Planar harmonic and monogenic polynomials of type A

Harmonic and Monogenic Potentials in Low Dimensional Euclidean Half-Space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^{m+1} was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary R^{m} were also determined. In this paper the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down.

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…

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 …

Fischer Decomposition for osp(4|2)-monogenics in Quaternion Clifford Analysis

Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp$( p)$. These Fischer decompositions involve spaces of homogeneous, so-called $\mathfrak{osp}(4|2)$-monogenic polynomials, the Lie superalgebra $\mathfrak{osp}(4|2)$ being the Howe dual partner to the symplectic group Sp$( p)$. In order to obtain Sp$( p)$-irreducibility this new concept of $\mathfrak{osp}(4|2)$-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined …

全纯与 Hermitian 单演函数的 Radon 型变换

Radon-type transforms for holomorphic and Hermitian monogenic functions

本文针对单位球上的调和、全纯及 Hermitian 单演函数，引入余维数为 2 的 Radon 型变换，基于对偶变换导出反演公式，并推广至 Fock 空间与 L²(Rⁿ)，细化了 Szegö-Radon 变换。

Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space $\mR^{m+1}$ was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at the boundary $\mR^{m}$ are crucial. The remarkable relationship between these distributional boundary values and four basic pseudodifferential operators linked with the Dirac and Laplace operators is studied.