Weyl Uncertainty Research Articles

Heisenberg–Pauli–Weyl uncertainty principle for ambiguity function

In this paper, we prove a stronger type of Heisenberg–Pauli–Weyl uncertainty principle for ambiguity function. An example is used to demonstrate the obtained result, and related uncertainty principles for short-time Fourier transform and Wigner distribution are given, respectively. In addition, we introduce the so-called K -ambiguity function, and obtain the related Heisenberg–Pauli–Weyl uncertainty principle.

Clifford‐Valued Shearlet Transforms on Cl(P,Q)‐Algebras

The shearlet transform is a promising and powerful time‐frequency tool for analyzing nonstationary signals. In this article, we introduce a novel integral transform coined as the Clifford‐valued shearlet transform on Cl(p,q) algebras which is designed to represent Clifford‐valued signals at different scales, locations, and orientations. We investigated the fundamental properties of the Clifford‐valued shearlet transform including Parseval’s formula, isometry, inversion formula, and characterization of range using the machinery of Clifford Fourier transforms. Moreover, we derived the pointwise convergence and homogeneous approximation properties for the proposed transform. We culminated our investigation by deriving several uncertainty principles such as the Heisenberg–Pauli–Weyl uncertainty inequality, Pitt’s inequality, and logarithmic and local‐type uncertainty inequalities for the Clifford‐valued shearlet transform.

Sufficient Criteria for Obtaining Hardy Inequalities on Finsler Manifolds

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if \(\rho \) is a nonnegative function and \(-\varvec{\Delta } \rho \ge 0\) in weak sense, where \(\varvec{\Delta }\) is the Finsler–Laplace operator defined by \( \varvec{\Delta } \rho = \mathrm {div}(\varvec{\nabla } \rho )\), then we obtain the generalization of some Riemannian Hardy inequalities given in D’Ambrosio and Dipierro (Ann Inst H Poincaré Anal Non Linéaire 31(3):449–475, 2014). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo–Nirenberg inequality and a Heisenberg–Pauli–Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler–Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy inequality to a class of Finsler manifolds of bounded geometry.

The continuous quaternion algebra-valued wavelet transform and the associated uncertainty principle

This article aims to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform. We study some fundamental properties of this extension such as scaling, translation, rotation, and Parseval’s identity, then we derive the associated Heisenberg–Pauli–Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP to the CQWT domain.

Weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform

The classical uncertainty principle plays an important role in quantum mechanics, signal processing and applied mathematics. With the development of novel signal processing methods, the research of the related uncertainty principles has gradually been one of the most hottest research topics in modern signal processing community. In this paper, the weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform (LCT) have been investigated in detail. Firstly, the Plancherel–Parseval–Rayleigh identities associated with the LCT are derived. Secondly, the weighted Heisenberg–Pauli–Weyl uncertainty principles in the LCT domain are investigated based on the derived identities. The signals that can achieve the lower bound of the uncertainty principle are also obtained. The classical Heisenberg uncertainty principles in the Fourier transform (FT) domain are shown to be special cases of our achieved results. Thirdly, examples are provided to show that our weighted Heisenberg–Pauli–Weyl uncertainty principles are sharper than those in the existing literature. Finally, applications of the derived results in time frequency resolution analysis and signal energy concentrations are also analyzed and discussed in detail.

Heisenberg–Pauli–Weyl inequality for connected nilpotent Lie groups

The purpose of this paper is to formulate and prove an analogue of the classical Heisenberg–Pauli–Weyl uncertainty inequality for connected nilpotent Lie groups with noncompact center. Representation theory and a localized Plancherel formula play an important role in the proof.

Uncertainty principles and time frequency analysis related to the Riemann–Liouville operator

We define and study the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators defined on the half plane $$]0,+\infty [\times \mathbb {R}$$ . We prove a Plancherel theorem and an inversion formula that we use to establish the classical Heisenberg uncertainty principle. Next, we study this transform on subsets of $$([0,+\infty [\times \mathbb {R})^2$$ with finite measures, in particular we establish a well generalized Heisenberg–Pauli–Weyl uncertainty principle for this transform (with general magnitude). Also, we check a local uncertainty principle and we give nice applications.

Quaternionic shearlet transform

The shearlet transform has been shown to be a valuable and powerful time–frequency analyzing tool for optics and non-stationary signal processing. In this article, we propose a novel transform called quaternionic shearlet transform which is designed to represent quaternion-valued signals at different scales, locations and orientations. We investigate the fundamental properties of quaternionic shearlet transform including Parseval's formula, Moyal's principle, inversion formula, and characterization of its range using the machinery of quaternion Fourier transform and quaternion convolution. We conclude our investigation by deriving an analogue of the classical Heisenberg–Pauli–Weyl uncertainty inequality and the associated logarithmic version for the quaternionic shearlet transform.

Sharp uncertainty principles on Riemannian manifolds: the influence of curvature

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg–Pauli–Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:(a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn.Since the sharp HPW principle and the Hardy–Poincaré inequality are endpoints of the Caffarelli–Kohn–Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan–Hadamard manifolds.

Uncertainty principles for spherical mean $$L^2$$ L 2 -multiplier operators

We study the spherical mean \(L^2\)-multiplier operators, and for these operators we establish the Heisenberg–Pauli–Weyl uncertainty principle and Donoho–Stark’s uncertainty principle.

Uncertainty Inequality for Radon Transform on the Heisenberg Group

This article presents the Heisenberg–Pauli–Weyl uncertainty inequality for the Radon transform on the Heisenberg group, which indicates that the Radon transform and the Fourier transform of a nonzero function can not both be sharply localized. The proof is mainly based on some estimates related to the heat kernel, together with the relation between the sublaplacian and the group Fourier transform.

Uncertainty principles for the Fourier transform with numerical aspect

In this paper, we establish local uncertainty principle for the Fourier transform; and we deduce version of Heisenberg–Pauli–Weyl uncertainty principle. We use also the local uncertainty principle, the partial Fourier integrals and the techniques of Donoho–Stark, we present two uncertainty principles of concentration type in the theory, when . Some numerical applications are given.

An Uncertainty Principle on Compact Manifolds

Breitenberger’s uncertainty principle on the torus $$\mathbb {T}$$ and its higher-dimensional analogue on $$\mathbb {S}^{d-1}$$ are well understood. We describe an entire family of uncertainty principles on compact manifolds $$(M,g)$$ , which includes the classical Heisenberg–Weyl uncertainty principle (for $$M=B(0,1) \subset \mathbb {R}^d$$ the unit ball with the flat metric) and the Goh–Goodman uncertainty principle (for $$M=\mathbb {S}^{d-1}$$ with the canonical metric) as special cases. This raises a new geometric problem related to small-curvature low-distortion embeddings: given a function $$f:M \rightarrow \mathbb {R}$$ , which uncertainty principle in our family yields the best result? We give a (far from optimal) answer for the torus, discuss disconnected manifolds and state a variety of other open problems.

Qualitative uncertainty principles for the hypergeometric Fourier transform

We prove an Lpversion of the Donoho–Stark’s uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\). Next, using the ultracontractive properties of the semigroups generated by the Heckman–Opdam Laplacian operator, we obtain an Lp Heisenberg–Pauli–Weyl uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\).

Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform

The quaternion Fourier transform – a generalized form of the classical Fourier transform – has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt's inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form f=f1+if2+f3j+if4j of quaternion from Hitzer and the novel module or Lp-norm of the quaternion Fourier transform fˆ, then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg–Weyl's uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt's inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform.

A general form of Heisenberg–Pauli–Weyl uncertainty inequality for the Dunkl transform

In this paper, we give a generalization of the Heisenberg–Pauli–Weyl uncertainty inequality for the Dunkl transform on ℝ d .

Uncertainty inequalities for the Heisenberg group

We establish the Heisenberg–Pauli–Weyl uncertainty inequalities for Fourier transform and the continuous wavelet transform on the Heisenberg group.

On the fractional minimal length Heisenberg–Weyl uncertainty relation from fractional Riccati generalized momentum operator

It was showed that the minimal length Heisenberg–Weyl uncertainty relation may be obtained if the ordinary momentum differentiation operator is extended to its fractional counterpart, namely the generalized fractional Riccati momentum operator of order 0 < β ⩽ 1. Some interesting consequences are exposed in concordance with the UV/IR correspondence obtained within the framework of non-commutative C-space geometry, string theory, Rovelli loop quantum gravity, Amelino-Camelia doubly special relativity, Nottale scale relativity and El-Naschie Cantorian fractal spacetime. The fractional theory integrates an absolute minimal length and surprisingly a non-commutative position space.

Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth

In its simpler form, the Heisenberg–Pauli–Weyl inequality says that ‖ f ‖ 2 4 ⩽ C ( ∫ R n | x | 2 | f ( x ) | 2 d x ) ( ∫ R n | ( − Δ ) 1 2 f ( x ) | 2 d x ) . In this paper, we extend this inequality to positive self-adjoint operators L on measure spaces with a “gauge function” such that (a) measures of balls are controlled by powers of the radius (possibly different powers for large and small balls); (b) the semigroup generated by L satisfies ultracontractive estimates with polynomial bounds of the same type. We give examples of applications of this result to sub-Laplacians on groups of polynomial volume growth and to certain higher-order left-invariant hypoelliptic operators on nilpotent groups. We finally show that these estimates also imply generalized forms of local uncertainty inequalities.