Extremal functional interpolation in the mean with least value of the $n$-th derivative for large averaging intervals

The smallest number $A<\infty$ is found such that for any sequence $Y=\{y_k,k\in\mathbb Z\}$ with $|\Delta^ny_k|\le1$ there exists a $u(t)$, $|u(t)|\le A$, for which the equation $y^n(t)=u(t)$ ($-\infty<t<\infty$) has a solution satisfying the conditions
$$ y_k=\frac 1h\int_{-h/2}^{h/2}y(k+1)\,dt, $$
where $k\in\mathbb Z$, $1<h<2$. A similar problem is treated in $L_p(-\infty,\infty)$. It is shown that for $h=2m$ ($m$ a natural number) no such finite $A$ exists.