Extremal Interpolation in the Mean in the Space $L_1(\mathbb R)$ with Overlapping Averaging Intervals

On a uniform grid on the real axis $\mathbb R$, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space $L_1(\mathbb R)$ of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator $\mathcal L_n$ of order $n$ with constant real coefficients. This problem is considered for the class of sequences for which the generalized finite differences of order $n$ corresponding to the operator $\mathcal L_n$ are bounded in the space $l_1$. In this paper, the least value of the norm is calculated exactly if the grid step $h$ and the averaging step $h_1$ of the function to be interpolated in the mean are related by the inequalities $h<h_1\leqslant 2h$. The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the $n$-times differentiation operator $\mathcal L_n(D)=D^n$.