On the Geometric Interpretation of the Complex Fourier Transforms of a Class of Exponential Functions

TL;DR

This paper explores the geometric properties of a specific class of complex Fourier transforms of exponential functions, focusing on those with zeros on the real line, belonging to the Laguerre-Pólya class, and proves all zeros are simple.

Contribution

It provides a geometric interpretation of these Fourier transforms and proves the simplicity of their zeros, advancing understanding of their zero distribution.

Findings

Zeros are all on the real line for this class.

All zeros are simple.

Transforms belong to the Laguerre-Pólya class.

Abstract

A class of complex Fourier Transforms of exponential functions which have all their zeros on the real line is explored from a geometric perspective. These transforms belong to the Laguerre - Polya class, and it is proved that all the zeros are simple.