Статья: INTERPOLATION WITH MINIMUM VALUE OF L2-NORM OF DIFFERENTIAL OPERATOR

For the class of bounded in l2-norm interpolated data, we consider a problem of interpolation on a finite interval [a, b] ⊂ ℝ with minimal value of the L2-norm of a differential operator applied to interpolants. Interpolation is performed at knots of an arbitrary N-point mesh ΔN: a ≤ x1 < x2 < < xN ≤ b. The extremal function is the interpolating natural -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 l2-norm interpolated data, the minimal value of the L2-norm of the differential operator on the interpolants is represented through the largest eigenvalue of the matrix of a certain quadratic form.