On an optimal interpolation formula in K 2 (P 2 ) space.

Abstract

The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{\beta=0}^NC_{\beta}(z)\varphi(x_\beta)$ of given values of a function $\varphi$ from the space $K_2(P_2)$. 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 $\sin\omega x$ and $\cos\omega x$. At the end of the paper, we give some numerical results which confirm our theoretical results.