On Heisenberg and local uncertainty principles for the multivariate continuous quaternion Shearlet transform

In this paper, we generalize the continuous quaternion shearlet transform on R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{2}$$\end{document} to R2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{2d}$$\end{document}, 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, 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 the Heisenberg uncertainty principle. Last we study the multivariate two sided continuous quaternion shearlet transform on subset of finite measures.