Frank Sommen

Models for irreducible representations ofSpin(m)

In this paper we consider harmonic and monogenic polynomials of simplicial type. It is proved that these polynomials provide explicit realizations of all irreducible representations ofSpin(m).

The Dirac Complex on Abstract Vector Variables: Megaforms

In this paper, we propose a method to obtain the syzygies of the Dirac complex defined on abstract vector variables. We propose a generalized theory of differential forms which acts as a de Rham-like sequence for the Dirac complex and we show that closure in this complex is equivalent to the syzygies for the Dirac complex.

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.

Fundamental solutions for the super Laplace and Dirac operators and all their natural powers

The fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the framework of Clifford analysis.

Explicit solutions of the inhomogeneous dirac equation

In this paper we establish a general principle which may be used to construct many explicit solutions to special inhomogeneous Dirac equations with distributional right-hand side. These solutions are presented as series of products of Clifford algebra valued functions which themselves satisfy Dirac equations in a lower dimension. We also present several special examples, including plane waves, zonal functions, Cauchy kernels and electromagnetic fields.

A Boundary Value Problem for Hermitian Monogenic Functions

We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in . Necessary and sufficient conditions for the solvability of this problem are obtained.

The general quadratic Radon transform

The general quadratic Radon transform in two dimensions is investigated. Whereas the classical Radon transform of a smooth function represents the integration over all lines, the general quadratic Radon transform integrates over all conic sections. First, the parabolic isofocal Radon transform, i.e. the restriction of the general quadratic Radon transform to all parabolae with focus in the origin, is defined and illustrated. We show its intense relation to the classical Radon transform, deduce a support theorem, formulate an extension of the support theorem and derive an inversion formula. The natural extension to a more general class of isofocal quadratic Radon transforms is outlined. We show how the general quadratic Radon transform can be derived from the integrals over all parabolae by solving the related Cauchy problem. Finally, we introduce an entirely geometrical definition of a generalized Radon transform, the oriented generalized Radon transform.

Hypercomplex analysis and applications

The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields. The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, including mathematical analysis, mathematical physics, algebra.

Integral Theorems for Solutions of the Complex Hodge-Dolbeault System

The Cauchy-Riemann condition of the real Hodge-Dolbeault system is generalized and several facts for solutions of the complex Hodge-Dolbeault system are given explaining, as well, how they are obtained from the general theory.

Analysis of Functions and Differential Forms in C m

The class of holomorphic functions of several complex variables is extended to a general set of matrix-valued functions and differential forms having a deep analogy to the set of holomorphic functions of one variable. Representation formulas of Borel-Pompeiu type are developed.