Fred Brackx - Academia.edu (original) (raw)

Papers by Fred Brackx

Simon Stevin a Quarterly Journal of Pure and Applied Mathematics, 1993

Mathematical Methods in the Applied Sciences, 2016

ABSTRACT Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valu... more ABSTRACT Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp$( p)$. These Fischer decompositions involve spaces of homogeneous, so-called mathfrakosp(4∣2)\mathfrak{osp}(4|2)mathfrakosp(4∣2)-monogenic polynomials, the Lie superalgebra mathfrakosp(4∣2)\mathfrak{osp}(4|2)mathfrakosp(4∣2) being the Howe dual partner to the symplectic group Sp$( p)$. In order to obtain Sp$( p)$-irreducibility this new concept of mathfrakosp(4∣2)\mathfrak{osp}(4|2)mathfrakosp(4∣2)-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator mathbbE\mathbb{E}mathbbE underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator PPP underlying the decomposition of spinor space into symplectic cells. These operators mathbbE\mathbb{E}mathbbE and PPP, and their hermitian conjugates, arise naturally when constructing the Howe dual pair mathfrakosp(4∣2)times\mathfrak{osp}(4|2) \timesmathfrakosp(4∣2)times Sp$( p)$, the action of which will make the Fischer decomposition multiplicityfree.

ABSTRACT Taylor &amp; Francis makes every effort to ensure the accuracy of all the inform... more ABSTRACT Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the &quot;Content&quot;) contained in the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &amp;

n het academiejaar 2013-2014 is, op initiatief van directeur onderwijsaangelegenheden prof. dr. K... more n het academiejaar 2013-2014 is, op initiatief van directeur onderwijsaangelegenheden prof. dr. K. Versluys, aan de Universiteit Gent een universiteitsbreed honoursprogramma van start gegaan. Zij was de eerste en is tot hiertoe de enige Belgische universiteit die zoiets heeft gedaan. Het programma richt zich op studenten die, naast het volgen van hun regulier programma, een bijkomende intellectuele uitdaging willen aangaan. Vanuit het principe van gelijke kansen wil de Universiteit Gent ook aan goede studenten mogelijkheden bieden om het maximum uit hun capaciteiten te halen.

$Research paper thumbnail of A Hilbert transform for hermitean matrix functions on fractal domains$

Complex Analysis and Operator Theory, 2012

Journal of Mathematical Imaging and Vision

Bulletin of the Belgian Mathematical Society Simon Stevin, 2008

Clifford analysis is a higher dimensional function theory offering a refinement of classical harm... more Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing a higher dimensional continuous wavelet transform theory. In this setting a very specific construction of the wavelets has been established, encompassing all dimensions at once as opposed to the usual tensorial approaches, and being based on generalizations to higher dimension of classical orthogonal polynomials on the real line, such as the radial Clifford-Hermite polynomials, leading to Clifford-Hermite wavelets. More recently, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the orthogonal case. In this new setting a Clifford-Hermite continuous wavelet transform has already been introduced in earlier work, its norm preserving character however being expressed in terms of suitably adapted scalar valued inner products on the respective L 2 -spaces of signals and of transforms involved. In this contribution we present an alternative Hermitian Clifford-Hermite wavelet theory with Clifford algebra valued inner products, based on an orthogonal decomposition of the space of square integral functions, which is obtained by introducing a new Hilbert transform in the Hermitian setting.

Journal of Physics: Conference Series, 2015

Among the mathematical models suggested for the receptive field profiles of the hu- man visual sy... more Among the mathematical models suggested for the receptive field profiles of the hu- man visual system, the Gabor model is well-known and widely used. Another less used model that agrees with the Gaussian derivative model for human vision is the Hermite model, which is based on analysis filters of the Hermite transform. It offers some advantages like being an orthogonal basis and having better match to experimental physiological data. In our earlier research both filter models, Gabor and Hermite, have been developed in the framework of Clifford analysis. Clifford analysis offers a direct, elegant and powerful general- ization to higher dimension of the theory of holomorphic functions in the complex plane. In this paper we expose the construction of the Hermite and Gabor filters, both in the classical and in the Clifford analysis framework. We also generalize the concept of complex Gaussian derivative filters to the Clifford analysis setting. Moreover, we present further properties of...

Advances in Applied Clifford Algebras, Feb 1, 2001

Specific wavelet kernel functions for a continuous wavelet transform in Euclidean space are const... more Specific wavelet kernel functions for a continuous wavelet transform in Euclidean space are constructed in the framework of Clifford analysis. Their relationship with the heat equation and a newly introduced wavelet differential equation is established.

The authors describe the construction of a wide class of specific multidimensional wavelet kernel... more The authors describe the construction of a wide class of specific multidimensional wavelet kernel functions within the framework of Clifford analysis. The presented bi-axial Clifford-Hermite wavelets have an elliptic shape with two parameters and are a refinement of the previously introduced circular Clifford-Hermite wavelets. The corresponding continuous wavelet transforms allow an improved shape and orientation analysis, e.g., in image processing.

In this paper boundary value problems for quaternionic Hermitian monogenic functions are presente... more In this paper boundary value problems for quaternionic Hermitian monogenic functions are presented using a circulant matrix approach. MSC: 30G35

In this paper we devise a new multi-dimensional integral transform within the Clifford analysis s... more In this paper we devise a new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier-Bessel transform. It appears that in the two-dimensional case, it coincides with the Clifford-Fourier and cylindrical Fourier transforms introduced ear- lier. We show that this new integral transform satisfies operational formulae which are similar to those of the classical tensorial Fourier transform. Moreover theL2-basis elements consis- ting of generalized Clifford-Hermite functions appear to be eigenfunctions of the Fourier-Bessel transform.

Advances in Applied Clifford Algebras, 2011

The polynomial null solutions are studied of the higher spin Dirac operator Q k,l acting on funct... more The polynomial null solutions are studied of the higher spin Dirac operator Q k,l acting on functions taking values in an irreducible representation space for Spin(m) with highest weight (

Euclidean Clifford analysis is a higher dimensional functio n theory offering a refine- ment of c... more Euclidean Clifford analysis is a higher dimensional functio n theory offering a refine- ment of classical harmonic analysis. The theory is centeredaround the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential op- erator called the Dirac operator, which factorizes the Lapla cian. More recently, Hermitean Clifford analysis has emerged as a new and successful branch o f Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simu ltaneous null solutions, called Hermitean (or h-) monogenic functions, of two Hermitean Dira c operators which are invari- ant under the action of the unitary group. In Euclidean Cliffo rd analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the functi on theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Previously, a Her- mitean Clifford-Cauchy integral formula has been establishe d...