The following problem is considered. For a class of interpolated sequences $y=\{y_{k}\}_{k=-\infty}^{+\infty}$ of real numbers such that their third-order divided difference constructed for arbitrary knots $\{x_{k}\}_{k=-\infty}^{+\infty}$ are bounded in absolute value by a fixed positive number, it is required to find a function $f$ having the third derivative almost everywhere and such that $f(x_{k})=y_{k}\ (k\in\mathbb{Z})$ and the third derivative has the smallest $L_{\infty}$-norm. The problem is solved on the positive semiaxis $\mathbb{R}_{+}=(0,+\infty)$ for geometric grids in which the sequence of steps $h_{k}=x_{k+1}-x_{k}$ $(k\in\mathbb{Z})$ is a geometric progression with ratio~$p$ $(p>1)$; i.e., $h_{k+1}/h_{k}=p$. In the case of a uniform grid $x_{k}=kh\ (h>0,k\in\mathbb{Z})$ on the whole axis $\mathbb{R}$ (i.e., for $p=1$), this problem was solved by Yu.N. Subbotin in 1965 and is known as the Yanenko-Stechkin-Subbotin problem of extremal function interpolation.
Keywords: interpolation, divided difference, splines, difference equation
Received September 30, 2020
Revised October 23, 2020
Accepted November 2, 2020
Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center.
Sergey Igorevich Novikov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: Sergey.Novikov@imm.uran.ru
Valerii Trifonovich Shevaldin, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics 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: S.I. Novikov, V.T. Shevaldin. Extremal interpolation on the semiaxis with the smallest norm of the third derivative, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 4, pp. 210–223.
