Heisenberg-type Uncertainty Principle Research Articles

Time-frequency analysis of (k,a)-generalized wavelet transform and applications

The (k, a)-generalized wavelet transform is a novel addition to the class of wavelet transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short period of time. Since the study of time-frequency analysis is both theoretically interesting and practically useful, in this article, we investigated several subjects of time-frequency analysis for the (k, a)-generalized wavelet transform. First, we analyze the concentration of this transform on sets of finite measure. In particular, we prove Donoho–Stark and Benedicks-type uncertainty principles. We prove several versions of Heisenberg-type uncertainty principles for this transformation. Furthermore, involving the reproducing kernel and spectral theories, we investigate the time frequency and study the scalogram for the same wavelet transform. Finally, we provide Shapiro’s mean dispersion type theorems at the end.

Logarithm Sobolev and Shannon’s Inequalities Associated with the Deformed Fourier Transform and Applications

By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash’s inequality, Carlson’s inequality and Sobolev’s embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the L∞-norm. Finally, we obtain a logarithmic Sobolev inequality in Lp-spaces, from which we derive an Lp-Heisenberg-type uncertainty inequality and an Lp-Nash-type inequality for the Dunkl transform.

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

Shannon's inequality for the Rényi entropy and an application to the uncertainty principle

We consider Shannon's inequality for the Rényi entropy. The Rényi entropy corresponds to a nonlinear diffusion equation, for example, the porous-medium equation. Shannon's inequality implies the positivity of the relative Lyapunov functional for a nonlinear Fokker–Planck type equation. We derive Shannon's inequality for the Rényi entropy and give the other proof of this positivity. Moreover, by combining the Shannon and the logarithmic Sobolev inequalities for the Rényi entropy, we obtain the Heisenberg type uncertainty principle.

Bendlet transforms: a mathematical perspective

Recently, Lessig et al. [Appl Comput Harmon Anal. 2019;46:384–399] introduced the notion of bendlets; a shearlet-like system that is based on anisotropic scaling, translation, shearing and bending of a compactly supported generator. The theoretical framework of the bendlet transform is yet to be explored exclusively. Taking this opportunity, our aim is to investigate the mathematical properties of the bendlet transform such as the Rayleigh theorem, inversion formula, characterization of range and the pointwise convergence of the inversion formula. In continuation, we obtain the Heisenberg-type uncertainty principle and the Pitt's inequality for the bendlet transform. Subsequently, we employ the bendlet transform for obtaining the solutions of the wave and Laplace equations. Finally, we introduce the notion of quaternionic bendlet transform and also study its fundamental properties.

On uncertainty principle for the two-sided quaternion linear canonical transform

The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose five different forms of uncertainty principles for the two-sided QLCT, including logarithmic uncertainty principle, Heisenberg-type uncertainty principle, local uncertainty principle, Benedicks–Amrein–Berthier uncertainty principle and entropic uncertainty principle. These consequences actually describe the quantitative relationships of a quaternion-valued signal in arbitrary two different QLCT domains, and they have great applications in signal recovery and physical quantum.

An Intertwining of Curvelet and Linear Canonical Transforms

In this article, we introduce a novel curvelet transform by combining the merits of the well-known curvelet and linear canonical transforms. The motivation towards the endeavour spurts from the fundamental question of whether it is possible to increase the flexibility of the curvelet transform to optimize the concentration of the curvelet spectrum. By invoking the fundamental relationship between the Fourier and linear canonical transforms, we formulate a novel family of curvelets, which is comparatively flexible and enjoys certain extra degrees of freedom. The preliminary analysis encompasses the study of fundamental properties including the formulation of reconstruction formula and Rayleigh’s energy theorem. Subsequently, we develop the Heisenberg-type uncertainty principle for the novel curvelet transform. Nevertheless, to extend the scope of the present study, we introduce the semidiscrete and discrete analogues of the novel curvelet transform. Finally, we present an example demonstrating the construction of novel curvelet waveforms in a lucid manner.

A convolution-based special affine wavelet transform

ABSTRACT In the article ‘Convolution and product theorems for the special affine Fourier transform’ [In: Nashed MZ, Li X, editors. Frontiers in orthogonal polynomials and q-series. World Scientific; 2018. p. 119–137], a convolution structure is presented in the realm of the special affine Fourier transform. In continuation of the study, we introduce a novel integral transform coined the special affine wavelet transform by combining the merits of the well-known special affine Fourier and wavelet transforms via the special affine convolution. The preliminary analysis encompasses the derivation of fundamental properties, Moyal's principle, inversion formula and range theorem. Subsequently, we obtain a mild extension of Heisenberg's uncertainty principle and also develop an analogue of Pitt's inequality for the special affine Fourier transform. In addition, we derive a Heisenberg-type uncertainty principle for the special affine wavelet transform. Finally, we extend the scope of the present study by introducing the notion of composition of special affine wavelet transforms.

A sharp Clifford wavelet Heisenberg-type uncertainty principle

In the present work, we are concerned with the development of a new uncertainty principle based on the wavelet transform in the Clifford analysis/algebra framework. We precisely derive a sharp Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform.

Clifford Valued Shearlet Transform

This paper deals with the construction of $$n=3 \text{ mod } 4$$ Clifford algebra $$Cl_{n,0}$$ -valued admissible shearlet transform using the shearlet group $$(\mathbb {R}^* < imes \mathbb {R}^{n-1}) < imes \mathbb {R}^n$$ , a subgroup of affine group of $${\mathbb {R}}^n$$ . The admissibility conditions for a nonzero Clifford valued square integrable function have been obtained. Various properties such as reconstruction formula, orthogonality relation, isometry and reproducing kernel have been dealt. As an application, Heisenberg type uncertainty principle for Clifford shearlet transform has been derived.

Uncertainty principles for the windowed Hankel transform

ABSTRACT The aim of this paper is to prove some new uncertainty principles for the windowed Hankel transform. They include uncertainty principle for orthonormal sequence, local uncertainty principle, logarithmic uncertainty principle and Heisenberg-type uncertainty principle. As a side result, we obtain the Shapiro's dispersion theorem for the windowed Hankel transform.

Heisenberg type uncertainty principle for the Gabor spherical mean transform

We prove a Heisenberg type uncertainty principle for the Gabor spherical mean transform, and we study its generalization. Next, we extend local uncertainty principle for sets of finite measure to the latter transform.

Clifford Wavelet Transform and the Uncertainty Principle

The present paper lies in the whole topic of wavelet harmonic analysis on Clifford algebras. In which we derive a Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform. A brief review of Clifford algebra/analysis, wavelet transform on $$\mathbb {R}$$ and Clifford Fourier transform and their properties is conducted. Next, such concepts are applied to develop an uncertainty principle based on Clifford wavelets.

New results on the continuous Weinstein wavelet transform

We consider the continuous wavelet transform mathcal{S}_{h}^{W} associated with the Weinstein operator. We introduce the notion of localization operators for mathcal {S}_{h}^{W}. In particular, we prove the boundedness and compactness of localization operators associated with the continuous wavelet transform. Next, we analyze the concentration of mathcal{S}_{h}^{W} on sets of finite measure. In particular, Benedicks-type and Donoho-Stark’s uncertainty principles are given. Finally, we prove many versions of Heisenberg-type uncertainty principles for mathcal{S}_{h}^{W}.

New Studies on the Fock Space Associated to the Generalized Airy Operator and Applications

In this work, we introduce the Fock space $$F_\nu (\mathbb {C})$$ associated to the Airy operator $$L_\nu $$ , and we establish Heisenberg-type uncertainty principle for this space. Next, we study the Toeplitz operators, the Hankel operators and the translation operators on this space. Furthermore, we give an application of the theory of extremal function and reproducing kernel of Hilbert space, to establish the extremal function associated to a bounded linear operator $$T{:}\,F_\nu (\mathbb {C})\rightarrow H$$ , where H be a Hilbert space. Finally, we come up with some results regarding the extremal functions, when T is the difference operator and the Dunkl-difference operator, respectively.

Time–Frequency Concentration, Heisenberg Type Uncertainty Principles and Localization Operators for the Continuous Dunkl Wavelet Transform on $$\mathbb {R}^{d}$$ R d

We consider the continuous Dunkl wavelet transform $$\Phi ^{D}_{h} $$ associated with the Dunkl operators on $$\mathbb {R}^{d}$$ . We analyze the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks type uncertainty principles are given. Next, we prove many versions of Heisenberg type uncertainty principles for $$\Phi ^{D}_{h} $$ . Quantitative Shapiro’s dispersion uncertainty principle and umbrella theorem are proved for the Dunkl continuous wavelet transform. Finally, we investigate the localization operators for $$\Phi ^{D}_{h} $$ , in particular we prove that they are in the Schatten-von Neumann class.

The Donoho–Stark, Benedicks and Heisenberg type uncertainty principles, and the localization operators for the Heckman–Opdam continuous wavelet transform on $${\mathbb {R}}^{d}$$ R d

We consider the continuous wavelet transform \(\Phi _{h}^{W}\) associated with the Heckman–Opdam operators on \(\mathbb {R}^{d}\). We analyse the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks-type uncertainty principles are given. Next, we prove many versions of Heisenberg-type uncertainty principles for \(\Phi _{h}^{W}\). Finally, we investigate the localization operators for \(\Phi _{h}^{W}\), in particular we prove that they are in the Schatten–von Neumann class.

Uncertainty Principles for the Dunkl-Wigner Transforms

We prove a version of Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude s&gt;0; and we deduce a local uncertainty principle for this transform.

Analytical and Numerical Approximation Formulas on the Dunkl-Type Fock Spaces

In this work, we establish some versions of Heisenberg-type uncertainty principles for the Dunkl-type Fock space \(F_{k}(\mathbb {C}^{d})\). Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for operator \(L:F_{k}(\mathbb {C}^{d})\rightarrow H\), where H is a Hilbert space. Finally, we come up with some results regarding the Tikhonov regularization problem and the Heisenberg-type uncertainty principle for the Dunkl-type Segal-Bargmann transform \(\mathcal {B}_{k}\). Some numerical applications are given.

Heisenberg type uncertainty principle for continuous shearlet transform

Heisenberg type uncertainty principle for continuous shearlet transform