Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L2(m)(0,1)$L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the L2(m)(0,1)$L_{2}^{(m)}(0,1)$ 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+1źm, 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 L2(m)(0,1)$L_{2}^{(m)}(0,1)$ space are exact for Pmź1(x), where Pmź1(x) is a polynomial of degree mź1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.