Uncertainty Principles For The Continuous Quaternion Shearlet Transform

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

“…We have the following Theorem 1. With R 2n replacing R 2 in [11] we get the results, we will not repeat the proof here.…”

“…For a quaternion function f ∈ L 2 (R 2d , H) and a non zero quaternion function g ∈ L 2 (R 2d , H) called a quaternion shearlet . The aim of this paper is to generalize the continuous quaternion shearlet transform on R 2 to R 2d , called the multivariate two sided continuous quaternion shearlet transform which has been started in [11]. Our purpose in this work is to prove the Lieb uncertainty principle for the multivariate continuous quaternion shearlet transform.…”

Uncertainty Principles for the Continuous Shearlet Transforms in Arbitrary Space Dimensions

“…Very recently, Nefzi et al [26] generalized the results of Su [25] for the multivariate shearlet transform and analyze the net concentration of these transforms on sets of finite measure using the machinery of projection operators. Recent results in this direction can be found in [27,28].…”

A variation of the Lq,p‐uncertainty inequalities of Heisenberg‐type for symmetric q‐integral transforms

“…Generalizations of this result in both classical and quantum analysis have been treated and many versions of Heisenberg-Pauli-Weyl type uncertainty inequalities have been obtained for several generalized Fourier transforms, see previous studies. [15][16][17][18][19][20][21][22][23][24][25][26][27][28] Recently, S. Ghobber 18 proved a variation on Heisenberg's uncertainty inequality for a wide variety of integral operators, including the Fourier transform, the Fourier-Bessel transform, the generalized Fourier transform and the G-transform. The proof of the new Heisenberg's uncertainty inequality follows by combining the Nash and Carlson's inequalities for these transformations.…”

“…Every discussion of the uncertainty principle must necessarily begin with the classical uncertainty principle which also called the Heisenberg‐Pauli‐Weyl uncertainty inequality 14 in which concentration is measured in terms of dispersions. This leads to this quantitative formulation in the form of a lower bound of the product of the dispersions of a nonzero signal f and its Fourier transform : where for Generalizations of this result in both classical and quantum analysis have been treated and many versions of Heisenberg‐Pauli‐Weyl type uncertainty inequalities have been obtained for several generalized Fourier transforms, see previous studies 15–28 …”