Interpolation with minimum value of $L_{2}$-norm of differential operator

For the class of bounded in $l_{2}$-norm interpolated data, we consider a problem of interpolation on a finite interval $[a,b]\subset\mathbb{R}$ with minimal value of the $L_{2}$-norm of a differential operator applied to interpolants. Interpolation is performed at knots of an arbitrary $N$-point mesh $\Delta_{N}:\ a\leq x_{1}<x_{2}<\cdots <x_{N}\leq b$. The extremal function is the interpolating natural ${\mathcal L}$-spline for an arbitrary fixed set of interpolated data. For some differential operators with constant real coefficients, it is proved that on the class of bounded in $l_{2}$-norm interpolated data, the minimal value of the $L_{2}$-norm of the differential operator on the interpolants is represented through the largest eigenvalue of the matrix of a certain quadratic form.