On a space of entire functions rapidly decreasing on ${\mathbb R}^n$ and its Fourier transformation

TL;DR

This paper studies a space of entire functions that decrease rapidly on real n-dimensional space, providing new characterizations via derivatives and a Paley-Wiener type theorem, enhancing understanding of their Fourier transforms.

Contribution

It introduces a new class of entire functions with rapid decay, offering an equivalent description through derivative estimates and establishing a Paley-Wiener type theorem for this space.

Findings

Equivalent description via partial derivatives

Paley-Wiener type theorem established

Characterization of Fourier transforms of these functions

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 a help of a family of weight functions is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are obtained.