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The strong converse inequality for Bernstein-Kantorovich operators. (English) Zbl 0842.41019

The aim of this paper is the following Theorem. There exists an absolute positive constant \(C\) such that for all \(f\in L_p [0,1 ]\), \(1\leq p\leq \infty\), there holds \[ C^{-1} K(f, n^{-{1\over 2}})_p\leq |f- K_n f|_p\leq CK (f, n^{-{1\over 2}})_p, \] where \[ K_n (f; x):= (n+1) \sum^n_{k=0} p_{n,k} (x) \int^{(k+1)/ (n+1)}_{k/ (r+1)} f(t) dt, \quad n\in \mathbb{N}, \] are Kantorovich operators, \(K(f,t)_p:= \inf \{|f-g |_p+ t^2|P(D) g|_p: g\in C^2 [0,1 ]\}\), \(P(D) f:= (h^2 f')'\), \(h^2 (x):= x(1- x)\), for every \(f\in C^2 [0,1 ]\).
Reviewer: I.Badea (Craiova)

MSC:

41A35 Approximation by operators (in particular, by integral operators)
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[1] Ditzian, Z.; Ivanov, K. G., Strong converse inequalities, J. d’Analyse Mat., 61, 61-111 (1993) · Zbl 0798.41009
[2] Knoop, H.-B.; Zhou, X.-l., The Lower Estimate for Linear Positive Operators (1992), Schriftenreihe des Fachbereichs Mathematik: Schriftenreihe des Fachbereichs Mathematik Duisburg, SM-DU-201
[3] Totik, V., Approximation by Bernstein polynomials (1992), Manuscript
[4] Ditzian, Z.; Totik, V., Moduli of Smoothness (1987), Springer: Springer New York · Zbl 0666.41001
[5] Totik, V., Approximation by algebraic polynomials, (Cheney, E. W.; Chui, C. K.; Schumaker, L. L., Approximation Theory VII (1992), Academic Press: Academic Press New York), 227-249 · Zbl 0767.41014
[6] Ditzian, Z.; Zhou, X.-l., Kantorovich-Bernstein polynomials, Constr. Approx., 6, 421-435 (1990) · Zbl 0743.41022
[7] Berens, H.; Xu, Y., On Bernstein-Durrmeyer polynomials with Jacobi weights, (Chui, C. K., Approximation Theory and Functional Analysis (1990), Academic Press: Academic Press New York), 25-46 · Zbl 0715.41013
[8] Derriennic, M. M., On multivariate approximation by Bernstein-type polynomials, J. Approx. Theory, 45, 155-166 (1985) · Zbl 0578.41010
[9] Gonska, H. H.; Zhou, X.-l., A global inverse theorem on simultaneous approximation by Bernstein-Durrmeyer operators, J. Approx. Theory, 67, 284-302 (1991) · Zbl 0756.41027
[10] Lorentz, G. G., Bernstein Polynomials, (Mathematical Expositions, No. 8 (1953), University of Toronto Press: University of Toronto Press Toronto) · Zbl 0989.41504
[11] Bergh, J.; Löfström, J., Interpolation Spaces (1976), Springer: Springer New York · Zbl 0344.46071
[12] Bojanic, R.; Shisha, O., Degree of \(L^1\) approximation to integrable functions by modified Bernstein polynomials, J. Approx. Theory, 13, 66-72 (1975) · Zbl 0305.41009
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