. App. Math. and Comput 2008, 197(1):172-178. 10.1016/j.amc.2007.07.056 · Zbl 1142.41008 · doi:10.1016/j.amc.2007.07.056
[11] Gupta V, Sharma H: Recurrence formula and better approximation for q-Durrmeyer operators. Lobachevskii J. Mathematics 2011, 32(2):140-145. 10.1134/S1995080211020065 · Zbl 1255.41010 · doi:10.1134/S1995080211020065
[12] Kac V, Cheung P: Quantum calculus. Springer, New York; 2002. · Zbl 0986.05001 · doi:10.1007/978-1-4613-0071-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.