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Said Fahlaoui

Publications and source records attributed to Said Fahlaoui.

16 recordsLinked to original sources
arXiv2025

The Two-Sided Clifford Dunkl Transform and Miyachi's Theorem

Recent advances have extended the Dunkl transform to the setting of Clifford algebras. In particular, the two-sided quaternionic Dunkl transform has been introduced as a Dunkl analogue of the two-dimensional quaternionic Fourier transform. In this paper, we develop the two-sided Clifford Dunkl transform, defined using two square roots of -1 in Cl_{p,q}. We establish its fundamental properties, including the inversion and Plancherel formulas, and provide two explicit expressions for the associated translation operator. Moreover, we prove an analogue of Miyachi's theorem for this transform, thereby extending a classical result in harmonic analysis to the Clifford-Dunkl framework.

math.FA↗
arXiv2021

Titchmarsh theorem associated with QFT

In this paper, we prove both of the Titchmarsh theorems associated with Two-Sided Quaternionic Fourier Transform and we conclude about Short-Time Two-Sided Quaternionic Fourier Transform.

math.CA↗
arXiv2020

Benedicks-Amrein-Berthier type theorem related to the two-sided Quaternion Fourier transform

The main objective of the present paper is to establish a new uncertainty principle (UP) for the two-sided quaternion Fourier transform (QFT). This result is an extension of a result of Benedicks, Amrein and Berthier, which states that a nonzero function in $L^1\left({\mathbb{R}}^2, {\mathbb{H}}\right)$ and its two-sided QFT cannot both have support of finite measure.

math.CA↗
arXiv2020

The Continuous quaternion Algebra-Valued Wavelet Transform and the Associated Uncertainty Principle

The purpose of this article is to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform (QFT). We study some fundamental properties of this extension such as scaling, translation, rotation, Parseval's identity, inversion theorem, and a reproducing kernel, 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 and Hardy's UP to the CQWT domain.

math.CA↗
arXiv2019

Generalized Uncertainty Principles associated with the Quaternionic Offset Linear Canonical Transform

The quaternionic offset linear canonical transform (QOLCT) can be thought as a generalization of the quaternionic linear canonical transform (QLCT). In this paper we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact we prove the Plancherel formula, and some properties related to the QOLCT, then we generalize some different uncertainty principles (UPs), including Heisenberg-Weyls UP, Hardys UP, Beurlings UP, and logarithmic UP to the QOLCT domain in a broader sense

math.CA↗
arXiv2019

Uncertainty Principles For the continuous Gabor quaternion linear canonical transform

Gabor transform is one of the performed tools for time-frequency signal analysis. The principal aim of this paper is to generalize the Gabor Fourier transform to the quaternion linear canonical transform. Actually, this transform gives us more flexibility to studied nonstationary and local signals associated with the quaternion linear canonical transform. Some useful properties are derived, such as Plancherel and inversion formulas. And we prove some uncertainty principles: those including Heisenberg's, Lieb's and logarithmic inequalities. We finish by analogs of concentration and Benedick's type theorems.

math.CA↗
arXiv2019

Donoho-Stark's Uncertainty Principles in Real Clifford Algebras

The Clifford Fourier transform (CFT) has been shown to be a powerful tool in the Clifford analysis. In this work, several uncertainty inequalities are established in the real Clifford algebra $Cl_{(p,q)}$, \ including the Hausdorf-Young inequality, and three qualitative uncertainty principles of Donoho-Stark.

math.CA↗
arXiv2018

Helgason Gabor Fourier transform and uncertainty principles

Windowing a Fourier transform is a useful tool, which gives us the similarity between the signal and time frequency signal, and it allows to get sense when/where ceratin frequencies occur in the input signal, this method is introduced by Dennis Gabor. In this paper, we generalize the classical Gabor-Fourier transform(GFT) to the Riemannian symmetric space called the Helgason Gabor Fourier transform (HGFT). We continue with proving several important properties of HGFT, like the reconstruction formula, the Plancherel formula, and Parseval formula. Finally we establish some local uncertainty principle such as Benedicks-type uncertainty principle

math.CA↗
arXiv2018

A heat kernel version of Miyachi's Theorem for the Laguerre hypergroup

Let $\mathbb{K}=[0,+\infty[\times\mathbb{R}$ the Laguerre Hypergroup. In this paper, we are going to formulate and prove an analogue of Miyachi's uncertainty principle for the Laguerre-Hypergroup Fourier transform. Our version will be in terms of the heat kernel associated to the radial part of the sub-Laplacian on the Heisenberg group.

math.CA↗
arXiv2018

Wigner-Ville distribution associated with the quaternion offset linear canonical transforms

The Wigner-Ville distribution (WVD) and quaternion offset linear canonical transform (QOLCT) are a useful tools in signal analysis and image processing. The purpose of this paper is to define the Wigner-Ville distribution associated with quaternionic offset linear canonical transform (WVD-QOLCT). Actually, this transform combines both the results and flexibility of the two transform WVD and QOLCT. We derive some important properties of this transform such as inversion and Plancherel formulas, we establish a version of Heisenberg inequality, Lieb's theorem and we give the Poisson summation formula for the WVD-QOLCT.

math.CA↗
arXiv2018

The two-sided Gabor quaternion Fourier transform and some uncertainty principles

In this paper, we define a new transform called the Gabor quaternionic Fourier transform (GQFT), which generalizes the classical windowed Fourier transform to quaternion valued-signals, we give several important properties such as the Plancherel formula and inversion formula. Finally, we establish the Heisenberg uncertainty principles for the GQFT.

math.CA↗
arXiv2017

Beurling's Theorem for the Two-sided Quaternion Fourier Transform

The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling's theorem, Hardy, Cowling-Price and Gelfand-Shilov theorems, is obtained for the two-sided quaternion Fourier transform.

math.CA↗