The Clifford Deformation of the Hermite Semigroup

Bie, Hendrik De; Ørsted, Bent; Somberg, Petr; Souček, Vladimír

doi:10.3842/sigma.2013.010

Cited by 10 publications

(18 citation statements)

References 35 publications

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“…So far, we have applied these ideas in 3 different directions of hypercomplex FTs, namely -k-vector Fourier transforms ( [14]) -radially deformed Fourier transforms ( [15,16]) -Clifford-Fourier transforms ( [17,13]). …”

Section: Overview Of Recent Resultsmentioning

confidence: 99%

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Fractional Fourier transforms of hypercomplex signals

Bie

Schepper

2012

SIViP

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An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal.Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given.

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Section: Overview Of Recent Resultsmentioning

confidence: 99%

“…In recent work (see [13][14][15][16][17]) we have developed a different methodology: we start from a list of properties or general mathematical principles we want a hypercomplex Fourier transform to have, and then determine all kernels that satisfy these properties.…”

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confidence: 99%

Fractional Fourier transforms of hypercomplex signals

Bie

Schepper

2012

SIViP

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“…It is well-known that the classical Dirac operator ∂ x = m i=1 e i ∂ x i and its Fourier symbol x = m i=1 e i x i generate via Clifford multiplication a natural Lie superalgebra osp(1|2) contained in the Clifford-Weyl algebra. More surprisingly, this carries over to a natural family of deformations of the Dirac operator (see [9,11]) which was inspired by the scalar results in [1,20]. Indeed, the radially deformed Dirac operator…”

Section: Introductionmentioning

confidence: 94%

“…The investigation of this radially deformed Fourier transform was continued in [11], using a group theoretical approach. The analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the holomorphic Hermite semigroup were established.…”

Section: Introductionmentioning

confidence: 99%

New Results on the Radially Deformed Dirac Operator

Bie

Schepper

Eelbode

2016

Complex Anal. Oper. Theory

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In recent work a deformation of the classical Dirac operator in R m was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra osp(1|2), by means of a so-called radially deformed Dirac operator D depending on a deformation parameter c, such that for c = 0 the classical Dirac operator is reobtained. In this paper, we investigate various properties of this deformation. We first determine the conformal structure of D and obtain a version of Stokes' theorem. Subsequently we derive an explicit form for the kernel of the associated Fourier transform in terms of trigonometric functions.

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“…(v) the Clifford-Fourier transform and the fractional Clifford-Fourier transform, both already mentioned above; meanwhile an entire class of CliffordFourier transforms has been thoroughly studied in [36]; (vi) the radially deformed hypercomplex Fourier transform, which appears as a special case in the theory of radial deformations of the Lie algebra osp(1|2), see [38,37], and is a topic of current research, see [32].…”

Section: The Clifford Fourier Transform In the Light Of Clifford Analmentioning

confidence: 99%

Quaternion and Clifford Fourier Transforms and Wavelets

Hitzer¹,

Sangwine²

2013

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Abstract. We survey the historical development of quaternion and Clifford Fourier transforms and wavelets. Keywords. Quaternions, Clifford Algebra, Fourier Transforms, Wavelet Transforms.The development of hypercomplex Fourier transforms and wavelets has taken place in several different threads, reflected in the overview of the subject presented in this chapter. We present in Section 1 an overview of the development of quaternion Fourier transforms, then in Section 2 the development of Clifford Fourier transforms. Finally, since wavelets are a more recent development, and the distinction between their quaternion and Clifford algebra approach has been much less pronounced than in the case of Fourier transforms, Section 3 reviews the history of both quaternion and Clifford wavelets.We recognise that the history we present here may be incomplete, and that work by some authors may have been overlooked, for which we can only offer our humble apologies. [90,91] on Clifford Fourier and Laplace transforms further explained in Section 2.2. Ernst and Delsuc's quaternion transforms were two-dimensional (that is they had two independent variables) and proposed for application to nuclear magnetic resonance (NMR) imaging. Written in terms of two independent time variables 1 t 1 and t 2 , the forward transforms 1 The two independent time variables arise naturally from the formulation of twodimensional NMR spectroscopy. Quaternion Fourier Transforms (QFT)1

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The Clifford Deformation of the Hermite Semigroup

Cited by 10 publications

References 35 publications

Fractional Fourier transforms of hypercomplex signals

Fractional Fourier transforms of hypercomplex signals

New Results on the Radially Deformed Dirac Operator

Quaternion and Clifford Fourier Transforms and Wavelets

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