On a class of infinitely differentiable functions in ${\mathbb{R}}^n$ admitting holomorphic extension in ${\mathbb{C}}^n$

A space $G(M,\Phi )$ of infinitely differentiable functions in ${\mathbb R}^n$constructedwithahelpofafamily$\varPhi={\varphi_m}_{m=1}^{\infty}$ofreal-valuedfunctions$\varphi_m \in~C({\mathbb R}^n)$andalogarithmicallyconvexsequence$M$ of positive numbers is considered in the article. In view of conditions on $M$ each function of $G(M,\Phi )$ can be extended to an entire function in ${\mathbb C}^n$\mathrm{.}Imposedconditionson$M$and$\varPhi$ allow to describe the space of such extensions.