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... 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 $\mathfrak{osp}(4|2)$-monogenic polynomials, the Lie superalgebra $\mathfrak{osp}(4|2)$ being the Howe dual partner to the symplectic group Sp$( p)$. In order to obtain Sp$( p)$-irreducibility this new concept of $\mathfrak{osp}(4|2)$-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 $\mathbb{E}$ underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator $P$ underlying the decomposition of spinor space into symplectic cells. These operators $\mathbb{E}$ and $P$, and their hermitian conjugates, arise naturally when constructing the Howe dual pair $\mathfrak{osp}(4|2) \times$ Sp$( p)$, the action of which will make the Fischer decomposition multiplicityfree.
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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... 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...
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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... 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.
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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... 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.
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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... 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...
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Clifford analysis may be regarded as a higher-dimensional analogue of the theory of holomorphic functions in the complex plane. It has proven to be an appropriate framework for higher-dimensional continuous wavelet transforms, based on... more
Clifford analysis may be regarded as a higher-dimensional analogue of the theory of holomorphic functions in the complex plane. It has proven to be an appropriate framework for higher-dimensional continuous wavelet transforms, based on specific types of multi-dimensional orthogonal polynomials, such as the Clifford-Hermite polynomials, which form the building blocks for so-called Clifford-Hermite wavelets, offering a refinement of the traditional Marr wavelets. In this paper, a generalization of the Clifford-Hermite polynomials to a two-parameter family is obtained by taking the double monogenic extension of a modulated Gaussian, i.e. the classical Morlet wavelet. The eventual goal being the construction of new Clifford wavelets refining the Morlet wavelet, we first investigate the properties of the underlying polynomials.
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This is a very readable exposition of the Hilbert transform on smooth closed hypersurfaces in higher dimensions, containing not only an overview but also results by the authors using Clifford analysis. The Hilbert transform on a smooth... more
This is a very readable exposition of the Hilbert transform on smooth closed hypersurfaces in higher dimensions, containing not only an overview but also results by the authors using Clifford analysis. The Hilbert transform on a smooth hypersurface is defined via the Cauchy integral and this leads to non-tangential boundary values of monogenic (holomorphic) functions in the inner and outer domain of the surface. New operators similar to the Cauchy transform can be defined and new relations between these operators are given, parallel to the classical Kerzman-Stein formulas. At last also the Dirichlet problem is dealt with, it is closely related to the Hilbert transform and the Hardy space. A harmonic function in a neighborhood of the surface is constructed, taking given values on the surface, but a Poisson formula could not be proved.
Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has... more
Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.
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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... more
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.
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An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centred around the simultaneous null... more
An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centred around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators.
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1 Facultad de Informática y Matemática, Universidad de Holguın, Holguın 80100, Cuba 2 Departamento de Matemática, Universidad de Oriente, Santiago de Cuba 90500, Cuba 3 Department of Mathematical Analysis, Faculty of Engineering, Ghent... more
1 Facultad de Informática y Matemática, Universidad de Holguın, Holguın 80100, Cuba 2 Departamento de Matemática, Universidad de Oriente, Santiago de Cuba 90500, Cuba 3 Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Ghent, Belgium 4 ...
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Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane.... more
Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane. So-called quaternionic monogenic functions satisfy a system of first order linear differential equations expressed in terms of four interrelated Dirac operators. The conceptual significance of quaternionic Clifford analysis is unraveled by showing that quaternionic monogenicity can be characterized by means of generalized gradients in the sense of Stein and Weiss. At the same time, connections between quaternionic monogenic functions and other branches of Clifford analysis, viz Hermitian monogenic and standard or Euclidean monogenic functions are established as well.
Hermitean Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean conjugate complex Dirac operators. As an essential step towards... more
Hermitean Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitean monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitean monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitean monogenics, i.e. homogeneous Hermitean monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.
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The Mehler formula for the Hermite polynomials allows for an integral representation of the one–dimensional Fractional Fourier transform. In this paper, we introduce a multi–dimensional Fractional Fourier transform in the framework of... more
The Mehler formula for the Hermite polynomials allows for an integral representation of the one–dimensional Fractional Fourier transform. In this paper, we introduce a multi–dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials of Clifford analysis.
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In this note we describe explicitly irreducible decompositions of kernels of the Hermitean Dirac Operators. In [6], it is shown that these decompositions are essential for a construction of orthogonal (or even Gelfand-Tsetlin) bases of... more
In this note we describe explicitly irreducible decompositions of kernels of the Hermitean Dirac Operators. In [6], it is shown that these decompositions are essential for a construction of orthogonal (or even Gelfand-Tsetlin) bases of homogeneous Hermitean monogenic polynomials.