Fred Brackx

Learn More
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(More)
Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces of L 2-functions
Recommended by Colin Rogers We consider H ¨ older continuous circulant 2 × 2 matrix functions G 1 2 defined on the Ahlfors-David regular boundary Γ of a domain Ω in R 2n. The main goal is to study under which conditions such a function G G 1 2 ± are extendable to two-sided H-monogenic functions in the interior and the exterior of Ω, respectively.(More)
In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again(More)
Euclidean Clifford analysis is a higher dimensional function theory studying so–called monogenic functions, i.e. 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(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 earlier. We show that this new integral transform satisfies operational formulae(More)
We consider Hölder continuous circulant 2 × 2 matrix functions G2 defined on the fractal boundary Γ of a domain Ω in R2n. The main goal is to study under which conditions such a function G2 can be decomposed as G 1 2 G 1 2 − G 1− 2 , where the components G 1± 2 are extendable toH-monogenic functions in the interior and the exterior of Ω,(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 Clif-ford algebra-valued functions. The method consists in transforming the(More)
  • 1