Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space
In this work, we construct an optimal quadrature formula in the sense of Sard based on a functional approach for numerical calculation of integrals of rapidly oscillating functions. To solve this problem, we will use Sobolev’s method.To do this, we first solve the boundary value problem for an extre...
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Elsevier
2025-05-01
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| Series: | Results in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2590037425000330 |
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| author | Abdullo Hayotov Samandar Babaev Abdimumin Kurbonnazarov |
| author_facet | Abdullo Hayotov Samandar Babaev Abdimumin Kurbonnazarov |
| author_sort | Abdullo Hayotov |
| collection | DOAJ |
| description | In this work, we construct an optimal quadrature formula in the sense of Sard based on a functional approach for numerical calculation of integrals of rapidly oscillating functions. To solve this problem, we will use Sobolev’s method.To do this, we first solve the boundary value problem for an extremal function. To solve the boundary value problem, we use direct and inverse Fourier transforms and find the fundamental solution of the given differential operator. Using the extremal function, we find the norm of the error functional. For the given nodes, we find the minimum value of the error functional norm along the coefficients.This quadrature formula is exact for the hyperbolic functions sinh(x),cosh(x) and a constant term. In this work, we consider the case ωh∉Z and ω∈R in the Hilbert space K2(3)(0,1).We apply the constructed quadrature formula for reconstruction of a Computed Tomography image. |
| format | Article |
| id | doaj-art-6d81717df0b54c39a280c42c94ccd7c4 |
| institution | DOAJ |
| issn | 2590-0374 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Applied Mathematics |
| spelling | doaj-art-6d81717df0b54c39a280c42c94ccd7c42025-08-20T03:04:47ZengElsevierResults in Applied Mathematics2590-03742025-05-012610056910.1016/j.rinam.2025.100569Optimization of approximate integrals of rapidly oscillating functions in the Hilbert spaceAbdullo Hayotov0Samandar Babaev1Abdimumin Kurbonnazarov2V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University str., Tashkent, 100174, Uzbekistan; Central Asian University, 264, Milliy bog str., Barkamol MFY, M.Ulugbek district, Tashkent, 111221, Uzbekistan; Tashkent International University, 7, Kichik khalka yoli str., Tashkent, 100084, Uzbekistan; Bukhara State University, 11, M.Ikbol str., Bukhara, 200114, UzbekistanV.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University str., Tashkent, 100174, Uzbekistan; Tashkent International University, 7, Kichik khalka yoli str., Tashkent, 100084, Uzbekistan; Bukhara State University, 11, M.Ikbol str., Bukhara, 200114, Uzbekistan; Corresponding author at: V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University str., Tashkent, 100174, Uzbekistan.V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University str., Tashkent, 100174, Uzbekistan; Termiz State University, 43, Barkamol Avlod str., Termiz, 190111, Uzbekistan; Tashkent State Technical University, 2A, University str., Tashkent, 100095, UzbekistanIn this work, we construct an optimal quadrature formula in the sense of Sard based on a functional approach for numerical calculation of integrals of rapidly oscillating functions. To solve this problem, we will use Sobolev’s method.To do this, we first solve the boundary value problem for an extremal function. To solve the boundary value problem, we use direct and inverse Fourier transforms and find the fundamental solution of the given differential operator. Using the extremal function, we find the norm of the error functional. For the given nodes, we find the minimum value of the error functional norm along the coefficients.This quadrature formula is exact for the hyperbolic functions sinh(x),cosh(x) and a constant term. In this work, we consider the case ωh∉Z and ω∈R in the Hilbert space K2(3)(0,1).We apply the constructed quadrature formula for reconstruction of a Computed Tomography image.http://www.sciencedirect.com/science/article/pii/S2590037425000330Quadrature formulaFourier coefficientsError functionalExtremal functionRapidly oscillation integral |
| spellingShingle | Abdullo Hayotov Samandar Babaev Abdimumin Kurbonnazarov Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space Results in Applied Mathematics Quadrature formula Fourier coefficients Error functional Extremal function Rapidly oscillation integral |
| title | Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space |
| title_full | Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space |
| title_fullStr | Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space |
| title_full_unstemmed | Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space |
| title_short | Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space |
| title_sort | optimization of approximate integrals of rapidly oscillating functions in the hilbert space |
| topic | Quadrature formula Fourier coefficients Error functional Extremal function Rapidly oscillation integral |
| url | http://www.sciencedirect.com/science/article/pii/S2590037425000330 |
| work_keys_str_mv | AT abdullohayotov optimizationofapproximateintegralsofrapidlyoscillatingfunctionsinthehilbertspace AT samandarbabaev optimizationofapproximateintegralsofrapidlyoscillatingfunctionsinthehilbertspace AT abdimuminkurbonnazarov optimizationofapproximateintegralsofrapidlyoscillatingfunctionsinthehilbertspace |