Abstract
We obtained that for any n ∈ N, C = 1 is the smallest constant for which the inequality ||B n (f) - f|| ≤ C ⋅ ω2(f, 1/√n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B n is the Bernstein operators of degree n, ω2 is the second order modulus and || ⋅ || is the sup-norm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Yu. A. Brudnyi, On a certain method for approximation of bounded functions, given on an segment (Russian), In: “Studies in Contemporary Problems in Constructive Theory of Functions” (Proc. Second All-Union Conf. Baku 1962; ed. by I. I. Ibragimov), Baku: Izdat. Akad. Nauk Azerbaidzan, 40–45 (1965.
Jia-ding Cao, On linear approximation methods (in Chinese), Acta Sci. Natur. Univ. Fudan 9, 43–52 (1964).
C. G. Esseen, Uber die asymptotisch beste Approximation stetiger Functionen mit Hilfe von Bernstein-Polynomen, Numer. Math. 2, 206–213 (1960).
H. Gonska, Quantitative Aussagen zur Approximation durch positive lineare Operatoren, Dissertation, Universitaat Duisburg, (1979).
H. Gonska, Two problemson best constants in direct estimates, Problem Section of Proc. Edmonton Conf. Approximation Theory (Z. Ditzian et al. eds.), 194 Providence, RI. Amer. Math. Soc. (1983).
H. Gonska and R.K. Kovacheva, The second order modulus revisited: remarks, applications, problems, Conf. Sem. Mat. Univ. Bari 257, 1–32 (1994).
H. Gonska and J. Meier, On approximation by Bernstein type operators: best constants, Studia Sci. Matyh. Hungar. 22, 287–297 (1987).
H. Gonska and D-x. Zhou, On an extremal problem concerning Bernstein operators, Serdica Math. J. 137–150 (1995).
H. Gonska and X-l. Zhou, Polynomial approximation with side conditions: recent results and open problems, In, Proc. of the First International Colloquim on Numerical Analysys (Plovdiv (1992), (D. Bainov and V. Covachev eds.), 61–71, Zeist / The Nederlands: VPS International Science Publishers, 1993.
D. K. Kacso, Approximation by means of piecewise linear functions, Result. Math. 35, 89–102 (1999).
G. G. Lorentz: Bernstein Polynomials, Toronto: University of Toronto Press, (1953).
R. Paltanea, Improved constantin approximation with Bernstein operators, In, Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca 1988), Preprint 88–6, Univ. Babes-Bolyai, Cluj-Napoca (1988), 261–268.
R. Paltanea, General estimatesfor linear positive operators that preserve linear functions, Anal. Numér. Théor. Approx. 18(2), 147–159 (1989).
R. Paltanea, On theestimate of the pointwise approximation of functions by linear positive functionals, Studia Univ. Babeş-Bolyai, 53(1), 11–24 (1990).
R. Paltanea, Best constantin estimates with second order moduli of continuity, In, Approximation Theory, (Proc. Int. Dortmund Meeting on Approximation Theory 1995, M. W. Muller et al. eds.), Akad Verlag, Berlin 1995, pp. 251–275.
R. Paltanea, New secondorder moduli of continuity, in, Approximation and optimization (Proc. Int. Conf. Approximation and Optimization, Cluj-Napoca 1996, D. D. Stancu et al. eds.), vol I, Transilvania Press, Cluj-Napoca 1997, pp. 327–334.
R. Paltanea, Optimal estimateswith moduli of continuity, Result. Math. 32, 318–331 (1997).
R. Paltanea, On an optimal constant in approximation by Bernstein operators, Rend. Circ. Mat. Palermo (2), suppl. 52, 663–686 (1998).
R. Paltanea, An improvedestimate with the second order modulus of continuity, in Proc. of the “Tiberiu Popoviciu” Itinerant Seminar of Functional Equations, Approximation and Convexity, (E. Popoviciu ed.), Srima Press, Cluj-Napoca 2000, pp. 167–171.
T. Popoviciu, Sur l'approximationdes fonctions convexes d'ordre superieur, Mathematica (Cluj) 10, 49–54 (1935).
P. C. Sikkema, Uber den Grad der Approximation mit Bernstein-Polynomen, Numerische Math. 1, 221–239 (1959).
P. C Sikkema, Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, Numerische Math. 3, 107–116 (1961).
F. Schurer and F. W. Steutel, On the degree of approximation of functions in C 1[0; 1] by Bernstein polynomials, Th-Report 75-WSK-07 (Onderafdeling der Wiskunde, Technische Hogeschool Eindhoven) 1975.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Păltănea, R. Optimal Constant in Approximation by Bernstein Operators. Journal of Computational Analysis and Applications 5, 195–235 (2003). https://doi.org/10.1023/A:1022898728718
Issue Date:
DOI: https://doi.org/10.1023/A:1022898728718