On Heisenberg and local uncertainty principles for the multivariate continuous quaternion shearlet transform. (English) Zbl 1498.42009
Summary: 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, 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.
MSC:
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 42C15 | General harmonic expansions, frames |
| 46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
| 44A35 | Convolution as an integral transform |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
Keywords:
quaternion Fourier transform; shearlet; continuous quaternion shearlet transform; uncertainty principleReferences:
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