Kolmogorov-type inequalities and the best formulas for numerical differentiation

For a certain class of complex-valued functions $f(x)$, $-\infty<x<\infty$, is found the best approximation
$$ u_N=\inf_{\|A\|\le N}\sup_{\|f^{(n)}\|_{L_2}\le1}\|f^{(k)}-A(f)\|C $$
of a differential operator by linear operators $A$ with the norm $\|A\|_{L_2}^C\le N$, $N>0$. Using the value $u_N$, the smallest constant $Q$ in the inequality
$$ \|f^{(k)}\|_Q\le Q\|f\|_{L_2}^\alpha\|f^{(n)}\|^\beta_{L_2} $$
is found.