An exact solution is found to the problem of interpolation on a finite interval $[a,b]$ with the smallest $L_{2}$-norm of the $r$th-order derivative $(r\geq 2)$ by functions $f$: $[a,b]\to \mathbb{R}$ with absolutely continuous $(r-1)$th-order derivatives for finite collections of data from the unit ball of the space $l_{2}^{N}$. Interpolation is performed at nodes of an arbitrary grid $\Delta _{N}$: $a=x_{1}<x_{2}<\cdots<x_{N}=b$. The smallest value of the $L_{2}$-norm on the class of interpolated data is expressed in terms of the largest eigenvalue of a certain square matrix and its determinant. The paper improves the classical results of spline theory related to the minimum norm property, which were originally obtained by J. Holladay and then developed by J. Ahlberg, E. Nilson, and J. Walsh, as well as by V.N. Malozemov and A.B. Pevnyi.
Keywords: interpolation, natural splines, matrix eigenvalue
Received June 9, 2023
Revised June 30 2023
Accepted July 3, 2023
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
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Cite this article as: S.I. Novikov. Optimal interpolation on an interval with the smallest mean-square norm of the rth derivative. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 217–228.
