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

Kh. M. Shadimetov   ; A. K. Boltaev   ; R. I. Parovik

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 38 (2022) no. 1, pp. 131-146

Résumé

The paper is devoted to derivation of the optimal interpolation formula in $W_2^{(0,2)}(0,1)$ Hilbert space by Sobolev's method. Here the interpolation formula consists of a linear combination $\sum\limits_{\beta=0}^N C_\beta \phi (x_\beta)$ of the given values of a function $\phi$ from the space $W_2^{(0,2)}(0,1)$. 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 $C_\beta (z)$ of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions $\sin (x)$ and $\cos (x)$. At the end of the paper we give some numerical results which confirm the numerical convergence of the optimal interpolation formula.

Détail

Keywords: extremal function, error functional, Hilbert space, Sobolev's method.
Mots-clés : optimal interpolation formula, optimal coefficients

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