Estimating the Norm of the Error Functional in Sobolev Space of Periodic Functions
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Abstract
This article is devoted to obtaining an upper estimate of the error of an optimal quadrature formula for approximating the integral of periodic functions in the Sobolev space \(\widetilde{W_{2}^{(2,1,0)}}(0,1]\). In the quadrature formulas, a complex exponential weight function of the form \({{e}^{2\pi i\omega x}}\) is used. To minimize the norm of the error functional of the quadrature formula, a corresponding extremal function is found, and using it, an expression for the norm of the error functional is derived. The optimal coefficients that give the smallest value to this norm are obtained. Using Fourier analysis and the extremal function method, explicit formulas for the optimal coefficients are derived. These results extend the classical theory of quadrature formulas to the exponential-weight and oscillatory cases, providing efficient schemes for the numerical integration of periodic functions.
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References
- K.M. Shadimetov, A.R. Hayotov, Construction of Interpolation Splines Minimizing Semi-Norm in $w_{2}^{(m,m-1)}(0,1)$ Space, BIT Numer. Math. 53 (2013), 545–563. https://doi.org/10.1007/s10543-012-0407-z.
- A.R. Hayotov, G.V. Milovanović, K.M. Shadimetov, Optimal Quadratures in the Sense of Sard in a Hilbert Space, Appl. Math. Comput. 259 (2015), 637–653. https://doi.org/10.1016/j.amc.2015.02.093.
- A. Hayotov, S. Jeon, C. Lee, K. Shadimetov, Optimal Quadrature Formulas for Non-Periodic Functions in Sobolev Space and Its Application to CT Image Reconstruction, Filomat 35 (2021), 4177–4195. https://doi.org/10.2298/fil2112177h.
- K. Shadimetov, S. Azamov, On the Construction of Optimal Quadrature Formulas with Equally Spaced Nodes, Filomat 38 (2024), 10279–10295. https://doi.org/10.2298/fil2429279s.
- K. Shadimetov, A. Hayotov, U. Khayriev, Optimal Quadrature Formulas for Approximating Strongly Oscillating Integrals in the Hilbert Space ${W}_2^{(m,m-1)}$ of Periodic Functions, J. Comput. Appl. Math. 453 (2025), 116133. https://doi.org/10.1016/j.cam.2024.116133.
- Shadimetov, Kh.M., Azamov, K.M. Shadimetov, S.S. Azamov, Discrete Analogues of High Order Differential Operators, J. Math. Sci. 284 (2024), 253–265. https://doi.org/10.1007/s10958-024-07347-y.
- S.S. Azamov, On the Construction of Optimal Quadrature Formulas With Equally Spaced Nodes, Uzbek Math. J. 67 (2023), 41–48.
- K. Shadimetov, A. Adilkhodjaev, O. Gulomov, Optimal Quadrature Formulas for Approximate Calculation of Rapidly Oscillating Integrals, Results Appl. Math. 27 (2025), 100627. https://doi.org/10.1016/j.rinam.2025.100627.
- A.R. Hayotov, U.N. Khayriev, Construction of an Optimal Quadrature Formula in the Hilbert Space of Periodic Functions, Lobachevskii J. Math. 43 (2022), 3151–3160. https://doi.org/10.1134/s199508022214013x.
- K. Shadimetov, F. Nuraliev, S. Kuziev, Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions, Int. J. Anal. Appl. 22 (2024), 25. https://doi.org/10.28924/2291-8639-22-2024-25.
- S.L. Sobolev, Introduction to the Theory of Cubature Formulas, Nauka, (1974).
- M.A. Chakhkiev, Linear Differential Operators With Real Spectrum, and Optimal Quadrature Formulas, Math. USSR-Izv. 25 (1985), 391–417. https://doi.org/10.1070/im1985v025n02abeh001289.
- M. Shadimetov, S. Azamov, H. Qobilov Optimality of Approximate Integration Formulas for Classes of Periodic Functions, Bull. Inst. Math. 8 (2025), 218–225.