On the Zeros of the Complex Fourier Transforms of a Class of Exponential Functions

TL;DR

This paper proves that the complex Fourier transforms of a specific class of exponential functions have all their zeros on the real line, using a class theorem and convolution methods.

Contribution

It introduces a class theorem establishing the real-line zeros of Fourier transforms for a class of exponential functions, extending the class through convolutions.

Findings

Fourier transforms of the studied exponential functions have zeros only on the real line

A class theorem is proved to support this property

The class of functions is extended via convolution methods

Abstract

A class theorem is presented and proved: the complex Fourier transforms of a certain class of exponential functions have all their zeros on the real line. A class of basis functions is first considered, and the class is then extended via the method of convolutions.