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
1. Gel’fond A.O. Calculus of finite differences. Delhi: Hindustan Publ. Corp., 1971, Ser. International Monographs on Advanced Mathematics and Physics, 451 p. Original Russian text published in Gel’fond A.O. Ischislenie konechnykh raznostei. Moscow: Nauka Publ., 1967. 376 p.
2. Subbotin Yu.N. On the connection between finite differences and corresponding derivatives. Proc. Steklov Institute Math., 1965, vol. 78, pp. 24–42 (in Russian).
3. Subbotin Yu.N. Functional interpolation in the mean with smallest n derivative. Proc. Steklov Inst. Math., 1967, vol. 88, pp. 31–63.
4. Subbotin Yu.N. Extremal problems of functional interpolation, and mean interpolation splines. Proc. Steklov Inst. Math., 1977, vol. 138, pp. 127–185.
5. Kunkle Th. Favard’s interpolation problem in one or more variables. Constructive Approxim. 2002, vol. 18, pp. 467–478. doi: 10.1007/s00365-001-0015-7 .
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.
