Lieb, Entropy and Logarithmic uncertainty principles for the multivariate continuous quaternion Shearlet Transform

TL;DR

This paper extends the continuous quaternion shearlet transform to higher dimensions, deriving key properties and applying them to establish various uncertainty principles including Lieb, logarithmic, and Beckner's entropy-based principles.

Contribution

It introduces the multivariate two-sided continuous quaternion shearlet transform and derives fundamental properties and uncertainty principles for this new transform.

Findings

Established Lieb uncertainty principle for the transform.

Proved logarithmic uncertainty principle in the quaternion shearlet context.

Analyzed Beckner's entropy-based uncertainty principle for the transform.

Abstract

In this paper, we generalize the continuous quaternion shearlet transform on $\mathbb{R}^{2}$ to $\mathbb{R}^{2d}$, called the multivariate two sided continuous quaternion shearlet transform. Using the two sided quaternion Fourier transform, we derive several important properties such as (reconstruction formula, reproducing kernel, plancherel's formula, etc.). We present several example of the multivariate two sided continuous quaternion shearlet transform. We apply the multivariate two sided continuous quaternion shearlet transform properties and the two sided quaternion Fourier transform to establish Lieb uncertainty principle and the Logarithmic uncertainty principle. Last we study the Beckner's uncertainty principle in term of entropy for the multivariate two sided continuous quaternion shearlet transform.