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

Hardy spaces of solutions of generalized Riesz and Moisil-Teodorescu systems

Hardy spaces of solutions of generalized Riesz and generalized Moisil-Teodorescu systems in half space Rm+1,+ , and of their non-tangential L2-boundary values in Rm are characterized

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 R^(m+1), including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary R^(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

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

Orthogonal bases of Hermitean monogenic polynomials : an explicit construction in complex dimension 2

In this contribution we construct an orthogonal basis of Hermitean monogenic polynomials for the specific case of two complex variables. The approach combines group representation theory, see [5], with a Fischer decomposition for the kernels of each of the considered Dirac operators, see [4], and a Cauchy-Kovalevskaya extension principle, see [3]

A Goursat decomposition for polyharmonic functions in Euclidean space

The Goursat representation formula in the complex plane, expressing a real–valued biharmonic function in terms of two holomorphic functions and their anti–holomorphic complex conjugates, is generalized to Euclidean space, expressing a real–valued polyharmonic function of order p in terms of p so–called monogenic functions of Clifford analysis

On primitives and conjugate harmonic pairs in hermitian Clifford analysis

The notion of a conjugate harmonic pair in the context of Hermitian Clifford analysis is introduced as a pair of specific harmonic functions summing up to a Hermitian monogenic function in an open region of . Hermitian monogenic functions are special monogenic functions, which are at the core of so-called Clifford analyis, a straightforward generalization to higher dimension of the holomorphic functions in the complex plane. Under certain geometric conditions on the conjugate harmonic to a given specific harmonic is explicitly constructed and the potential or primitive of a Hermitian monogenic function is determined