Optimal quadrature formulas for Fourier coefficients in W_2^{(m,m-1)} space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W2(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 ≥ 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 W2(m,m-1)[0,1] space is exact for exp(-x) and Pm-2(x), where Pm-2(x) is a polynomial of degree m -2. Furthermore, we present some numerical results, which confirm the obtained theoretical results