On extension theorems in spaces of infinitely differentiable functions

Conditions on a sequence $\{f_\omega(x)\}$ of functions sufficient for there to exist an extension in the space
$$W^\infty\{a_\alpha,p,r\}\equiv\biggl\{u(x)\in C^\infty(G),\quad\rho(u)\equiv\sum_{|\alpha|=0}^\infty a_\alpha\|D^\alpha u\|_r^p <\infty\biggr\}$$
are established in the one-dimensional case $G\equiv(a,b)$ and also in the multidimensional strip $G\equiv\mathbf R^\nu\times[a, b]$. The conditions obtained reduce matters to a study of convergence of numerical series, and in a number of cases are not only sufficient but also necessary.
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