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

S. I. Novikov ; V. T. Shevaldin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 210-223

Résumé

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.

Détail

Mots-clés : interpolation
Keywords: divided difference, splines, difference equation.