Yu. N. Subbotin’s Method in the Problem of Extremal Interpolation in the Mean in the Space $$L_p(\mathbb R)$$ with Overlapping Averaging Intervals

Abstract

On a uniform grid on the real axis, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space \(L_p(\mathbb R)\), \(1<p<\infty\), 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. In case of even \(n\), the value of the least norm in the space \(L_p(\mathbb R)\), \(1<p<\infty\), of the extremal interpolant is calculated exactly if the grid step \(h\) and the averaging step \(h_1\) are related by the inequality \(h<h_1\le 2h\).