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Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space \(L_{2}^{(m)}(0,1)\)

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Abstract

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P m−1(x), where P m−1(x) is a polynomial of degree m−1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.

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Authors and Affiliations

  1. Institute of Mathematics, National University of Uzbekistan, 29, Do‘rmon yo‘li str., Tashkent, 100125, Uzbekistan

    N. D. Boltaev, A. R. Hayotov & Kh. M. Shadimetov

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  1. N. D. Boltaev
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  2. A. R. Hayotov
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  3. Kh. M. Shadimetov
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Correspondence to A. R. Hayotov.

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Boltaev, N.D., Hayotov, A.R. & Shadimetov, K.M. Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space \(L_{2}^{(m)}(0,1)\) . Numer Algor 74, 307–336 (2017). https://doi.org/10.1007/s11075-016-0150-7

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