On Yu. N. Subbotin's Circle of Ideas in the Problem of Local Extremal Interpolation on the Semiaxis

Subbotin's problem of extremal functional interpolation of numerical sequences $\{y_k\}_{k=0}^{\infty}$ such that their first terms $y_0,y_1,\ldots,y_{s-1}$ are given and the $n$th-order divided differences are bounded is considered on an arbitrary grid $\Delta=\{x_k\}_{k=0}^{\infty}$ of the semiaxis $[x_0;+\infty)$. It is required to find an $n$-times differentiable function $f$ with the smallest norm of the $n$th-order derivative in the space $L_{\infty}$ such that $f(x_k)=y_k$ $(k \in \mathbb{Z}_+)$. Subbotin formulated and studied this problem only for a uniform grid on the semiaxis $[0;+\infty)$. We prove the finiteness of the smallest norm for $s\ge n$ if the smallest step of the interpolation grid $\underline{h}=\inf\limits_k(x_{k+1}-x_{k})$ is bounded away from zero and the largest step $\overline{h}=\sup\limits_k(h_{k+1}-h_k)$ is bounded away from infinity. In the case of the second derivative (i.e., for $n=2$), the required value is calculated exactly for $s=2$ and is estimated from above for $s\ge 3$ in terms of the grid steps.