Abstract
We present a generalized sequence of Durrmeyer type operators that allows to summarize different formulas and results for different particular cases. We show for this sequence, several localization and Voronovskaja type results.
This work is partially supported by Junta de Andalucía Research Project FQM-178, by Research Projects DGA (E-64), MTM2015-67006-P (Spain) and by FEDER founds.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
U. Abel, E.E. Berdysheva, Complete asymptotic expansion for multivariate Bernstein–Durrmeyer operators and quasi-interpolants. J. Approx. Theory 162, 201–220 (2010)
U. Abel, V. Gupta, M. Ivan, Asymptotic approximation of functions and their derivatives by generalized Baskakov-Százs-Durrmeyer operators. Anal. Theory Appl. 21(1), 15–26 (2005)
U. Abel, V. Gupta, R.N. Mohapatra, Local approximation by a variant of Bernstein-Durrmeyer operators. Nonlinear Anal. 68, 3372–3381 (2008)
T. Acar, V. Gupta, A. Aral, Rate of convergence for generalized Szász operators. Bull. Math. Sci. 1, 99–113 (2011)
J. Bustamante, J.M. Quesada, A property of Ditzian-Totik second order moduli. Appl. Math. Lett. 23(5), 576–580 (2010)
D. Cárdenas-Morales, P. Garrancho, Local saturation of conservative operators. Acta Math. Hung. 100(1–2), 83–95 (2003)
D. Cárdenas-Morales, V. Gupta, Two families of Bernstein-Durrmeyer type operators. Appl. Math. Comput. 248, 342–353 (2014)
D. Cárdenas-Morales, P. Garrancho, I. Rasa, Approximation properties of Bernstein-Durrmeyer type operators. Appl. Math. Comput. 232, 1–8 (2014)
N. Deo, Direct result on exponential-type operators. Appl. Math. Comput. 204(1), 109–115 (2008)
M.M. Derriennic, Sur l’approximation des fonctions integrables sur \([0,1]\) par des polynomes de Bernstein modifies. J. Approx. Theory 31, 325–343 (1981)
R.A. DeVore, G.G. Lorentz, Constructive Approximation. A Series of Comprehensive Studies in Mathematics, vol. 303 (Springer, Berlin, 1993)
J.L. Durrmeyer, Une formule d’inversion de la transformqeé de Laplace: applications à la théorie des moments, Thése de 3e cycle, Faculté des Sciences de l’Université de Paris (1967)
V. Gupta, P.N. Agrawal, A.R. Gairola, On the integrated Baskakov type operators. Appl. Math. Comput. 213(2), 419–425 (2009)
V. Gupta, M.K. Gupta, Rate of convergence for certain families of summation-integral type operators. J. Math. Anal. Appl. 296, 608–618 (2004)
V. Gupta, M.K. Gupta, V. Vasishtha, Simultaneous approximation by summation-integral type operators. Nonlinear Funct. Anal. Appl. 8(3), 399–412 (2003)
V. Gupta, G.S. Srivastava, A. Sahai, On simultaneous approximation by Szász-beta operators. Soochow J. Math. 21(1), 1–11 (1995)
V. Gupta, Rate of approximation by a new sequence of linear positive operators. Comput. Math. Appl. 45(12), 1895–1904 (2003)
V. Gupta, R. Yadav, Rate of convergence for generalized Baskakov operators. Arab J. Math. Sci. 18(1), 39–50 (2012)
V. Gupta, R.N. Mohapatra, Z. Finta, A certain family of mixed summation-integral type operators. Math. Comput. Model. 42(1–2), 181–191 (2005)
V. Gupta, A.-J. López-Moreno, J.-M. Latorre-Palacios, On simultaneous approximation of the Bernstein Durrmeyer operators. Appl. Math. Comput. 213(1), 112–120 (2009)
V. Gupta E. Erkus, On a hybrid family of summation integral type operators. J. Inequal. Pure Appl. Math. 7(1) (2006), Article 23, 11 pp
M. Heilmann, M.W. Müller, Direct and converse results on simultaneous approximation by the method of Bernstein-Durrmeyer operators, in Algorithms for Approximation I, ed. by J.C. Mason, M.G. Cox (Chapman & Hall, London, 1989), pp. 107–116
M. Heilmann, M.W. Müller, On simultaneous approximation by the method of Baskakov-Durrmeyer operators. Numer. Funct. Anal. Optim. 10(1–2), 127–138 (1989)
A.-J. López-Moreno, J.-M. Latorre-Palacios, Localization results for generalized Baskakov/Mastroianni and composite operators. J. Math. Anal. Appl. 380(2), 425–439 (2011)
A.-J. López-Moreno, F.-J. Muñoz-Delgado, Asymptotic expansion of multivariate conservative linear operators. J. Comput. Appl. Math. 150(2), 219–251 (2003)
P.C. Sikkema, On some linear positive operators. Indag. Math. (Proc.) 73, 327–337 (1970)
H.M. Srivastava, V. Gupta, A certain family of summation-integral type operators. Math. Comput. Model. 37(12–13), 1307–1315 (2003)
A. Ulrich, Asymptotic approximation by Bernstein-Durrmeyer operators and their derivatives. Approx. Theory Appl. 16(2), 1–12 (2000)
G. Ulusoy, E. Deniz, A. Aral, Simultaneous approximation with generalized Durrmeyer operators. Appl. Math. Comput. 260, 126–134 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
López-Moreno, AJ. (2020). Expressions, Localization Results, and Voronovskaja Formulas for Generalized Durrmeyer Type Operators. In: Deo, N., Gupta, V., Acu, A., Agrawal, P. (eds) Mathematical Analysis I: Approximation Theory . ICRAPAM 2018. Springer Proceedings in Mathematics & Statistics, vol 306. Springer, Singapore. https://doi.org/10.1007/978-981-15-1153-0_1
Download citation
DOI: https://doi.org/10.1007/978-981-15-1153-0_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1152-3
Online ISBN: 978-981-15-1153-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)