Science and Education Publishing American Journal of Numerical Analysis 2372-2126 4 1 2016 1 27 Construction of Optimal Quadrature Formula for Numerical Calculation of Fourier Coefficients in Sobolev Space L2(1) 1 7 EN Nurali D. Boltaev Abdullo R. Hayotov Department of Comptational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan Kholmat M. Shadimetov AJNA2016411 10.12691/ajna-4-1-1 2015 9 8 2015 12 30 2016 1 25 In the present paper the optimal quadrature formula for approximate evaluation of Fourier coefficients is constructed for functions of the space . At the same time the explicit formulas for optimal coefficients, which are very useful in applications, are obtained. The obtained formula is exact for constant. In particular, as consequences of the main result the new optimal quadrature formulas for approximate evaluation of integrals and are obtained. Furthermore, the order of convergence of the constructed optimal quadrature formula is studied.