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

Abstract

A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{\varphi_m\}_{m=1}^{\infty}$ of real-valued functions $\varphi_m \in~C({\mathbb R}^n)$ and a logarithmically convex sequence $M$ of positive numbers is considered in the article. In view of conditions on $M$ each function of $G(M, \varPhi)$ can be extended to an entire function in ${\mathbb C}^n$. Imposed conditions on $M$ and $\varPhi$ allow to describe the space of such extensions.