Construction of an Optimal Quadrature Formula in the Hilbert Space of Periodic Functions

T. K. Yuldashev and B. J. Kadirkulov, ‘‘Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions,’’ Lobachevskii J. Math. 42, 649–662 (2021).

T. K. Yuldashev, B. J. Kadirkulov, and R. A. Bandaliyev, ‘‘On a mixed problem for Hilfer type fractional differential equation with degeneration,’’ Lobachevskii J. Math. 43, 263–274 (2022).

C. Li and F. Zeng, Numerical Methods for Fractional Calculus (CRC, New York, 2015).

D. Baleanu, K. Diethelm, E. Scalesm, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, 2nd ed. (World Scientific, Singapore, 2016), Vol. 5.

A. Lapin and E. Laitinen, ‘‘A numerical model for steel continuous casting problem in a time-variable domain,’’ Lobachevskii J. Math. 41, 2664–2672 (2020).

A. Lapin and K. O. Levinskaya, ‘‘Numerical solution of a quasilinear parabolic equation with a fractional time derivative,’’ Lobachevskii J. Math. 41, 2673–2686 (2020).

A. Lapin, S. Lapin, and S. Zhang, ‘‘Approximation of a mean field game problem with Caputo time-fractional derivative,’’ Lobachevskii J. Math. 42, 2876–2889 (2021).

S. I. Solov’ev, ‘‘Quadrature finite element method for elliptic eigenvalue problems,’’ Lobachevskii J. Math. 38, 856–863 (2017).

A. M. Burden, J. D. Faires, and R. L. Burden, Numerical Analysis, 10th ed. (Cengage Learning, Boston, MA, 2016).

A. Sard, ‘‘Best approximate integration formulas; best approximation formulas,’’ Am. J. Math. 71, 80–91 (1949).

S. M. Nikolskii, Quadrature Formulas (Nauka, Moscow, 1988) [in Russian].

G. V. Demidenko and V. L. Vaskevich, Selected Works of S. L. Sobolev (Springer, New York, 2006).

S. L. Sobolev and V. L. Vaskevich, The Theory of Cubature Formulas (Kluwer Academic, Dordrecht, 1997).

A. Baboş and A. M. Acu, ‘‘Note on corrected optimal quadrature formulas in the sense Nikolski,’’ Appl. Math. Inform. Sci. Int. J. 9, 1231–1238 (2015).

A. R. Hayotov, G. V. Milovanović, and Kh. M. Shadimetov, ‘‘Optimal quadratures in the sense of Sard in a Hilbert space,’’ Appl. Math. Comput. 259, 637–653 (2015).

Kh. M. Shadimetov and B. S. Daliev, ‘‘Optimal formulas for the approximate-analytical solution of the general Abel integral equation in the Sobolev space,’’ Results Appl. Math. 15, 100276 (2022).

B. G. Gabdulkhaev, ‘‘Continuity and compactness of singular integral operators,’’ Russ. Math. 53, 1–7 (2009).

Kh. M. Shadimetov, A. R. Hayotov, and D. M. Akhmedov, ‘‘Optimal quadrature formulas for Cauchy type singular integrals in Sobolev space,’’ Appl. Math. Comput. 263, 302–314 (2015).

Kh. M. Shadimetov and D. M. Akhmedov, ‘‘Approximate solution of a singular integral equation using the Sobolev method,’’ Lobachevskii J. Math. 43, 496–505 (2022).

A. R. Hayotov, S. Jeon, C.-O. Lee, and Kh. M. Shadimetov, ‘‘Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction,’’ Filomat 35, 4177–4195 (2021).

A. R. Hayotov, S. Jeon, and Kh. M. Shadimetov, ‘‘Application of optimal quadrature formulas for reconstruction of CT images,’’ J. Comput. Appl. Math. 388, 113313 (2021).

S. S. Babaev, A. R. Hayotov, and U. N. Khayriev, ‘‘On an optimal quadrature formula for approximation of Fourier integrals in the space $$W_{2}^{(1,0)}$$,’’ Uzbek Math. J., тДЦ 2, 23–36 (2020).

A. R. Hayotov and S. S. Babaev, ‘‘Optimal quadrature formulas for computing of Fourier integrals in $$W_{2}^{(m,m-1)}$$ space,’’ AIP Conf. Proc. 2365, 020021 (2021).

A. R. Hayotov and U. N. Khayriev, ‘‘Optimal quadrature formulas in the space $$\widetilde{W}_{2}^{(1,0)}$$ of periodic functions,’’ Uzbek Math. J. 65 (3), 93–100 (2021).

Sh. Maqsudov, M. S. Salokhitdinov, and S. H. Sirojiddinov, The Theory of Complex Variable Functions (FAN, Tashkent, 1976) [in Russian].