On finding the coefficients of the optimal interpolation formula in the space of S.L. Sobolev Ŵ(m) (T1)

A typical approximation problem is the interpolation problem. The classical method for solving it is to construct an interpolation polynomial. However, polynomials have a number of disadvantages, such as being a tool for approximating functions with singularities and functions with not very high smoothness. In practice, in order to approximate functions well, instead of constructing a high-degree interpolation polynomial, splines are used, which are very convenient to use. This paper examines the construction of interpolation splines using the Sobolev method, minimizing the norm in a certain Hilbert space. For the first time, S.L. Sobolev \cite{6} posed the problem of finding the extremal function for the interpolation formula and calculating the norm of the error functional in the Sobolev space. In this work, the extremal function of the interpolation formula is found in explicit form in the Sobolev space $W_{2}^{\left( m \right)}\left( {{R}^{n}} \right)$; a function whose generalized derivatives of order $m$ are square integrable. Basically, the problem of constructing optimal interpolation formulas in the space of S.L. Sobolev $\tilde{W}_{2}^{\left( m \right)}\left( {{T}_{1}} \right )$ for $m=4$ is considered.