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Volume 7, Number 4, 2017, Pages 1233-1266                                                                DOI:10.11948/2017076
Optimal quadrature formulas for Fourier coefficients in $W_2^{(m,m-1)}$ space
Nurali Boltaev,Abdullo Hayotov,Gradimir Milovanovic,Kholmat Shadimetov
Keywords:Fourier coefficients, optimal quadrature formulas, the error functional, extremal function, Hilbert space.
      This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}[0,1]$ space for calculating Fourier coefficients. Using S.~L.\ Sobolev''s method we obtain new optimal quadrature formulas of such type for $N 1\geq m$, 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 $W_2^{(m,m-1)}[0,1]$ 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.
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