On a new family of generalized Bernstein operators
DOI:
https://doi.org/10.24193/subbmath.2022.3.12Keywords:
Bernstein-type operators, global smoothness preservation, second order modulus of continuity.Abstract
In this paper we remark that α-Bernstein operators, introduced by X. Y. Chen et al., are combinations of two known operators (Stancu and Bernstein operators) and we establish the preservation of global smoothness properties by these linear operators, the global smoothness being expressed by a Lipschitz condition with a certain second order modulus of continuity.
Mathematics Subject Classification (2010): 41A36, 41A17.
Received 27 June 2020; Accepted 29 August 2020.
References
Adell, J.A., de la Cal, J., Preservation of moduli of continuity by Bernstein-type operators, in: Approximation, Probability and Related Fields (G.A. Anastassiou and S.T. Rachev Eds.), Plenum Press, New York, (1994), 1-18.
Anastassiou, G.A., Cottin, C., Gonska, H.H., Global smoothness of approximating functions, Analysis (Munich), 11(1991), 43-57.
Anastassiou, G.A., Gal, S., Approximation Theory: Moduli of Continuity and Global Smoothness Preservation, Birkha¨user, Boston, 2000.
Brown, B.M., Elliot, D., Paget, D.F., Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory, 49(1987), 196-199.
Chen, X, Tan, J., Liu, Z., Xie, J., Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl., 450(2017), 244-261.
Cottin, C., Gonska, H.H., Simultaneous approximation and global smoothness preservation, Rend. Circ. Mat. Palermo (2) Suppl., 33(1993), 259-279.
Hajek, D., Uniform polynomial approximation, Amer. Math. Monthly, 72(1965), 681. [8] Lindvall, T., Bernstein polynomials and the law of large numbers, Math. Sci., 7(1982), 127-139.
P˘altanea, R., Improved constant in approximation with Bernstein operators, Research Seminars Fac. Math. Babe¸s-Bolyai Univ., Cluj-Napoca, 6(1988), 261-268.
P˘altanea, R., Approximation Theory Using Positive Linear Operators, Birkha¨user, 2004.
Stancu, D.D., A note on a multiparameter Bernstein-type approximating operator, Mathematica (Cluj), 26(49)(1984), no. 2, 153-157.
Talpau Dimitriu, M., On global smoothness preservation by Bernstein-type operators, Stud. Univ. Babe¸s-Bolyai Math., 60(2015), no. 2, 303-310.
Talpau Dimitriu, M., Global smoothness preservation for the Stancu operators on simplex, An. Univ. Oradea Fasc. Mat., 23(2016), no. 2, 49-56.
Zhou, D.-X., On a problem of Gonska, Results Math., 28(1995), 169-183.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
