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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

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An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials

Ulrich AbelORCID iD: http://orcid.org/0000-0003-1889-4850 / Hartmut Siebert
Published Online: 2018-02-07 | DOI: https://doi.org/10.1515/gmj-2017-0065


In this paper, we deal with improvements on the constant Mn(γ) in the so-called Videnskiĭ inequality


for a fixed constant γ1 and x[0,1], where Bnf is the Bernstein polynomial, fC2[0,1] and ω is the first order modulus of continuity. Let M(γ)=supnMn(γ). We prove the Videnskiĭ inequality for arbitrary γ1. In particular, we improve the constant M(2)=0.9 (Gonska and Ra şa [8], 2008) to M(2)=0.6875. Finally, we consider Mn(1) for small values of n.

Keywords: Approximation by positive operators; rate of convergence; degree of approximation

MSC 2010: 41A36; 41A25


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About the article

Received: 2016-08-18

Revised: 2016-12-19

Accepted: 2016-12-29

Published Online: 2018-02-07

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2017-0065.

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