arXiv subjects

Hendrik De Bie

Publications and source records attributed to Hendrik De Bie.

At least 55 records · Page 3Linked to original sources

arXiv2015

Slice Fourier transform and convolutions

Recently the construction of various integral transforms for slice monogenic functions has gained a lot of attention. In line with these developments, the article at hand introduces the slice Fourier transform. In the first part, the kernel function of this integral transform is constructed using the Mehler formula. An explicit expression for the integral transform is obtained and allows for the study of its properties. In the second part, two kinds of corresponding convolutions are examined: Mustard convolutions and convolutions based on generalised translation operators. The paper finishes by demonstrating the connection between both.

math.CV↗

arXiv2015

Algebraic approach to slice monogenic functions

In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.

math.CV↗

arXiv2015

A new construction of the Clifford-Fourier kernel

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.

math.CA↗

arXiv2015

Connecting spatial and frequency domains for the quaternion Fourier transform

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined `Mustard' convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution.

math.CA↗

arXiv2015

The higher spin Laplace operator

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as well as a second order analogue of the Rarita-Schwinger operator. To construct these operators, we will use the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the orthogonal group can be realised in terms of polynomials satisfying a system of differential equations. As a consequence, the functions on which this particular class of operators act are functions taking values in the space of harmonics homogeneous of degree k. We prove the ellipticity of these operators and use this to investigate their kernel, focusing on both polynomial solutions and the fundamental solution.

math-ph↗

arXiv2015

A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra

The Dirac-Dunkl operator on the 2-sphere associated to the $\mathbb{Z}_2^3$ reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

math-ph↗

arXiv2014

The Bannai-Ito algebra and some applications

The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the $sl_{-1}(2)$ algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the 2-sphere are surveyed.

math-ph↗

arXiv2014

Representation of Distributions by Harmonic and Monogenic Potentials in Euclidean Space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of (m+1)-dimensional Euclidean space was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane, and their distributional boundary values were computed. In this paper we determine these potentials in lower half-space, and investigate whether they can be extended through the boundary R^m. This is a stepping stone to the representation of a doubly infinite sequence of distributions in R^m, consisting of positive and negative integer powers of the Dirac and the Hilbert-Dirac operators, as the jump across R^m of monogenic functions in the upper and lower half-spaces, in this way providing a sequence of interesting examples of Clifford hyperfunctions.

math.FA↗

arXiv2013

The kernel of the radially deformed Fourier transform

The radially deformed Fourier transform, introduced in [S. Ben Said, T. Kobayashi and B. Orsted, Laguerre semigroup and Dunkl operators, Compositio Math.], is an integral transform that depends on a numerical parameter $a \in R^{+}$. So far, only for $a=1$ and $a=2$ the kernel of this integral transform is determined explicitly. In the present paper, explicit formulas for the kernel of this transform are obtained when the dimension is even and $a = 2/n$ with $n \in N$. As a consequence, it is shown that the integral kernel is bounded in dimension 2.

math.CA↗

arXiv2013

Generating functions of orthogonal polynomials in higher dimensions

In this paper two important classes of orthogonal polynomials in higher dimensions using the framework of Clifford analysis are considered, namely the Clifford-Hermite and the Clifford-Gegenbauer polynomials. For both classes an explicit generating function is obtained.

math.CV↗

arXiv2013

Conformal symmetries of the super Dirac operator

In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair osp(1|2) x osp(m|2n) < osp(m + 4n|2m + 2n). Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries.

math-ph↗

arXiv2013

Convolution products for hypercomplex Fourier transforms

Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.

math.CA↗

arXiv2013

The Clifford Deformation of the Hermite Semigroup

This paper is a continuation of the paper [arXiv:0911.4725], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [arXiv:0907.3749]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.

math.CA↗

arXiv2012

Harmonic and Monogenic Potentials in Low Dimensional Euclidean Half-Space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^{m+1} was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary R^{m} were also determined. In this paper the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down.

math.CA↗

arXiv2012

On a Chain of Harmonic and Monogenic Potentials in Euclidean Half-space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space $\mR^{m+1}$, including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary $\mR^{m}$ turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit.

math.CA↗

arXiv2012

Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space $\mR^{m+1}$ was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at the boundary $\mR^{m}$ are crucial. The remarkable relationship between these distributional boundary values and four basic pseudodifferential operators linked with the Dirac and Laplace operators is studied.

math.CA↗

arXiv2012

Clifford algebras, Fourier transforms and quantum mechanics

In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel and connection with various special functions.

math-ph↗

arXiv2012

The fractional Clifford-Fourier transform

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.

math.CA↗