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This paper deals with the construction of an optimal quadrature formula forthe approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ ofnon-periodic, complex valued functions which are square integrable with firstorder derivative. Here the quadrature sum consists of linear combination of thegiven function values in a uniform grid. The difference between the integraland the quadrature sum is estimated by the norm of the error functional. Theoptimal quadrature formula is obtained by minimizing the norm of the errorfunctional with respect to coefficients. Analytic formulas for optimalcoefficients can also be obtained using discrete analogue of the differentialoperator $d^2/d x^2$. In addition, the convergence order of the optimalquadrature formula is studied. It is proved that the obtained formula is exactfor all linear polynomials. Thus, it is shown that the convergence order of theoptimal quadrature formula for functions of the space $C^2[a,b]$ is $O(h^2)$.Moreover, several numerical results are presented and the obtained optimalquadrature formula is applied to reconstruct the X-ray Computed Tomographyimage by approximating Fourier transforms.

On an optimal quadrature formula for approximation of Fourier integrals in the space $L_2^{(1)}$