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

TL;DR

This paper develops an optimal quadrature formula for approximating Fourier integrals in Sobolev space, providing explicit coefficients, analyzing convergence, and demonstrating applications in image reconstruction.

Contribution

It introduces a new optimal quadrature formula for Fourier integrals in Sobolev space with explicit coefficients and convergence analysis.

Findings

The formula is exact for all linear polynomials.

Convergence order is O(h^2) for functions in C^2[a,b].

Numerical results confirm the effectiveness of the formula in applications.

Abstract

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.