  This paper deals with the construction of an optimal quadrature formula for
the approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ of
non-periodic, complex valued functions which are square integrable with first
order derivative. Here the quadrature sum consists of linear combination of the
given function values in a uniform grid. The difference between the integral
and the 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. Analytic formulas for optimal
coefficients can also be obtained using discrete analogue of the differential
operator $d^2/d x^2$. In addition, the convergence order of the optimal
quadrature formula is studied. It is proved that the obtained formula is exact
for all linear polynomials. Thus, it is shown that the convergence order of the
optimal quadrature formula for functions of the space $C^2[a,b]$ is $O(h^2)$.
Moreover, several numerical results are presented and the obtained optimal
quadrature formula is applied to reconstruct the X-ray Computed Tomography
image by approximating Fourier transforms.
