Extremal interpolation in the mean with overlapping averaging intervals and the smallest norm of a linear differential operator

V. T. Shevaldin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 1, pp. 219-232

Résumé

The Yanenko–Stechkin–Subbotin problem of extremal functional interpolation in the mean is considered for sequences infinite in both directions on a uniform grid of the numerical axis with the smallest norm in the space $L_p(R)$ $(1 $ of a linear differential operator $\mathcal{L}_n$ with constant coefficients. It is assumed that the generalized finite differences of each sequence corresponding to the operator $\mathcal{L}_n$ are bounded in the space $l_p$, the grid step $h$ and the averaging step $h_1$ are related by the inequality $h$, and the operator $\mathcal{L}_n$ is formally self-adjoint. Under these assumptions, in the case of odd $n$, the smallest norm of the operator is found exactly, and the extremal function is a generalized $\mathcal{L}$-spline whose knots coincide with the interpolation nodes. This work continues the research of this problem by Yu. N. Subbotin and the author started by Subbotin in 1965.

Détail

Keywords: extremal interpolation, splines, uniform grid, formally self-adjoint differential operator, splines.
Mots-clés : minimum norm