Vestnik КRAUNC. Fiz.-Mat. nauki. 2022. vol. 38. no. 1. P. 131-146. ISSN 2079-6641

Contents of this issue

Read Russian Version US Flag

MSC 65D30, 65D32

Research Article

Construction of optimal interpolation formula exact for trigonometric functions by Sobolev’s method

Kh. M. Shadimetov1,2, A. K. Boltaev2,3, R. I. Parovik3,4

1Tashkent State Transport University, 1 Odilxojaev str., Tashkent 100167, Uzbekistan

2V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b University str., Tashkent, 100174, Uzbekistan

3National University of Uzbekistan named after Mirzo Ulugbek, 4 University str., Tashkent, 100174, Uzbekistan

4Vitus Bering Kamchatka State University, 683032, Petropavlovsk-Kamchatskiy, Pogranichnaya str., 4, Russia

E-mail: aziz_boltayev@mail.ru, kholmatshadimetov@mail.ru

The paper is devoted to derivation of the optimal interpolation formula in W2(0,2)(0,1) Hilbert space by Sobolev’s method. Here the interpolation formula consists of a linear combination   ΣNβ=0Cβφ(xβ) of the given values of a function φ from the space W2(0,2)(0,1). The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients Cβ(z) of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions sinx and cosx. At the end of the paper we give some numerical results which confirm the numerical convergence of the optimal interpolation formula.

Key words: extremal function, error functional, Hilbert space, optimal interpolation formula, optimal coefficients, Sobolev’s method.

DOI: 10.26117/2079-6641-2022-38-1-131-146

Original article submitted: 16.02.2022

Revision submitted: 26.03.2022

For citation. Shadimetov Kh. M., Boltaev A. K., Parovik R. I. Construction of optimal interpolation formula exact for trigonometric functions by Sobolev’s method. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 38: 1, 131-146. DOI: 10.26117/2079-6641-2022-38-1-131-146

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Shadimetov Kh. M., Boltaev A. K., Parovik R. I., 2022

References

  1. Arqub O. Abu. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions, Computers and Mathematics with Applications, 2017. vol. 73, pp. 1243–1261 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  2. Arqub O. Abu, Maayah B.Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator Chaos, Solitons and Fractals, 2018. vol. 117, pp. 117–124 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  3. Arqub O. Abu, Al-Smadi M. Numerical algorithm for solving time-fractional partial integro differential equations subject to initial and Dirichlet boundary conditions, Numer Methods for Partial Differential Equations, 2018. vol. 34, pp. 1577–1597 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  4. Aronszajn N. Theory of reproducing kernels,Trans Am Math Soc, 1950. vol. 68, pp. 337–404 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  5. Berlinet A., Thomas-Agnan C. Reproducing Kernel Hilbert Space in Probability and Statistics. Dordrecht: Springer, 2004.
  6. Boltaev A., Shadimetov Kh., Nuraliev F. The extremal function of interpolation formulas in W2(0,2) space, Vestnik KRAUNC. Fiz.-mat. nauki, 2021. vol. 36, no. 3, pp. 123–132 DOI: 10.26117/2079-6641-2021-36-3-123-132.
  7. Cabada A., Hayotov A., Shadimetov Kh. Construction of Dm-splines in L2(m)(0;1) space by Sobolev method, Applied Mathematics and Computation, 2014. vol. 244, pp. 542–551 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  8. Hayotov A. The discrete analogue of a differential operator and its applications, Lithuanian Mathematical Journal, 2014. vol. 54, no. 2, pp. 290–307 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  9. Hayotov A. Construction of interpolation splines minimizing the semi-norm in the space K2(Pm), Journal of Siberian Federal University. Mathematics and Physics, 2018. vol. 11, pp. 383–396 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  10. Hayotov A., Milovanovi´c G., Shadimetov Kh. On an optimal quadrature formula in the sense of Sard, Numerical Algorithms, 2011. vol. 57, pp. 487–510 DOI: 10.26117/2079-6641-2020-32-3-42-54
  11.  Hayotov A.,  Milovanovi´c G., Shadimetov Kh. Optimal quadrature formulas and interpolation splines minimizing the semi-norm in K2(P2) space, Analytic Number Theory, Approximation Theory, and Special Functions, 2014, pp. 573–611 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  12. Hayotov A., Milovanovi´c G., Shadimetov Kh. Interpolation splines minimizing a seminorm, Calcolo, 2014. vol. 51, pp. 245–260 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  13. Hayotov A., Milovanovi´c G., Shadimetov Kh. Optimal quadratures in the sense of Sard in a Hilbert space, Applied Mathematics and Computation, 2015. vol. 259, pp. 637–653 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  14. Novak E., Ullrich M., Wo´zniakowski H., Zhang Sh. Reproducing kernels of Sobolev spaces on Rd and applications to embedding constants and tractability, Analysis and Applications, 2018. vol. 16, pp. 693–715 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  15. Shadimetov Kh., Hayotov A. Optimal quadrature formulas with positive coefficients in L2(m)(0;1) space, Journal of Computational and Applied Mathematics, 2011. vol. 235, pp. 1114–1128 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  16. Shadimetov Kh., Hayotov A. Construction of interpolation splines minimizing semi-norm in W2(m,m-1)(0;1) space, BIT Numer Math, 2013. vol. 53, pp. 545–563 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  17. Shadimetov Kh., Hayotov A. Optimal quadrature formulas in the sense of Sard in W2(m,m-1) space, Calcolo, 2014. vol. 51, pp. 211–243 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  18. Sobolev S. L. Introduction to the theory of cubature formulas. Moscow: Nauka, 1974. 808 pp. (In Russian)
  19. Sobolev S. L. On interpolation of functions of n variables, in: Selected works of S. L. Sobolev, Springer US, 2006, pp. 451–456 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  20. Sobolev S. L. Formulas of mechanical cubature in n-dimensional space, in: Selected Works of S. L. Sobolev, Springer US, 2006, pp. 445–450 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  21. Sobolev S. L. The coefficients of optimal quadrature formulas, in: Selected works of S. L. Sobolev, Springer US, 2006, pp. 561–566 DOI: 10.26117/2079-6641-2020-32-3-42-54.
  22. Sobolev S. L., Vaskevich V. L. The theory of cubature formulas. Dordrecht: Kluwer Academic Publishers Group, 1997. 418 pp.

Shadimetov Kholmatvay Makhkambaevich – D. Sci. (Phys. & Math.), Professor, Chief of the Department of Computer Science and computer graphics, Tashkent State Transport University, ORCID 0000-0002-4183-6184.

Boltaev Aziz Kuziyevich – Ph. D. (Phys. & Math.), senior staff scientist at the laboratory of Computational Mathematics, V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, ORCID 0000-0002-8329-4440.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Professor of the Depart. Math.& Phys., Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatskiy, Russia, ORCID 0000-0002-1576-1860.

Download article: Shadimetov Kh. M., Boltaev A. K., Parovik R. I. Construction of optimal interpolation formula exact for trigonometric functions by Sobolev’s method