On a space of entire functions and its Fourier transformation

TL;DR

This paper studies a space of entire functions of several complex variables that decrease rapidly on real space and have growth controlled by non-radial weight functions, providing new characterizations and a Paley-Wiener type theorem.

Contribution

It introduces a novel space of entire functions with non-radial weight controls and establishes equivalent derivative estimates and a Paley-Wiener theorem for this space.

Findings

Characterization of the function space via derivative estimates

Equivalent descriptions in terms of growth conditions

A Paley-Wiener type theorem for the space

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 (not radial in general) 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 given.