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

Abstract

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