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

TL;DR

This paper introduces a class of infinitely differentiable functions in real space, constructed via specific sequences and functions, which can be extended to entire functions in complex space, with conditions characterizing these extensions.

Contribution

The paper defines a new function space based on sequences and functions, and characterizes the conditions under which functions in this space extend holomorphically to complex space.

Findings

Functions in the space can be extended to entire functions in ${f C}^n$

Conditions on the sequence $M$ and family $oldsymbol{ m extPhi}$ characterize the extension space

The space $G(M, oldsymbol{ m extPhi})$ is explicitly described in terms of these conditions

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.