INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR

Abstract

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 \({\cal L}\)-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.

Keywords

• interpolation, natural \({\cal l}\)-spline, differential operator, reproducing kernel, quadratic form.