On a Space of Infinitely Differentiable Functions on an Unbounded Convex Set in ${\mathbb R}^n$ Admitting Holomorphic Extension in ${\mathbb C}^n$ and its Dual

TL;DR

This paper studies a space of infinitely differentiable functions on an unbounded convex set in real space, showing they can be extended holomorphically to complex space with specific growth, and characterizes continuous linear functionals via Fourier-Laplace transform.

Contribution

It introduces a new class of smooth functions on unbounded convex sets, characterizes their holomorphic extensions, and provides a Paley-Wiener-Schwartz type theorem for distributions.

Findings

Functions extend to entire functions with growth conditions

Linear functionals characterized by Fourier-Laplace transform

A variant of Paley-Wiener-Schwartz theorem established

Abstract

We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.