Abstract:
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.
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