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