On the dependence of the norm of a function on the norms of its derivatives of orders $k$ , $r - 2$ and $r , 0 < k < r - 2$

Abstract

We establish conditions for a system of positive numbers $M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}, \; 0 = k_1 < k2 < k3 = r − 2, k4 = r$, necessary and sufficient for the existence of a function $x \in L^r_{\infty, \infty}(R)$ such that $||x^{(k_i)} ||_{\infty} = M_{k_i},\quad i = 1, 2, 3, 4$