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

Abstract

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 s 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 \(h<h_1\le 2h\). 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 \(\mathcal L_n(D)=D^n\).