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

Contents of this issue

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

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

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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.