Cauchy-Kowalevski extensions and monogenic plane waves using spherical monogenics

Clifford analysis may be regarded as a direct and elegant generalization to higher dimensions of the theory of holomorphic functions in the complex plane, centred around the notion of monogenic function, i.e. a null solution of the Dirac operator. This paper dealswith axial and biaxialmonogenic functions containing sphericalmonogenics. They are constructed bymeans of two fundamentalmethods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves

Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6

Taylor series expansion in discrete Clifford analysis

Discrete Clifford analysis is a discrete higher-dimensional function theory which corresponds simultaneously to a refinement of discrete harmonic analysis and to a discrete counterpart of Euclidean Clifford analysis. The discrete framework is based on a discrete Dirac operator that combines both forward and backward difference operators and on the splitting of the basis elements into forward and backward basis elements . For a systematic development of this function theory, an indispensable tool is the Taylor series expansion, which decomposes a discrete (monogenic) function in terms of discrete homogeneous (monogenic) building blocks. The latter are the so-called discrete Fueter polynomials. For a discrete function, the authors assumed a series expansion which is formally equivalent to the Taylor series expansion in Euclidean Clifford analysis; however, attention needed to be paid to the geometrical conditions on the domain of the function, the convergence and the equivalence to the given discrete function. We furthermore applied the theory to discrete delta functions and investigated the connection with Shannon sampling theorem (Bell Sys Tech J 27:379-423, 1948). We found that any discrete function admits a series expansion into discrete homogeneous polynomials and any discrete monogenic function admits a Taylor series expansion in terms of the discrete Fueter polynomials, i.e. discrete homogeneous monogenic polynomials. Although formally the discrete Taylor series expansion of a function resembles the continuous Taylor series expansion, the main difference is that there is no restriction on discrete functions to be represented as infinite series of discrete homogeneous polynomials. Finally, since the continuous expansion of the Taylor series expansion of discrete delta functions is a sinc function, the discrete Taylor series expansion lays a link with Shannon sampling

Representation of Distributions by Harmonic and Monogenic Potentials in Euclidean Space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of (m+1)-dimensional Euclidean space was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane, and their distributional boundary values were computed. In this paper we determine these potentials in lower half-space, and investigate whether they can be extended through the boundary R^m. This is a stepping stone to the representation of a doubly infinite sequence of distributions in R^m, consisting of positive and negative integer powers of the Dirac and the Hilbert-Dirac operators, as the jump across R^m of monogenic functions in the upper and lower half-spaces, in this way providing a sequence of interesting examples of Clifford hyperfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1210.238

The class of Clifford-Fourier transforms

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see [F. Brackx et al., The Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.Comment: 30 pages, accepted for publication in J. Fourier Anal. App

On a Chain of Harmonic and Monogenic Potentials in Euclidean Half-space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space \mR^{m+1}, including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary \mR^{m} turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, 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

On a special type of solutions of arbitrary higher spin Dirac operators

In this paper an explicit expression is determined for the elliptic higher spin Dirac operator, acting on functions f(x) taking values in an arbitrary irreducible finite-dimensional module for the group Spin(m) characterized by a half-integer highest weight. Also a special class of solutions of these operators is constructed, and the connection between these solutions and transvector algebras is explained

On the structure of complex Clifford algebra

The structure of a complex Clifford algebra is studied by direct sum decompositions into eigenspaces of specific linear operators