Extremal functional interpolation in the mean with least value of the \(n\)th derivative for large averaging intervals (Q1358458)

Let \(n\) be a positive integer and \(1< h<2\). The smallest number \(A<\infty\) is found such that for any sequence \((y_k)_{k\in\mathbb{Z}}\) with \(|\Delta^n y_k|\leq 1\) there exists a control function \(u(t)\), \(|u(t)|\leq A\), \(t\in\mathbb{R}\), for which the equation \(y^{(n)}= u\) has a solution satisfying \[ y_k= (1/h)\int^{h/2}_{-h/2} y(k+t)dt,\quad k\in\mathbb{Z}. \] A similar problem is treated in \(L_p(\mathbb{R})\). It is shown that for \(h=2m\) (\(m\) a positive integer) no such finite \(A\) exists.