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

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