S.I. Novikov, V.T. Shevaldin. On the connection between the second divided difference and the second derivative ... P. 216-224

We formulate the general problem of the extremal interpolation of real-valued functions with the $n$th derivative defined almost everywhere on the axis $\mathbf R$ (for finite differences, this is the Yanenko-Stechkin-Subbotin problem). It is required to find the smallest value of this derivative in the uniform norm on the class of functions interpolating any given sequence $y=\{y_k\}_{k=-\infty}^{\infty}$ of real numbers on an arbitrary, infinite in both directions node grid $\Delta=\{x_k\}_{k=-\infty}^{\infty}$ for a class of sequences $Y$ such that the moduli of their $n$th-order divided differences on this node grid are upper bounded by a fixed positive number. We solve this problem in the case $n=2$. For the value of the second derivative according to Yu.N. Subbotin's scheme, we derive upper and lower estimates, which coincide for a geometric node grid of the form $\Delta_p=\{p^kh\}_{k=-\infty}^{\infty}$ ($h>0$, $p\ge 1$). The estimates are derived in terms of the ratios of neighboring steps of the gird and interpolated values.

Keywords: interpolation, divided difference, splines, derivatives.

REFERENCES

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Received March 25, 2020

Revised May 5, 2020

Accepted May 11, 2020

Sergey Igorevich Novikov, Cand. Phys.-Math. Sci., 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 of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia,
e-mail: Valerii.Shevaldin@imm.uran.ru.

Cite this article as: S.I.Novikov, V.T.Shevaldin. On the connection between the second divided difference and the second derivative. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 216–224.