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

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 $L_p(R)$ $(1 <p<\infty)$ 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 $h<h_1<2h$, 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.