Optimal quadrature formulas for Fourier coefficients in $W_2^{(m,m-1)}$ space |
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Authors: | Nurali Boltaev Abdullo Hayotov Gradimir Milovanovic and Kholmat Shadimetov |
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Institution: | Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan,Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan,Serbian Academy of Sciences and Arts, Beograd \& University of Ni\v s, Faculty of Sciences and Mathematics, Ni\v s, Serbia and Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan |
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Abstract: | 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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Keywords: | Fourier coefficients optimal quadrature formulas the error functional extremal function Hilbert space |
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