Fourier Transforms in Clifford Analysis

Bie, Hendrik De

doi:10.1007/978-3-0348-0667-1_12

Cited by 9 publications

(7 citation statements)

References 43 publications

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“…Indeed in this case it is not known if it is possible to write K − (x, y) as sums of Bessel functions. Moreover, it seems not possible to obtain an upper bound of the kernel as in (10). So in the rest of the paper we focus only on the even dimensions more than two.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

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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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Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Moreover, in [6] the authors explained that some hypercomplex signals are useful tools for extracting intrinsically 1D-features from images. De Bie H. explains in [10] that nowadays the emphasis of the research is on three different methods: the eigenfunction approach, the generalized roots of −1 approach and the spin group approach.…”

Section: Introductionmentioning

confidence: 99%

On the Clifford short-time Fourier transform and its properties

Martino¹

2021

Preprint

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“…Introduction. In this decade many integral transforms have been extended to the quaternionic and Clifford algebras, see for example [21,24,31,36]. One of the motivations behind the study of such integral transforms in non-commutative settings is that one can deal with n-dimensional signals.…”

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confidence: 99%

On the polyanalytic short-time Fourier transform in the quaternionic setting

Martino

Diki

2022

CPAA

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<p style='text-indent:20px;'>In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.</p>

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“…In this decade many integral transforms have been extended to the quaternionic and Clifford algebras, see for example [19,22,29,34]. One of the motivations behind the study of such integral transforms in non-commutative se ings is that one can deal with n-dimensional signals.…”

mentioning

confidence: 99%

On the polyanalytic short-time Fourier transform in the quaternionic setting

Martino¹,

Diki²

2021

Preprint

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A. In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. ese results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.

show abstract

Fourier Transforms in Clifford Analysis

Cited by 9 publications

References 43 publications

On the Clifford short-time Fourier transform and its properties

On the Clifford short-time Fourier transform and its properties

On the polyanalytic short-time Fourier transform in the quaternionic setting

On the polyanalytic short-time Fourier transform in the quaternionic setting

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