Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex algebra or spinor space, where h-monogenicity is expressed by means of two and mutually adjoint Dirac operators, which are invariant under the action realization unitary group. In part 1 article fundamental elements setting have been introduced natural way, i.e., introducing structure underlying vector eventually extended to whole . The operators then shown originate as generalized gradients when...

Clifford analysis offers a higher dimensional function theory studying the null solutions of rotation invariant, vector valued, first order Dirac operator \partial . In more recent branch Hermitean analysis, this rotational invariance has been broken by introducing complex structure J on Euclidean space and corresponding second \partial_J , leading to system equations f = 0 expressing so-called monogenicity. The is reduced unitary group. paper we show that choice fully justified. Indeed,...

In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means new "skew" type Weyl relations. We will show that overcomes difficulties working with standard relations case. A Fischer decomposition, Euler operator, monogenic projection, and basic homogeneous powers be constructed.

As is well-known, there a close and well-defined connection between the notions of Hilbert transform conjugate harmonic functions in context complex plane. This holds e.g. case on real line, which linked to harmonicity upper (or lower) half It also can be rephrased when dealing with boundary simply connected domain related harmonics its interior exterior). In this paper, we extend these principles higher dimensional space, more specifically, Clifford analysis setting. We will show that...

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

In this note, we describe the Gel'fand‐Tsetlin procedure for construction of an orthogonal basis in spaces Hermitean monogenic polynomials a fixed bidegree. The algorithm is based on Cauchy‐Kovalevskaya extension theorem and Fischer decomposition Clifford analysis.

In this paper, we show that a higher order Borel–Pompeiu (Cauchy–Pompeiu) formula, associated with an arbitrary orthogonal basis (called structural set) of Euclidean space, can be extended to the framework generalized Clifford analysis. Furthermore, in lower dimensional cases, as well for combinations standard sets, explicit expressions kernel functions are derived. Copyright © 2015 John Wiley & Sons, Ltd.

Hermitean Clifford analysis focusses on monogenic functions taking values in a complex algebra or spinor space. Here monogenicity is expressed by means of two mutually adjoint Dirac operators, which are invariant under the action representation unitary group. In this paper we have further developed theory introducing so-called zonal and studying plane wave null solutions operators. Moreover defined new Hermite polynomials setting them terms former Clifford-Hermite one-dimensional Laguerre...

We consider Hölder continuous circulant matrix functions defined on the Ahlfors-David regular boundary of a domain in . The main goal is to study under which conditions such function can be decomposed as , where components are extendable two-sided -monogenic interior and exterior respectively. -monogenicity concept from framework Hermitean Clifford analysis, higher dimensional theory centered around simultaneous null solutions two first-order vector-valued differential operators, called...

An explicit algorithmic construction is given for orthogonal bases spaces of homogeneous polynomials, in the context Hermitean Clifford analysis, which a higher dimensional function theory centered around simultaneous null solutions two conjugate complex Dirac operators. Copyright © 2011 John Wiley & Sons, Ltd.

Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though each these some traditional properties inevitably lost, new bounded singular operators emerge on or Sobolev spaces L 2 ‐functions.

Abstract A basic framework is derived for the development of a higher-dimensional discrete function theory in Clifford algebra context. The concept monogenic introduced as proper generalization holomorphic, or monodiffric, functions by Isaacs 1950s. concrete model provided definition corresponding Dirac operator. Keywords: Primary 30G35Clifford analysisdiscrete operatordiscrete