Continuous Inversion Formulas for Multi-Dimensional Stockwell Transforms

Riba, L.; Wong, M. W.

doi:10.1051/mmnp/20138117

Cited by 10 publications

(4 citation statements)

References 17 publications

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“…In this section, we present the Continuous quaternion stockwell transform and we establish some new results (Parseval formula, inversion formula,Lieb inequality,..). For more details on stockwell transform, the reader can see [7,8]. Let I 2 denote the (2, 2)-identity matrix and 0 2 , resp.…”

Section: Continuous Quaternion Stockwell Transformmentioning

confidence: 99%

Continuous quaternion Stockwell transform and Uncertainty principle

Kamel,

Tefjeni

2019

Preprint

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In this paper we propose a novel transform called continuous quaternionic stockwell transform. We express the admissibility condition in term of the (two-sided) quaternion Fourier transform . We show that its fundamental properties, such as Plancherel, Parseval and inversion formula, can be established whenever the continuous quaternion stockwell satisfy a particular admissibility condition. We present several examples of the continuous quaternion stockwell transform. We apply the continuous quaternion stockwell transform properties and the two sided quaternion Fourier transform to establish a number of uncertainty principle for these extended Stockwell .

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Section: Continuous Quaternion Stockwell Transformmentioning

confidence: 99%

Continuous quaternion Stockwell transform and Uncertainty principle

Kamel,

Tefjeni

2019

Preprint

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“…By adopting the progressive resolution of wavelets, the Stockwell transform is able to resolve a wider range of frequencies than the ordinary STFT and by using a Fourier-like basis and maintaining a phase of zero about the time t = 0, Fourier-based analysis could be performed locally. This unique feature of the Stockwell transform makes it a highly valuable tool for signal processing and is one of the hottest research areas of the contemporary era [52,40,41,45,46,51]. We note that this transform has been successfully used to analyse signals in numerous applications, such as seismic recordings, ground vibrations, geophysics, medical imaging, hydrology, gravitational waves, power system analysis and many other areas.…”

Section: Introductionmentioning

confidence: 99%

Generalized translation operator and uncertainty principles associated with the deformed Stockwell transform

Mejjaoli

2023

Revista De La Unión Matemática Argentina

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We study the generalized translation operator associated with the deformed Hankel transform on R. Firstly, we prove the trigonometric form of the generalized translation operator. Next, we derive the positivity of this operator on a suitable space of even functions. Making use of the positivity of the generalized translation operator we introduce and study the deformed Stockwell transform. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative uncertainty principles for this new integral transform. Firstly, we mainly establish various versions of Heisenberg's uncertainty principles. Secondly, we derive some weighted uncertainty inequalities such as Pitt's and Beckner's uncertainty inequalities for the deformed Stockwell transform. We culminate our study by formulating several concentration-based uncertainty principles, including the Amrein-Berthier-Benedicks and local inequalities for the deformed Stockwell transform.

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“…Many extensions of the Stockwell transform have been proposed in the last years (see for example [3,7,8,9,12,13,14]). For other aspects or some applications of the Stockwell transform we refer to [1,2,5].…”

Section: Introductionmentioning

confidence: 99%

Multilinear Stockwell transforms

Catană

2018

Math. Model. Nat. Phenom.

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The main aim of this paper is to introduce multilinear versions of the Stockwell transforms (also named S-transforms) by using the fact that S-transforms can be written as convolution products. Further on we extend the multilinear S-transforms from the Schwartz class of rapidly decreasing functions to the space of tempered distributions. In the sequel we give a relation between multilinear S-transforms and multilinear pseudo-differential operators. We also state and prove some boundedness results regarding multilinear S-transforms on the Lebegue’s spaces Lp(Rn) and also on the Hörmander’s spaces Bp,k(Rn), where p ≥ 1 and k is a temperate weight function. In the end, a weak uncertainty principle for multilinear S-transforms and for its adjoint is also given.

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Continuous Inversion Formulas for Multi-Dimensional Stockwell Transforms

Cited by 10 publications

References 17 publications

Continuous quaternion Stockwell transform and Uncertainty principle

Continuous quaternion Stockwell transform and Uncertainty principle

Generalized translation operator and uncertainty principles associated with the deformed Stockwell transform

Multilinear Stockwell transforms

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