Abstract
This note presents the proof of monotonicity property for the sequence of classical Bernstein operators, involving divided differences and convex functions. As application, we get the form of remainder term associated to the classical Bernstein operators applying Popoviciu’s theorem. We also shall establish an upper bound estimation for the remainder term, when approximated function fulfills some given properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Acu, A.M.: Properties and applications of \(P_n\)-simple functionals. Positivity 21(1), 283–297 (2017)
Acu, A.M., Gonska, H.: Composite Bernstein Cubature. Banach J. Math. Anal. 10(2), 235–250 (2016)
Aramă, O.: Some properties concerning the sequence of polynomials of S.N. Bernstein (in Romanian). Studii şi Cerc. Mat. 8, 195–210 (1957)
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités. Commun. Soc. Math. Kharkow 13(2), 1–2 (1912–1913)
Bărbosu, D., Miclăuş, D.: On the composite Bernstein type quadrature formula. Rev. Anal. Numér. Théor. Approx. 39(1), 3–7 (2010)
Bărbosu, D., Deo, N.: Some Bernstein-Kantorovich operators. Automat. Comput. Appl. Math. 22(1), 15–21 (2013)
Bărbosu, D., Ardelean, G.: The Bernstein quadrature formula revised. Carpathian J. Math. 30(3), 275–282 (2014)
Dascioglu, A.A., Isler, N.: Bernstein collocation method for solving nonlinear differential equations. Math. Comput. Appl. 18(3), 293–300 (2013)
Gonska, H., Raşa, I.: On the sequence of composite Bernstein operators. Rev. Numér. Théor. Approx. 42(2), 151–160 (2013)
Ivan, M.: Elements of Interpolation Theory. Mediamira Science Publisher, Cluj-Napoca (2004)
Lupaş, A.: Mean value theorems for linear positive transformations (in Romanian). Rev. Anal. Numer. Teor. Aprox. 3(2), 121–140 (1974)
Miclăuş, D.: The generalization of Bernstein operator on any finite interval. Georgian Math. J. 24(3), 447–454 (2017)
Popoviciu, T.: Sur quelques propriété des fonctions d’une ou de deux variables réelles. Mathematica (Cluj) 8, 1–85 (1934)
Popoviciu, T.: Les Fonctions Convexes, Actualités Scientifiques et Industrielles, Fasc. 992, Paris (1944) (Publiés sous la direction de Paul Montel)
Popoviciu, T.: Sur le reste dans certaines formules linéaires d’approximation de l’analyse. Mathematica 1(24), 95–142 (1959)
Sofonea, D.F.: Evaluations of the remainder using divided differences. Gen. Math. 20(5), 117–124 (2012)
Stancu, D.D.: On the monotonicity of the sequence formed by the first derivatives of the Bernstein polynomials. Math. Zeitschrift 98, 46–51 (1967)
Temple, W.B.: Stieljes integral representation of convex functions. Duke Math. J. 21, 527–531 (1954)
Acknowledgements
The results presented in this paper were obtained with the support of the Technical University of Cluj-Napoca through the research Contract No. 2011 / 12.07.2017, Internal Competition CICDI-2017.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miclăuş, D. On the monotonicity property for the sequence of classical Bernstein operators. Afr. Mat. 29, 1141–1149 (2018). https://doi.org/10.1007/s13370-018-0602-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-018-0602-4