Grüss-type and Ostrowski-type inequalities in approximation theory

Springer Science and Business Media LLC - Tập 63 Số 6 - Trang 843-864 - 2011
Ana Maria Acu1, Heiner Gonska2, Ioan Raşa3
1Lucian Blaga University of Sibiu, Sibiu, Romania
2University of Duisburg-Essen, Duisburg, Germany
3Technical University, Cluj-Napoca, Romania

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Tài liệu tham khảo

A. M. Acu and H. Gonska, “Ostrowski-type inequalities and moduli of smoothness,” Results Math., 53, 217–228 (2009).

D. Andrica and C. Badea, “Grüss inequality for positive linear functionals,” Period. Math. Hung., 19, 155–167 (1988).

G. A. Anastassiou, “Ostrowski type inequalities,” Proc. Amer. Math. Soc., 123, 3775–3781 (1995).

P. L. Chebyshev, “Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites,” Proc. Math. Soc. Kharkov, 2, 93–98 (1882); French translation: Oeuvres, 2, 716–719 (1907).

X. L. Cheng, “Improvement of some Ostrowski–Grüss type inequalities,” Comput. Math. Appl., 42, 109–114 (2001).

S. S. Dragomir, “On the Ostrowski integral inequality for Lipschitzian mappings and applications,” Comput. Math. Appl., 38, 33–37 (1999).

S. S. Dragomir and S. Wang, “An inequality of Ostrowski–Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules,” Comput. Math. Appl., 33, No. 11, 15–20 (1997).

B. Gavrea and I. Gavrea, “Ostrowski type inequalities from a linear functional point of view,” J. Inequal. Pure Appl. Math., 1, Article 11 (2000).

H. Gonska, “Quantitative Approximation in C(X),” Habilitationsschrift, University of Duisburg (1986).

S. J. Goodenough and T. M. Mills, “A new estimate for the approximation of functions by Hermite–Fejér interpolation polynomials,” J. Approxim. Theory, 31, 253–260 (1981).

G. Grüss, “Über das Maximum des absoluten Betrages von $ \frac{1}{{b - a}}\int_a^b {f(x)g(x)dx - \frac{1}{{{{\left( {b - a} \right)}^2}}}} \int_a^b {f(x)dx} \int_a^b {g(x)dx} $ ,” Math. Z., 39, 215–226 (1935).

J. Karamata, “Inégalités relatives aux quotients et à la différence de $ \int {fg\;{\text{et}}\;} \int f \int g $ ,” Bull. Acad. Serbe. Sci. Math. Natur. A, 131–145 (1948).

O. Kiš, “Remarks on the rapidity of convergence of Lagrange interpolation,” Ann. Univ. Sci. Budapest. Sec. Math., 11, 27–40 (1968).

E. Landau, “Über einige Ungleichungen von Herrn G. Grüss,” Math. Z., 39, 742–744 (1935).

H. G. Lehnhoff, “A simple proof of A. F. Timan’s theorem,” J. Approxim. Theory, 38, 172–176 (1983).

Y. Matsuoka, “On the degree of approximation of functions by some positive linear operators,” Sci. Rep. Kagoshima Univ., 9, 11–16 (1960).

A. Mc. D. Mercer and P. R. Mercer, “New proofs of the Grüss inequality,” Austral. J. Math. Anal. Appl., 1, Issue 2, 1–6 (2004).

R. N. Misra, “On the rate of convergence of Hermite–Fejér interpolation polynomials,” Period. Math. Hung., 13, 15–20 (1982).

D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer, Dordrecht (1993).

A. Ostrowski, “Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert,” Comment. Math. Helv., 10, 226–227 (1938).

A. Ostrowski, “On an integral inequality,” Aequat. Math., 4, 358–373 (1970).

B. G. Pachpatte, “A note on Ostrowski like inequalities,” J. Inequal. Pure Appl. Math., 6, Article 114 (2005).

B. G. Pachpatte, “A note on Grüss type inequalities via Cauchy’s mean-value theorem,” Math. Inequal. Appl., 11, No. 1, 75–80 (2007).

E. M. Semenov and B. S. Mitjagin, “Lack of interpolation of linear operators in spaces of smooth functions,” Mat. USSR. Izv., 11, 1229–1266 (1977).