On an arbitrary grid $\Delta=\{x_k\}_{k=0}^{\infty}$ of the half-line $[x_0;+\infty)$, we consider Yu.N. 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. 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 half-line $[0;+\infty)$. 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.
Keywords: local interpolation, semiaxis, arbitrary grid, divided differences
Received February 17, 2022
Revised August 19, 2022
Accepted August 22, 2022
Valerii Trifonovich Shevaldin, Dr.Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: Valerii.Shevaldin@imm.uran.ru
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Cite this article as: V.T. Shevaldin. On Yu.N.Subbotin’s circle of ideas in the problem of local extremal interpolation on the semiaxis. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 237–249; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S229–S241.
