Benedicks-Amrein-Berthier type theorem related to the two-sided Quaternion Fourier transform

TL;DR

This paper extends a classical uncertainty principle to the two-sided quaternion Fourier transform, showing that a nonzero function and its QFT cannot both be supported on finite measure sets, thus advancing harmonic analysis in quaternionic settings.

Contribution

It establishes a new uncertainty principle for the two-sided quaternion Fourier transform, generalizing a classical result to quaternionic functions.

Findings

Nonzero functions and their two-sided QFTs cannot both have finite measure support.

Extends classical uncertainty principles to quaternionic Fourier analysis.

Provides theoretical foundation for quaternionic harmonic analysis.

Abstract

The main objective of the present paper is to establish a new uncertainty principle (UP) for the two-sided quaternion Fourier transform (QFT). This result is an extension of a result of Benedicks, Amrein and Berthier, which states that a nonzero function in $L^1\left({\mathbb{R}}^2, {\mathbb{H}}\right)$ and its two-sided QFT cannot both have support of finite measure.