eng Science and Education Publishing American Journal of Numerical Analysis 2372-2126 2016-01-27 4 1 1 7 10.12691/ajna-4-1-1 AJNA2016411 article Nurali D. Boltaev 1 Abdullo R. Hayotov hayotov@mail.ru 1 Kholmat M. Shadimetov 1 Department of Comptational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan 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. http://pubs.sciepub.com/ajna/4/1/1/ajna-4-1-1.pdf optimal quadrature formula extremal function error functional hilbert space