On an optimal quadrature formula for approximation of Fourier integrals in the space L2(1)

Abstract

This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space L (1) 2 [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 /dx 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.

Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction

“…2 [a, b] the coefficients of the optimal quadrature formulas (33) have the same forms as derived in [20], where it was shown that for functions with a continuous second derivative the convergence order of this optimal quadrature formula is O(h 2 ).…”

“…In this section, we deal with the second and third order optimal quadrature formulas only. For the numerical results with the first order optimal quadrature formula, see [20].…”

“…Then to approximate reconstruction of CT image, we apply the obtained optimal quadrature formulas to the Fourier transform and its inversion in (1). Recently we got the results when m = 1 [20]. There is a work by Reider and Faridani [31] for an optimal reconstruction algorithm, where the optimal L 2 −convergence rates of filtered back-projection (FBP) algorithm was provided with assumption that the Radon transform was analytically computable.…”