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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2020 Volume 26, Number 4, Pages 210–223 (Mi timm1776)

This article is cited in 1 paper

Extremal interpolation on the semiaxis with the smallest norm of the third derivative

S. I. Novikov, V. T. Shevaldin

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: 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.

UDC: 519.65

MSC: 41A15

Received: 09.09.2020
Revised: 23.10.2020
Accepted: 02.11.2020

DOI: 10.21538/0134-4889-2020-26-4-210-223



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