Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L 2 ( m ) ( 0 , 1 ) $L_{2}^{(m)}(0,1)$

Boltaev, None Nurali D.; Hayotov, Abdullo R.; Shadimetov, Kh. M.

doi:10.1007/s11075-016-0150-7

Cited by 40 publications

(15 citation statements)

References 20 publications

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“…It should be noted that the problem of construction of lattice optimal cubature formulas in the space L (m) 2 of multi-variable functions was first stated and investigated by Sobolev [43,45]. Further, in this section, based on the results of the work [8], we solve Sard's problem on construction of optimal quadrature formulas of the form (6) for ω ∈ R with ω 0, first for the interval [0, 1] and then using a linear transformation for the interval [a, b]. For this we use the following auxiliary results.…”

Section: Optimal Quadrature Formulas Formentioning

confidence: 88%

“…Here we obtain optimal quadrature formulas of the form (6) for the interval [0, 1] when ω ∈ R and ω 0. In the space L (m) 2 [0, 1], using the results of Sections 2, 3 and 5 of [8], for the coefficients of the optimal quadrature formulas in the sense of Sard of the form…”

Section: Construction Of Optimal Quadrature Formulas For the Interval [0 1]mentioning

confidence: 99%

“…Since then, for integrals of different types of highly oscillating functions many special effective methods have been developed, such as Filon-type method, Clenshaw-Curtis-Filon type method, Levin type methods, modified Clenshaw-Curtis method, generalized quadrature rule, and Gauss-Laguerre quadrature (see, for example, [1, 3-5, 21, 26, 29, 46, 47], for more review see, for instance, [14,27,30] and references therein). Recently, in [6][7][8], based on Sobolev's method, the problem of construction of optimal quadrature formulas in the sense of Sard for numerical calculation of integrals ( 4 . Here, we consider the Sobolev space L (m) 2 [a, b] of non-periodic, complex-valued functions defined on the interval [a, b], which possess an absolute continuous (m − 1)-th derivative on [a, b], and whose m-th order derivative is square integrable [43,45].…”

Section: Introductionmentioning

confidence: 99%

“…. The aim of the present work is to construct optimal quadrature formulas in the sense of Sard in the Sobolev space L (m) 2 for numerical integration of the integral (4) with real ω using the results of the work [8]. Then to approximate reconstruction of CT image, we apply the obtained optimal quadrature formulas to the Fourier transform and its inversion in (1).…”

Section: Introductionmentioning

confidence: 99%

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

Hayotov

Jeon

Lee

et al. 2021

Filomat

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In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral ?ba e2?i?x?(x)dx with ? ? R in the Sobolev space L(m)2 [a,b] of complexvalued functions which are square integrable with m-th order derivative. Here, using the discrete analogue of the differential operator d2m/dx2m, the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is O(hm). As an application, we implement the filtered back-projection (FBP) algorithm, which is a well-known image reconstruction algorithm for computed tomography (CT). By approximating Fourier transforms and its inversion using the proposed optimal quadrature formula of the second and third orders , we observe that the accuracy of the reconstruction algorithm is improved. In numerical experiments, we compare the quality of the reconstructed image obtained by using the proposed optimal quadrature formulas with the conventional FBP, in which fast Fourier transform is used for the calculation of Fourier transform and its inversion. In the noise test, the proposed algorithm provides more reliable results against the noise than the conventional FBP.

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Section: Optimal Quadrature Formulas Formentioning

confidence: 88%

Section: Construction Of Optimal Quadrature Formulas For the Interval [0 1]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Hayotov

Jeon

Lee

et al. 2021

Filomat

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“…Based on Sobolev's method, the problem of the construction of optimal quadrature formulas for numerical calculation of Fourier coefficients (1.1) with ω ∈ Z in Hilbert spaces L was studied in [6] and [7], respectively. In these works, explicit formulas of optimal coefficients were obtained for m ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

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

Hayotov

Jeon

Lee

2020

Journal of Computational and Applied Mathematics

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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.

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Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method

Boltaev

Hayotov

Shadimetov

2021

Acta. Math. Sin.-English Ser.

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Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L 2 ( m ) ( 0 , 1 ) $L_{2}^{(m)}(0,1)$

Cited by 40 publications

References 20 publications

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

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

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

Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method

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