Miyachi's Theorem for the Quaternion Fourier Transform

TL;DR

This paper extends Miyachi's uncertainty principle to the quaternion Fourier transform, demonstrating that similar limitations on simultaneous localization in time and frequency domains hold for quaternion-valued functions.

Contribution

The paper establishes Miyachi's theorem specifically for the quaternion Fourier transform, filling a gap in the understanding of uncertainty principles in quaternion analysis.

Findings

Miyachi's theorem is proven for the quaternion Fourier transform.

Uncertainty principles analogous to the classical case are valid in quaternion settings.

The results deepen the theoretical understanding of quaternion Fourier analysis.

Abstract

The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Euclidean Fourier transform. In this paper, we establish Miyachi's theorem for this transform.