OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W2(m,m-1) 2 SPACE

Nurali D. Boltaev; Abdullo R. Hayotov; Gradimir V. Milovanović; Kholmat M. Shadimetov

doi:10.11948/2017076

2017 Volume 7 Issue 4

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Nurali D. Boltaev, Abdullo R. Hayotov, Gradimir V. Milovanović, Kholmat M. Shadimetov. OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W2(m,m-1) 2 SPACE[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1233-1266. doi: 10.11948/2017076

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Nurali D. Boltaev, Abdullo R. Hayotov, Gradimir V. Milovanović, Kholmat M. Shadimetov. OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W₂^(m,m-1) 2 SPACE[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1233-1266. doi: 10.11948/2017076

OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W₂^(m,m-1) 2 SPACE

1 Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan;
2 Serbian Academy of Sciences and Arts, Beograd & University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia

Abstract

Abstract

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W₂^(m,m-1)[0,1] space for calculating Fourier coefficients. Using 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 the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for m=1. The obtained optimal quadrature formula in the W₂^(m,m-1)[0,1] space is exact for exp(-x) and P_m-2(x), where P_m-2(x) is a polynomial of degree m -2. Furthermore, we present some numerical results, which confirm the obtained theoretical results
- Fourier coefficients /
- optimal quadrature formulas /
- the error functional /
- extremal function /
- Hilbert space
MSC: 65D32

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References

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