EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 95[109], pp. 29–47 (2014)

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OPTIMAL QUADRATURE FORMULA IN THE SENSE OF SARD IN $\mathbf{K_2(P_3)}$ SPACE

Abdullo R. Hayotov, Gradimir V. Milovanovic, Kholmat M. Shadimetov

Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan Mathematical Institute of Serbian Academy of Sciences and Arts, Beograd, Serbia

Abstract: We construct an optimal quadrature formula in the sense of Sard in the Hilbert space $K_2(P_3)$. Using Sobolev's method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space $L_2^{(3)}(0,1)$. The obtained optimal quadrature formula is exact for the trigonometric functions $\sin x$, $\cos x$ and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

Keywords: optimal quadrature formulas, error functional, extremal function, Hilbert space, optimal coefficients

Classification (MSC2000): 65D32

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Electronic fulltext finalized on: 31 Mar 2014. This page was last modified: 2 Apr 2014.

© 2014 Mathematical Institute of the Serbian Academy of Science and Arts
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