Fred Brackx

The Two-Dimensional Clifford-Fourier Transform

Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel.In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L1 and in the L2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.

The Howe dual pair in Hermitean Clifford analysis

Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator ∂. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator ∂J , leading to the system of equations ∂f = 0 = ∂Jf, expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in thiscontext, which exactly coincide with the chosen ones.

Clifford-Hermite Wavelets in Euclidean Space

Specific kernel functions for the continuous wavelet transform in higher dimension and new continuous wavelet transforms are presented within the framework of Clifford analysis. Their relationship with the heat equation and the newly introduced wavelet differential equation is established.

On Harmonic Potential Fields and the Structure of Monogenic Functions

In specific open domains of Euclidean space, a correspondence is established between a monogenic function and a sequence of harmonic potential fields, leading to the construction of a unique vector-valued conjugate harmonic homogeneous polynomial to a given real-valued solid harmonic.

The generalized clifford-hermite continuous wavelet transform

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.

Duality in Hypercomplex Function Theory

Abstract Let A be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e1,…, en}, and e0 be the identity of A . Furthermore, let Mk(Ω; A ) be the set of A -valued functions defined in an open subset Ω of Rm+1 (1 ⩽ m ⩽ n) which satisfy Dkf = 0 in Ω, where D is the generalized Cauchy-Riemann operator D = ∑ i = 0 m e i ( ∂ ∂x i ) and k ϵ N. The aim of this paper is to characterize the dual and bidual of Mk(Ω; A ). It is proved that, if Mk(Ω; A ) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Frechet modules, which in its turn admits Mk(Ω; A ) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.

Chapter 2 The Fourier Transform in Clifford Analysis

Publisher Summary This chapter focuses on the Fourier transform in Clifford analysis. This chapter includes an introductory section on Clifford analysis, and each section starts with an introductory situation. This chapter presents the new Clifford–Fourier transform is given in terms of an operator exponential, or alternatively, by a series representation. Particular attention is directed to the two-dimensional (2D) case since then the Clifford–Fourier kernel can be written in a closed form. This chapter also discusses the fractional Fourier transform wherein, it is shown that the traditional and the Clifford analysis approach coincide. This chapter develops the theory for the Clifford–Hermite and Clifford–Gabor filters for early vision. This chapter faced with the following situation: In dimension greater than two, we have a first Clifford–Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension, two both transforms coincide.

Hermitean Cauchy Integral Decomposition of Continuous Functions on Hypersurfaces

We consider Hölder continuous circulant matrix functions defined on the Ahlfors-David regular boundary of a domain in . The main goal is to study under which conditions such a function can be decomposed as , where the components are extendable to two-sided -monogenic functions in the interior and the exterior of , respectively. -monogenicity is a concept from the framework of Hermitean Clifford analysis, a higher dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. -monogenic functions then are the null solutions of a matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context.

The Clifford–Gegenbauer polynomials and the associated continuous wavelet transform

In the framework of Clifford analysis, Clifford–Gegenbauer and generalized Clifford–Gegenbauer polynomials are constructed. Orthogonality relations and Rodrigues formulae are established. It is shown that they are the appropriate building blocks for new specific wavelet kernel functions for a higher dimensional continuous wavelet transform. E-mail: fs@cage.ugent.be E-mail: nds@cage.ugent.be

A bochner—martinelli formula for the biregular functions of clifford analysis

The biregular functions constitute in a certain way an extension to two higher dimensional variables of the one Clifford-variable monogenic functions. From a Bochner-Martinelli formula for C 1-functions in the biregular setting, a Cauchy integral formula with non-biregular kernels is obtained, which allows the investigation of the Taylor series of a biregular function in an open ball of .