The Weighted Optimal Quadrature Formula with Derivatives in the Hilbert Space
Keywords:
quadrature formula with derivatives, extremal function, error functional, optimal coefficient, Lagrange functionAbstract
In the present article, the problem of constructing the optimal quadrature formula in the sense of Sard is discussed for numerical integration of the weighted integrals in the Hilbert space of real-valued functions. Initially, the norm of the error functional is found using the extremal function of the error functional of the quadrature formula. Since the error functional is defined on the Hilbert space, the quadrature formula that we are constructing is exact for zeros of this space, that is, we have the conditions that the influence of the error functional on these functions is equal to zero. Then, the Lagrange function is constructed to find the conditional extremum of the error functional norm. Thereby, a system of linear equations is obtained for the coefficients of the optimal quadrature formula.
References
Akhmedov D.M., Kh.M. Shadimetov. 2022. Optimal quadrature formulas with derivative for Hadamard-type singular integrals. AIP Conference Proceedings, – Vol. 2365, – 020020 p.
Akhmedov D.M., Kh.M. Shadimetov. 2022. Optimal quadrature formulas for approximate solution of the first kind singular integral equation with Cauchy kernel. Studia Universitatis Babeş-Bolyai Mathematica, – Vol. 67, – No. 3, – P. 633–651.
Babaev S.S., A.R. Hayotov, and U.N. Khayriev 2020. On an optimal quadrature formula for approximation of Fourier integrals in the space ????(1,0) 2 . Uzbek Mathematical Journal arXiv:2102.07516, – No.2: – P. 23–36. doi: http://dx.doi.org/10.29229/uzmj.2020-2-3.
Babaev S.S., A.R. Hayotov 2019. Optimal interpolation formulas in the space ????(????,????−1) 2 . Calcolo, Springer International Publishing, 56(3): – P. 1–25. doi: http://dx.doi.org/10. 1007/s10092-019-0320-9.
Babaev S.S. 2024. Construction of an optimal quadrature formula for the approximation of fractional integrals. AIP Conference Proceedings, – Vol. 3004, – 060022 p. doi: http: //dx.doi.org/10.1063/5.0199596.
Blaga P., and Gh. Coman 2007. Some problems on optimal quadrature. Studia Universitatis Babeş-Bolyai Mathematica, – Vol. 52, – No. 4, – P. 21–44.
Bojanov B. 2005. Optimal quadrature formulas. Uspekhi Mat. Nauk, – Vol. 60, – No. 6(366), – P. 33–52. (in Russian).
Boytillayev B.A. 2023. Optimal formulas for the approximate solution of the generalized Abel integral equations in the Hilbert space. Problem of Computational and Applied Mathematics, – vol. 3/1(50), – P. 84–91. https://elibrary.ru/hlgvaq.
Hayotov A.R., S.S. Babaev 2021. Optimal quadrature formulas for computing of Fourier integrals in ????(????,????−1)2 space. AIP Conference Proceedings, – Vol. 2365, – 020021 p. doi: http://dx.doi.org/10.1063/5.0057127.
Hayotov A.R., S.S. Babaev. 2023. An optimal quadrature formula for numerical integration of the right Riemann–Liouville fractional integral. Lobachevskii Journal of Mathematics, – Vol. 44, – No. 10, – P. 4282–4293. doi: http://dx.doi.org/10.1134/S1995080223100165.
HayotovA.R., S.S. Babaev 2023. Optimal quadrature formula for numerical integration of fractional integrals in a Hilbert space. Journal of Mathematical Sciences, – Vol. 277, – No. 3, – P. 403–419. doi: http://dx.doi.org/10.1007/s10958-023-06844-w.
Hayotov A.R.m S. Jeon, and C.O. Lee. 2020. On an optimal quadrature formula for approximation of Fourier integrals in the space ????(1)2 . Journal of Computational and Applied Mathematics, – Vol. 388, – 112713 p. doi: http://dx.doi.org/10.1016/j.cam.2020.112713.
Hayotov A.R., S. Jeon, and Kh.M. Shadimetov 2021. Application of optimal quadrature formulas for reconstruction of CT images. Journal of Computational and Applied Mathematics, – Vol. 388, – 113313 p. doi: http://dx.doi.org/10.1016/j.cam.2020.113313.
Hayotov A.R., and R.G. Rasulov 2020. The order of convergence of an optimal quadrature formula with derivative in the space ????(2,1)2 (0, 1). Filomat, – Vol. 34, – No. 11. – P. 3835–3844. doi: http://dx.doi.org/10.2298/FIL2011835H.
Hayotov A.R., U.N. Khayriev, and F. Azatov 2023. Exponentially weighted optimal quadrature formula with derivative in the space ????(2) 2 . AIP Conference Proceedings, – Vol. 2781. – 020050 p. doi: http://dx.doi.org/10.1063/5.0144753.
Hayotov A.R., and U.N. Khayriev. 2022. Construction of an optimal quadrature formula in the Hilbert space of periodic functions. Lobachevskii Journal of Mathematics, – Vol. 11, – No. 43, – P. 3151–3160.
Sard A. 1949. Best approximate integration formulas; best approximation formulas. Amer. J. Math., – Vol. 71, – P. 80–91.
Shadimetov K.M., and D.M. Akhmedov 2022. Approximate solution of a singular integral equation using the Sobolev method. Lobachevskii Journal of Mathematics, – Vol. 43, – No. 2, – P. 496–505. doi: http://dx.doi.org/10.1134/S1995080222050249.
Shadimetov Kh.M. 2018. A method of construction of weight optimal quadrature formulas with derivatives in the Sobolev space. Uzbek Mathematical Journal, – No. 3, – P. 140–146.
Shadimetov Kh.M., A.R. Hayotov 2014. Optimal quadrature formulas in the sense of Sard in ????(????,????−1)2 (0, 1) space. Calcolo, – Vol. 51, – P. 211–243.
Shadimetov K., A. Hayotov, and B. Bozarov 2022. Optimal quadrature formulas for oscillatory integrals in the Sobolev space. Journal of Inequalities and Applications, 10.1186/s13660-022-02839-4.
Shadimetov Kh.M., and B.S. Daliev. 2022. Optimal formulas for the approximate-analytical solution of the general Abel integral equation in the Sobolev space. Results in Applied Mathematics, – Vol. 15, – 100276 p. 10.1016/j.rinam.2022.100276.
Zhang S., and E. Novak. 2019. Optimal quadrature formulas for the Sobolev space ????1. Journal of Scientific Computing, – Vol. 78, – P. 274–289.
