A new genuine Durrmeyer operator. (English) Zbl 1337.41009
Agrawal, P. N. (ed.) et al., Mathematical analysis and its applications. Proceedings of the international conference on recent trends in mathematical analyis and its applications, ICRTMAA 2014, Roorkee, India, December 21–23, 2014. New Delhi: Springer (ISBN 978-81-322-2484-6/hbk; 978-81-322-2485-3/ebook). Springer Proceedings in Mathematics & Statistics 143, 121-129 (2015).
Summary: The generalization of the Bernstein polynomials based on certain parameter was considered by D. D. Stancu [Rev. Roum. Math. Pures Appl. 13, 1173–1194 (1968; Zbl 0167.05001)]. Recently, V. Gupta and T. M. Rassias [Banach J. Math. Anal. 8, No. 2, 146–155 (2014; Zbl 1285.41008)] proposed a Durrmeyer-type modification of the Lupaş operators and established some results. Actually, the genuine operators are important as far as the approximation is concerned. Here we propose genuine Durrmeyer-type operators, which preserve linear functions. We establish moments using generalized hypergeometric function and obtain an asymptotic formula and a direct result in terms of second-order modulus of continuity. In the end we propose an open problem for the readers.
For the entire collection see [Zbl 1331.00047].
For the entire collection see [Zbl 1331.00047].
MSC:
| 41A36 | Approximation by positive operators |
References:
| [1] | Agrawal, PN; Gupta, V., Simultaneous approximation by linear combination of modified Bernstein polynomials, Bull. Greek Math. Soc., 39, 29-43 (1989) · Zbl 0747.41014 |
| [2] | DeVore, RA; Lorentz, GG, Constructive Approximation (1993), Berlin: Springer, Berlin · Zbl 0904.49005 · doi:10.1007/978-3-662-02888-9 |
| [3] | Derriennic, M.M.: Sur l’approximation de functionsintegrable sur [0, 1] par des polynomes de Bernstein modifies. J. Approx. Theory 31, 323-343 (1981) · Zbl 0475.41025 |
| [4] | Durrmeyer, J.L.: Une formule d’ inversion de la Transformee de Laplace: applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de l’ Universite de Paris (1967) |
| [5] | Finta, Z.; Gupta, V., Approximation by \(q\)-Durrmeyer operators, J. Appl. Math. Comput., 29, 1-2, 401-415 (2009) · Zbl 1198.41008 · doi:10.1007/s12190-008-0141-5 |
| [6] | Greubel, G.C.: Solution of the Problem 130 (Europeon Mathematical Socity Newsletter, March 2014). Eur. Math. Soc. Newsl, Sept (2014) |
| [7] | Gupta, V., Some approximation properties on \(q\)-Durrmeyer operators, Appl. Math. Comput., 197, 1, 172-178 (2008) · Zbl 1142.41008 · doi:10.1016/j.amc.2007.07.056 |
| [8] | Gupta, V., Europeon mathematical socity newsletter, Problem, 130, 64 (2014) |
| [9] | Gupta, V., Agarwal R.P.: Convergence Estimates in Approximation Theory. Springer, New York (2014) · Zbl 1295.41002 |
| [10] | Gupta, V.; Lopez-Moreno, AJ; Latorre-Palacios, J-M, On simultaneous approximation of the Bernstein Durrmeyer operators, Appl. Math. Comput., 213, 1, 112-120 (2009) · Zbl 0952.45005 · doi:10.1016/j.amc.2009.02.052 |
| [11] | Gupta, V.; Rassias, ThM, Lupaş-Durrmeyer operators based on Polya distribution, Banach J. Math. Anal., 8, 2, 146-155 (2014) · Zbl 1285.41008 · doi:10.15352/bjma/1396640060 |
| [12] | Lupaş, L.; Lupaş, A., Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica, 32, 4, 61-69 (1987) · Zbl 0659.41018 |
| [13] | Miclăuş, D., The revision of some results for Bernstein Stancu type operators, Carpathian J. Math., 28, 2, 289-300 (2012) · Zbl 1289.41038 |
| [14] | Stancu, DD, Approximation of functions by a new class of linear polynomial operators, Rew. Roum. Math. Pure. Appl., 13, 1173-1194 (1968) · Zbl 0167.05001 |
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.
