On a Fr\'echet space of entire functions rapidly decreasing on the real line

TL;DR

This paper studies a weighted space of entire functions that decrease rapidly on the real line, analyzing their growth along the imaginary axis and characterizing the space via derivative estimates, with a focus on Fourier transforms.

Contribution

It introduces a new weighted space of entire functions with controlled growth and provides an equivalent description through derivative estimates and Fourier transform analysis.

Findings

Characterization of the space via derivative estimates

Analysis of growth along the imaginary axis

Fourier transform properties of the functions

Abstract

A weighted space of entire functions rapidly decreasing on the real line is considered in the paper. A growth of these functions along the imaginary axis is controlled by some system of weight functions. The Fourier transform of functions of this space is studied. Equivalent description of the considered space in terms of estimates on derivatives of functions on real line is obtained.