On the extension of infinitely differentiable functions

Conditions on logarithmically convex sequences $\{M_n\}$ and $\{\widehat M_n\}$ are obtained under which, for every sequence $\{b_n\}$ with $|b_n|<C_1^nM_n$, $n=0,1,2,\dots$, there exists an infinitely differentiable function $f(x)$ such that $f_{(0)}^{(n)}=b_n$ and $\|f^{(n)}\|_{L_p(R)}\leqslant C_2^n\widehat M_n(p)$, $1\leqslant p\leqslant\infty$.
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