Beurling's Theorem for the Two-sided Quaternion Fourier Transform

TL;DR

This paper extends classical uncertainty principles, including Beurling's theorem, to the two-sided quaternion Fourier transform, broadening the mathematical understanding of quaternionic signal analysis.

Contribution

It generalizes key uncertainty principles like Beurling's theorem to the two-sided quaternion Fourier transform, a novel extension in quaternionic analysis.

Findings

Beurling's theorem is established for the quaternion Fourier transform.

Hardy, Cowling-Price, and Gelfand-Shilov theorems are extended to this setting.

The results deepen the theoretical foundation for quaternionic signal processing.

Abstract

The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling's theorem, Hardy, Cowling-Price and Gelfand-Shilov theorems, is obtained for the two-sided quaternion Fourier transform.