Optimal interpolation on an interval with the smallest mean-square norm of the $r$th derivative

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.