On the Durrmeyer-type variant and generalizations of Lototsky–Bernstein operators


The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.



Ulrich Abel
Fachbereich MND, Technische Hochschule Mittelhessen, Germany

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


Lototsky operator; Korovkin theorem; modulus of smoothness; K-functional; Durrmeyer extension


Cite this paper as:

U. Abel, O. Agratini, On the Durrmeyer-type variant and generalizations of Lototsky–Bernstein operators, Symmetry 2021, 13 (10),  https://doi.org/10.3390/sym13101841

About this paper



Publisher Name


Print ISSN

Not available yet.

Online ISSN


Google Scholar Profile


[1] King, J.P., The Lototsky transform and Bernstein polynomials. Canad. J. Math. 196618, 89–91. [Google Scholar] [CrossRef]
[2] Xu, X.-W.; Goldman, R., On Lototsky-Bernstein operators and Lototsky-Bernstein bases. Comput. Aided Geom. Design 201968, 48–59. [Google Scholar] [CrossRef]
[3] Goldman, R.; Xu, X.-W.; Zeng, X.-M., Applications of the Shorgin identity to Bernstein type operators. Results Math. 201873, 2. [Google Scholar] [CrossRef]
[4] Xu, X.-W.; Zeng, X.-M.; Goldman, R., Shape preserving properties of univariate Lototsky-Bernstein operators. J. Approx. Theory 2017224, 13–42. [Google Scholar] [CrossRef]
[5] Popa, D., Intermediate Voronovskaja type results for the Lototsky-Bernstein type operators. RACSAM 2020114, 12. [Google Scholar] [CrossRef]
[6] Eisenberg, S.; Wood, B., Approximation of analytic functions by Bernstein-type operators. J. Approx. Theory 19726, 242–248. [Google Scholar] [CrossRef]
[7] Abramowitz, M.; Stegun, I.A., (Eds.) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables; Series 55; National Bureau of Standards Applied Mathematics: Washington, DC, USA, 1964. Available online: https://eric.ed.gov/?id=ED250164 (accessed on 30 July 2021).
[8] Derriennic, M.M., Sur l’approximation de fonctions intégrables sur [0,1] par des polynômes de Bernstein modifies. J. Approx. Theory 198131, 325–343. [Google Scholar] [CrossRef]
[9] Shisha, O.; Mond, B., The degree of convergence of linear positive operators. Proc. Natl. Acad. Sci. USA 196860, 1196–1200. [Google Scholar] [CrossRef] [PubMed]
[10] DeVore, R.A.; Lorentz, G.G., Constructive Approximation; Grundlehren der Mathematischen Wissenschaften; Springer: Berlin, Germany, 1993; Volume 303. [Google Scholar]
[11] Mitjagin, B.S.; Semenov, E.M., Lack of interpolation of linear operators in spaces of smooth functions. Math. USSR. Izv. 197711, 1229–1266. [Google Scholar] [CrossRef]
[12] Johnen, H., Inequalities connected with the moduli of smoothness. Mat. Vesnik 19729, 289–305. [Google Scholar]
[13] Altomare, F., Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 20105, 92–164. [Google Scholar]
[14] Šaškin, Y.A., Korovkin systems in spaces of continuous functions. Izv. Akad. Nauk SSSR Ser. Mat. 196226, 495–512. translated in Amer. Math. Soc. Transl. 196654, 125–144. (In Russian) [Google Scholar]
[15] Censor, E., Quantitative results for positive linear approximation operators. J. Approx. Theory 19714, 442–450. [Google Scholar] [CrossRef]
[16] Deo, N.; Kumar, S., Durrmeyer variant of Apostol-Genocchi-Baskakov operators. Quaes. Math. 2020. [Google Scholar] [CrossRef]
[17] Garg, T.; Ispir, N.; Agrawal, P.N., Bivariate q-Bernstein-Chlodowsky-Durrmeyer type operators and associated GBS operators. Asian-Eur. J. Math. 202013, 2050091. [Google Scholar] [CrossRef]
[18] Kajla, A.; Mursaleen, M.; Acar, T., Durrmeyer type generalization of parametric Bernstein operators. Symmetry 202012, 1141. [Google Scholar] [CrossRef]
[19] Neer, T.; Acu, A.M.; Agrawal, P.N., Baskakov-Durrmeyer type operators involving generalized Appell polynomials. Math. Methods Appl. Sci. 202043, 2911–2923. [Google Scholar] [CrossRef]
[20] Abel, U.; Leviatan, D.; Rasa, I., On the q-monotonicity preservation of Durrmeyer-type operators. Mediterr. J. Math. 202118, 173. [Google Scholar] [CrossRef]
[21] Agratini, O.; Aral, A.,  Approximation of some classes of functions by Landau type operators. Results Math. 202176, 12. [Google Scholar] [CrossRef]


Related Posts