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This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W (m,m−1) 2 [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 W (m,m−1) 2 [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.

journal of applied analysis and computation

OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN &lt;i&gt;W&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;(&lt;i&gt;m&lt;/i&gt;,&lt;i&gt;m&lt;/i&gt;-1)&lt;/sup&gt; 2 SPACE