Jump to ContentJump to Main Navigation
Show Summary Details
More options …

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

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
See all formats and pricing
More options …
Ahead of print

Issues

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

Abstract

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

|(Bnf)(x)-f(x)-x(1-x)2nf′′(x)|Mn(γ)x(1-x)nω(f′′;γn)

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

References

  • [1]

    S. Beier, Die Konstante in der Videnskij-Ungleichung für Bernstein-Polynome, Diploma thesis, Fachhochschule Gießen–Friedberg, Friedberg, 2011. Google Scholar

  • [2]

    S. Bernstein, Complément à l’article de E. Voronovskaya “Détermination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein”, C. R. Dokl. Acad. Sci. URSS A 1932 (1932), 86–92. Google Scholar

  • [3]

    R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, Berlin, 1993. Google Scholar

  • [4]

    Z. Finta, On generalized Voronovskaja theorem for Bernstein polynomials, Carpathian J. Math. 28 (2012), no. 2, 231–238. Google Scholar

  • [5]

    M. S. Floater, On the convergence of derivatives of Bernstein approximation, J. Approx. Theory 134 (2005), no. 1, 130–135. CrossrefGoogle Scholar

  • [6]

    H. Gonska, On the degree of approximation in Voronovskaja’s theorem, Stud. Univ. Babeş-Bolyai Math. 52 (2007), no. 3, 103–115. Google Scholar

  • [7]

    H. Gonska, M. Heilmann and I. Raşa, Asymptotic behaviour of differentiated Bernstein polynomials revisited, Gen. Math. 18 (2010), no. 1, 45–53. Google Scholar

  • [8]

    H. Gonska and I. Raşa, Remarks on Voronovskaya’s theorem, Gen. Math. 16 (2008), no. 4, 87–97. Google Scholar

  • [9]

    H. Gonska and I. Raşa, Asymptotic behaviour of differentiated Bernstein polynomials, Mat. Vesnik 61 (2009), no. 1, 53–60. Google Scholar

  • [10]

    G. G. Lorentz, Bernstein Polynomials, Math. Exp. 8, University of Toronto Press, Toronto, 1953. Google Scholar

  • [11]

    T. Popoviciu, Sur l’approximation des fonctions convexes d’ordre supérieur, Mathematica Cluj 10 (1935), 49–54. Google Scholar

  • [12]

    P. C. Sikkema, Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, Numer. Math. 3 (1961), 107–116. CrossrefGoogle Scholar

  • [13]

    P. C. Sikkema and P. J. C. van der Meer, The exact degree of local approximation by linear positive operators involving the modulus of continuity of the pth derivative, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 1, 63–76. Google Scholar

  • [14]

    D. D. Stancu, The remainder of certain linear approximation formulas in two variables, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 1 (1964), 137–163. CrossrefGoogle Scholar

  • [15]

    G. T. Tachev, Voronovskaja’s theorem revisited, J. Math. Anal. Appl. 343 (2008), no. 1, 399–404. CrossrefWeb of ScienceGoogle Scholar

  • [16]

    V. S. Videnskiĭ, Linear Positive Operators of Finite Rank (in Russian), “A. I. Gerzen” State Pedagogical Institute, Leningrad, 1985. Google Scholar

  • [17]

    E. Voronovskaja, Détermination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein, C. R. Dokl. Acad. Sci. URSS A 1932 (1932), 79–85. Google Scholar

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.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[2]
José A. Adell and Daniel Cárdenas-Morales
Journal of Approximation Theory, 2018

Comments (0)

Please log in or register to comment.
Log in