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Optimal quadrature formulas for Fourier coefficients in W_2^{(m,m-1)} space
Keywords:Fourier coefficients, Optimal quadrature formulas, The error functional, Extremal function.
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.