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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:372

Article

Hayotov, Abdullo R.^1,2  Jeon, Soomin¹  Lee, Chang-Ock¹ 

[1] Korea Adv Inst Sci & Technol, Dept Math Sci, Daehak Ro 291, Daejeon 34141, South Korea

[2] Uzbek Acad Sci, VI Romanovskiy Inst Math, 81 M Ulugbek Str, Tashkent 100170, Uzbekistan

关键词: Optimal quadrature formula;    Square integrable function;    Error functional;    Fourier transform;    X-ray computed tomography image;   

DOI  :  10.1016/j.cam.2020.112713

来源: Elsevier

【 摘 要 】

This paper deals with the construction of an optimal quadrature formula for 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)/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 resudlts are presented and the obtained optimal quadrature formula is applied to reconstruct the X-ray Computed Tomography image by approximating Fourier transforms. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free