F. Brackx
Ghent University, Mathematical Analysis, Emeritus
ABSTRACT In earlier research generalized multidimensional Hilbert transforms have been constructed in $\mathbb{R}^m$ in the framework of Clifford analysis, a generalization to higher dimension of the theory of holomorphic functions in the... more
ABSTRACT In earlier research generalized multidimensional Hilbert transforms have been constructed in $\mathbb{R}^m$ in the framework of Clifford analysis, a generalization to higher dimension of the theory of holomorphic functions in the complex plane. These Hilbert transforms, obtained as part of the boundary value of an associated Cauchy transform in $\mathbb{R}^{m+1}$, might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one. In this paper we adopt the idea of a so--called anisotropic Clifford setting, leading to the introduction of a metric dependent Hilbert transform in $\mathbb{R}^m$, which formally shows similar properties as the isotropic one, but allows to adjust the co-ordinate system to preferential directions. A striking fact is that the associated Cauchy transform in $\mathbb{R}^{m+1}$ is no longer uniquely determined, but may correspond to various $(m+1)$--dimensional metrics.
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ABSTRACT The one-dimensional continuous wavelet transform (CWT) is a successful tool in signal and image analysis, with numerous applications (see e.g. [8, 9]). Standard (or orthogonal) Clifford analysis is a higher dimensional function... more
ABSTRACT The one-dimensional continuous wavelet transform (CWT) is a successful tool in signal and image analysis, with numerous applications (see e.g. [8, 9]). Standard (or orthogonal) Clifford analysis is a higher dimensional function theory which has proven to constitute an appropriate framework for developing higher dimensional CWTs, where all dimensions are encompassed at once, as opposed to tensorial approaches with products of onehyp-dimensional phenomena; the specific construction of higher dimensional wavelets is based on particular families of orthogonal polynomials, see e.g. [4, 5, 6, 7]. We explicitly mention the generalized Clifford-Hermite polynomials, introduced in [10] and applied to wavelet analysis in [7]. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, refining the orthogonal case, see [1]. Hermitean Clifford-Hermite polynomials and their associated families of wavelet kernels ! were constructed in [2, 3]. In this contribution, we introduce generalized Hermitean Clifford-Hermite polynomials, involving in their definition Hermitean spherical monogenics, the ultimate goal being new generalized continuous wavelet transforms.
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The basic object type in this program is a vector: a linear combination of the three unit vectors el, e2, e3 that form an orthonormal right-handed basis. A vector takes the form: al*e(1) +a2*e(2) +a3*e(3) where the aj, j E {I, 2, 3} are... more
The basic object type in this program is a vector: a linear combination of the three unit vectors el, e2, e3 that form an orthonormal right-handed basis. A vector takes the form: al*e(1) +a2*e(2) +a3*e(3) where the aj, j E {I, 2, 3} are scalar expressions. The unit vectors e(j) are defined by means of an operator, e, which is declared noncommutative for reasons that will be explained soon: operator e; noncom e; Extracting a cartesian co-ordinate out of a vector can be done using camp: procedure comp (j, u); df (u, e (j)) ; Next to addition and scalar multiplication, the ...
Research Interests: Mathematics and Geometry
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ABSTRACT We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic... more
ABSTRACT We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes’ theorem, Cauchy’s theorem and a Cauchy integral formula are established.
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ABSTRACT As an intrinsically multidimensional function theory, Clifford analysis offers a framework which is particularly suited for the integrated treatment of higher-dimensional phenomena. In this paper a detailed account is given of... more
ABSTRACT As an intrinsically multidimensional function theory, Clifford analysis offers a framework which is particularly suited for the integrated treatment of higher-dimensional phenomena. In this paper a detailed account is given of results connected to the Hilbert transform on the unit sphere in Euclidean space and some of its related concepts, such as Hardy spaces and the Cauchy integral, in a Clifford analysis context.
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Research Interests: Mathematics, Applied Mathematics, Pure Mathematics, Wavelet, Wavelet Analysis, and 10 moreHermite Polynomials, Mathematical Analysis and Applications, Technology and Engineering, Indexation, Multi Dimensional, Electrical And Electronic Engineering, Orthogonal Polynomial, Clifford analysis, Continuous wavelet transform, and building block
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A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions... more
A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. Consequently, appropriate orthogonal polynomials on the real axis give rise to Clifford algebra-valued orthogonal polynomials in the unit ball. Three specific examples of such orthogonal polynomials in the unit ball are discussed, namely, the generalized Clifford-Jacobi polynomials, the generalized Clifford-Gegenbauer polynomials, and the shifted Clifford-Jacobi polynomials.
Research Interests: Mathematics, Applied Mathematics, Mathematical Physics, Computer Science, Orthogonal polynomials, and 10 morePure Mathematics, Mathematics and Statistics, Clifford algebra, Euclidean space, Laguerre polynomial, Boolean Satisfiability, Orthogonal Polynomial, Clifford analysis, Continuous wavelet transform, and Jacobi polynomials
In the Clifford analysis context a specific type of solution for the higher spin Dirac operators [Formula: see text] is studied; these higher spin Dirac operators can be seen as generalizations of the classical Rarita–Schwinger operator.... more
In the Clifford analysis context a specific type of solution for the higher spin Dirac operators [Formula: see text] is studied; these higher spin Dirac operators can be seen as generalizations of the classical Rarita–Schwinger operator. To that end subspaces of the space of triple monogenic polynomials are introduced and their algebraic structure is investigated. Also a dimensional analysis is carried out.
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ABSTRACT Let Ω be a bounded, simply connected domain in the complex plane, with C smooth boundary and take u ∈ C∞(∂Ω) real-valued, then there exists a unique real-valued harmonic function for which the restriction to the boundary ∂Ω is... more
ABSTRACT Let Ω be a bounded, simply connected domain in the complex plane, with C smooth boundary and take u ∈ C∞(∂Ω) real-valued, then there exists a unique real-valued harmonic function for which the restriction to the boundary ∂Ω is precisely u. Let be the conjugate harmonic to U in the sense that F = U+iV is holomorphic in Ω, for which moreover V(a)=0, a ∈ Ω, then the restriction v of V to the boundary ∂Ω is called the Hilbert transform of u. In this article, we extend the principles of this construction to m-dimensional space , more specifically in a Clifford analysis setting, in order to define a Hilbert-like integral transform on the unit sphere S , based upon a specific notion of conjugate harmonic functions using spherical co-ordinates. ‡Dedicated to Richard Delanghe on the occasion of his 65th birthday.
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Research Interests: Mathematics, Operator Theory, Harmonic Analysis, Pure Mathematics, First-Order Logic, and 11 moreFundamental Solution, Mathematics and Statistics, Dirac operator, First Order Logic, Clifford algebra, Clifford analysis, Rotation Invariance, Spherical means, Distributions, Difference and differential operators, and Unitary group
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Research Interests: Mathematics, Harmonic Analysis, Several Complex Variables, Pure Mathematics, Hilbert transform, and 10 moreFirst-Order Logic, Mathematics and Statistics, Dirac operator, First Order Logic, Euclidean Geometry, Clifford analysis, Rotation Invariance, Difference and differential operators, Unitary group, and Bochner integral
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ABSTRACT Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator ¶ = åj = 1m ej ¶xj \underline \partial =... more
ABSTRACT Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator ¶ = åj = 1m ej ¶xj \underline \partial = \sum\nolimits_{j = 1}^m {} e_j \partial _{x_j } , where ( e1,¼, eme_{1},\ldots, e_m ) forms an orthogonal basis for the quadratic space \mathbbRm\mathbb{R}^m underlying the construction of the Clifford algebra \mathbbR0,m\mathbb{R}_{0,m} . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra \mathbbC2n\mathbb{C}_{2n} , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].
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