Uncertainty Principles for the Clifford–Fourier Transform

Kamel, Jamel El; Jday, Rim

doi:10.1007/s00006-017-0791-1

Cited by 6 publications

(7 citation statements)

References 20 publications

(24 reference statements)

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“…Lemma (El Kamel and Jday 13, Lemma 3.2 ) Let

m

be even and

C

a positive constant. Then

{\left\Vert {K}_{\pm}\left(x,y\right)\right\Vert}_c\le C{e}&#x0005E;{{\left\Vert x\right\Vert}_c{\left\Vert y\right\Vert}_c},\kern2em \forall x,y\in {\mathbb{R}}&#x0005E;m.

…”

Section: The Clifford‐fourier Transformunclassified

“…Theorem (El Kamel and Jday 13, Theorem 3.5 ) Let

a&gt;0

and

P\in {\mathcal{P}}_k

. Then, there exists

Q\in {\mathcal{P}}_k

satisfying

$$ {\mathcal{F}}_{\pm}\left(P(.

…”

Section: The Clifford‐fourier Transformunclassified

“…In this context, many uncertainty principles for the CFT were proved, such as the Hardy theorem and the Heisenberg inequality. [11][12][13] However, the Cowling and Price theorem still needs to be shown in the Clifford analysis setting, particularly in quaternionic analysis. Furthermore, Beurling's theorem, which is the most relevant one since it yields various other uncertainty principle theorems, is not provided.…”

Section: Introductionmentioning

confidence: 99%

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Beurling's theorem for the Clifford‐Fourier transform

Jday

Kamel

2023

Math Methods in App Sciences

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“…Lemma (El Kamel and Jday 13, Lemma 3.2 ) Let

m

be even and

C

a positive constant. Then

{\left\Vert {K}_{\pm}\left(x,y\right)\right\Vert}_c\le C{e}&#x0005E;{{\left\Vert x\right\Vert}_c{\left\Vert y\right\Vert}_c},\kern2em \forall x,y\in {\mathbb{R}}&#x0005E;m.

…”

Section: The Clifford‐fourier Transformunclassified

“…Theorem (El Kamel and Jday 13, Theorem 3.5 ) Let

a&gt;0

and

P\in {\mathcal{P}}_k

. Then, there exists

Q\in {\mathcal{P}}_k

satisfying

$$ {\mathcal{F}}_{\pm}\left(P(.

…”

Section: The Clifford‐fourier Transformunclassified

Section: Introductionmentioning

confidence: 99%

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Beurling's theorem for the Clifford‐Fourier transform

Jday

Kamel

2023

Math Methods in App Sciences

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“…where the kernel K ± (x, y) is given by an explicit expression. For this kind of Clifford-Fourier transform many generalizations were found, see [11,12,4,13,7] and some important properties such as the uncertainty principle and Riemann-Lebesgue lemma were proved [19,20].…”

Section: Introductionmentioning

confidence: 99%

On the Clifford short-time Fourier transform and its properties

Martino¹

2021

Preprint

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In this paper we investigate how the short-time Fourier transform can be extended in a Clifford setting. We prove some of the main properties of the Clifford short-time Fourier transform such as the orthogonality relation, the reconstruction property and the reproducing kernel formula. Moreover, we show the effects of modulating and translating the signal and the window function, respectively. Finally, we demonstrate the Lieb's uncertainty principle for the Clifford short-time Fourier transform.

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“…Miyachi's theorem has been extended in several different directions in recent years, including extensions to Dunkl transform [5], Clifford-Fourier transform [9], and much more generally, to nilpotent lie groups [1] and Heisenberg motion groups [2].…”

Section: Introductionmentioning

confidence: 99%

Miyachi’s Theorem for the Quaternion Fourier Transform

Haoui

Fahlaoui

2019

Circuits Syst Signal Process

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The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Euclidean Fourier transform. In this paper, we establish Miyachi's theorem for this transform and consequently generalize and prove the analogue of Hardy's theorem and Cowling-Price uncertainty principle in the QFT domain.1 done, that is the so-called left-sided, right-sided and the two-sided QFT. The QFT was introduced at first by Ell [10] for the analysis linear time-invariant partial differential systems and then applied in color image processing. Later, Blow [3] investigated the important properties of the two-sided QFT for real signals and applied it to signal and image processing. Furthermore, several uncertainty principles have been formulated for the quaternion Fourier transform.In [15], the authors generalized a component-wise UP for the right-sided QFT. The directional UP related to the two-sided QFT was proposed in [12]. Recently,in [4], the authors established logarithmic UP associated with the QFT. Meanwhile, Mawardi [16] obtained the connection between the QFT and quantum mechanics and then established the modified UP (full UP) for the two-sided QFT.Our contribution to these developments is that we propose a new UP for the QFT, namely Miyachi's UP. So far, no such uncertainty principle for the QFT (one-sided or two-sided) had been established. In our previous works, other UPs: Heisenberg, Hardy[7], and Beurling[8], have been extended for the two-sided QFT. Also, we derived in [8] the UPs of Cowling-Price and Hardy using the extension of Beurling theorem in a quaternion framework. Here, we will obtain, in a different way, by the main result of Miyachi, the same UPs of Cowling-Price and Hardy in QFT domain. The techniques used here are also applicable for the left-sided and the right-sided QFT as well. Our paper is organized as follows. In Sect. 2, we review some basic notions and notations related to the quaternion algebra. In Sect. 3, we recall the definition and some results for the quaternion Fourier transform useful in the sequel. In Sect. 4, we prove Miyachis theorem for the quaternion Fourier transform, and provide an extension of certain UPs to the quaternion Fourier transform domain. In Sect. 5, we conclude the paper.

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Uncertainty Principles for the Clifford–Fourier Transform

Abstract: Abstract. In this paper, we estabish an analogue of Hardy's theorem and Miyachi's theorem for the Clifford-Fourier transform.

Cited by 6 publications

References 20 publications

Beurling's theorem for the Clifford‐Fourier transform

Beurling's theorem for the Clifford‐Fourier transform

On the Clifford short-time Fourier transform and its properties

Miyachi’s Theorem for the Quaternion Fourier Transform

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