OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN <i>W</i><sub>2</sub><sup>(<i>m</i>,<i>m</i>-1)</sup> 2 SPACE

Abstract

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the <i>W</i>₂^{(<i>m</i>,<i>m</i>-1)}[0,1] space for calculating Fourier coefficients. Using S. L. Sobolev's method we obtain new optimal quadrature formulas of such type for <i>N</i> + 1 ≥ <i>m</i>, where <i>N</i> + 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 <i>m</i>=1. The obtained optimal quadrature formula in the <i>W</i>₂^{(<i>m</i>,<i>m</i>-1)}[0,1] space is exact for exp(-<i>x</i>) and <i>P</i>_<i>m</i>-2(<i>x</i>), where <i>P</i>_<i>m</i>-2(<i>x</i>) is a polynomial of degree <i>m</i> -2. Furthermore, we present some numerical results, which confirm the obtained theoretical results