# On a space of entire functions and its Fourier transform

## TL;DR

This paper studies a specific space of entire functions with rapid decay and controlled growth, providing new characterizations and a Paley-Wiener type theorem for these functions.

## Contribution

It introduces an equivalent description of the function space via derivative estimates and establishes a Paley-Wiener type theorem for it.

## Key findings

- Equivalent derivative estimates characterization
- Paley-Wiener type theorem established
- Space description in terms of growth and decay

## Abstract

A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on ${\mathbb R}^n$ and a Paley-Wiener type theorem are obtained.