The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. In addition, analytic formulas for optimal coefficients are obtained using the discrete analogue of the differential operator $d^2/d x^2-1$. Further, the order of convergence of the optimal quadrature formula is studied.

翻译：本文件专门论述为Hilbert空间Fourier整体体近似值$W_2 ⁇ (1,0)}[a,b]美元的非定期、复杂价值值函数[a,b]美元]构建一个最佳二次公式。这里的二次公式由统一网格上给定函数值的线性组合组成。集成和四方公式之间的差额由误差功能的规范估算。最佳二次公式是通过尽量减少系数方面差错功能的规范而获得的。此外,还利用差分操作员的离散模拟公式 $d ⁇ 2/d x ⁇ 2-1$获得最佳系数的分析公式。此外,还研究了最佳二次公式的趋同顺序。