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Ukrainian Mathematical Journal

, Volume 63, Issue 6, pp 843–864 | Cite as

Grüss-type and Ostrowski-type inequalities in approximation theory

  • A. M. Acu
  • H. Gonska
  • I. Raşa
Article

We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér operator.

Keywords

Approximation Theory Chebyshev Polynomial Type Inequality Interpolation Operator Positive Linear Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • A. M. Acu
    • 1
  • H. Gonska
    • 2
  • I. Raşa
    • 3
  1. 1.Lucian Blaga University of SibiuSibiuRomania
  2. 2.University of Duisburg-EssenDuisburgGermany
  3. 3.Technical UniversityCluj-NapocaRomania

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