F. Brackx

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2. During the final redaction of this article, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on 1 October 2006.

HILBERT-DIRAC OPERATORS IN CLIFFORD ANALYSIS

Around the central theme of square root of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.

Reproducing Bergman kernels in Clifford analysis

It is proved that the module of Clifford-algebra-valued square-integrable eigenfunctions of the Dirac-operator in an open subset Ω of R m is a Hilbert-module with reproducing kernel; this reproducing kernel for the case where Ω is the unit ball or the Euclidean space itself, is explicitly constructed. Also the module of square-integrable polymonogenic functions in R m is studied. It turns out that it is a Hilbert-module with reproducing kernel too.

Hp-spaces of monogenic functions

The classical theory of HP spaces (see e.g. [1] and [2]) combines the notions of holomorphy and boundary behaviour in Lp-sense. In this paper we introduce the concept of HP space in the unit ball of the Euclidean Space of arbitrary dimension, by considering the so called monogenic functions, which are a natural generalization to higher dimension of the holomorphic functions of one complex variable. Several possible definitions of the HP space of monogenic functions in the unit ball of Rm are given. The aim is to establish the equivalence of those definitions, providing in this way the basis properties of HP functions considered.

Clifford algebra-valued orthogonal polynomials in Euclidean space

In this paper a new method for constructing Clifford algebra-valued orthogonal polynomials in Euclidean space is presented. In earlier research, only scalar-valued weight functions were involved. Now the class of weight functions is enlarged with Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the Euclidean space 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 yield Clifford algebra-valued orthogonal polynomials in Euclidean space. Three specific examples of such orthogonal polynomials in Euclidean space are discussed, viz. the generalized Clifford-Hermite, the Clifford-Laguerre and the half-range Clifford-Hermite polynomials.

Reproducing Kernels on the Unit Sphere

Reproducing kernels are constructed for Hilbertmodules of those Clifford algebra-valued nullsolutions of certain differential operators in the unit ball of the Euclidean space R m , which moreover satisfy specific boundary conditions on the unit sphere. These nullsolutions are complexified to the Lie ball in C m , thus forming a closed sub-module of the Hilbertmodule (Math) which consists of holomorphic functions in the Lie ball with L 2-boundary behaviour on the Lie sphere. Projecting the so-called Cauchy-Hua kernel of (Math) on this closed submodule, followed by restricting the reproducing property to the Euclidean unit ball, leads to the reproducing kernel of the initial Hilbertmodule.

The Cylindrical Fourier Spectrum of an L2‐basis consisting of Generalized Clifford‐Hermite Functions

In this paper we devise a new so‐called cylindrical Fourier transform within the Clifford analysis setting by substituting for the standard inner product in the classical exponential Fourier kernel a wedge product of the old and new vector variable as argument. The cylindrical Fourier spectrum of an L2‐basis consisting of generalized Clifford‐Hermite functions is then expressed as a sum of generalized hypergeometric series.

Harmonic and Monogenic Potentials in Euclidean Halfspace

In the framework of Clifford analysis a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space Rm+1. Their distributional limits at the boundary are computed, obtaining in this way well‐known distributions in Rm such as the Dirac distribution, the Hilbert kernel, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit.