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On a certain sequence of positive linear operators and two optimization inequalities. (English) Zbl 0816.41014

Summary: In this paper the approximation of continuous functions by positive linear operators of Berstein type is investigated. The considered operators are constructed using a system of rational functions with prescribed matrix of real poles. A certain general problem of S. Bernstein concerning a scheme of construction of a sequence of positive linear operators is discussed. The answer on the Bernstein’s hypothesis is given. The optimal limiting relations for the norm of the second central moment of our sequence of operators are established.

MSC:

41A20 Approximation by rational functions
90C30 Nonlinear programming
41A25 Rate of convergence, degree of approximation
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References:

[1] Videnskii, V.S. 1991. On the approximation by positive linear operators constructed with rational functions.The Theory of Approximation by Rational Functions. Proceedings of the Conference on approximation of functions. 1991, Makhach-Kala, Russian. pp.6–8.
[2] Mencher, A.E. On exact estimations for rational function operators of Videnskii. The Constructive Theory of Functions.Thesis of reports of Conference in honour of 70-Years Professor Videnskii,V.S. pp.43–44. St.Petersburg State University, Russian
[3] Videnskii V.S., LGPI pp 68– (1985)
[4] Videnskii, V.S. 1985. ”On same new investigations on approximation by positive linear operators. The Theory of Functions, Functional Analysis and their Applications”. Vol. 43, 17–33. Kharkov, Russian English transl in Journ of Sov Math
[5] Videnskii V.S., LGPI pp 19– (1988)
[6] Videnskii V.S., Dekl.An.Arm.SSR 70 pp 145– (1980)
[7] Achieser, N.I. 1965. ”Lectures on Theory of Approximation”. 407Moscow, Russian
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