Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Families of Clifford Distributions Revisited

Fred Brackx

Abstract

Two specific families of distributions in harmonic and Clifford analysis are further studied through a spherical co-ordinates approach. In particular actions involving spherical co-ordinates, such as the radial derivative and the multiplication and division by the radial distance, are computed, giving rise to two families of so--called signumdistributions, i.e. bounded linear functionals on a space of test functions showing a singularity at the origin.

Two Families of Clifford Distributions Revisited

Abstract

Two specific families of distributions in harmonic and Clifford analysis are further studied through a spherical co-ordinates approach. In particular actions involving spherical co-ordinates, such as the radial derivative and the multiplication and division by the radial distance, are computed, giving rise to two families of so--called signumdistributions, i.e. bounded linear functionals on a space of test functions showing a singularity at the origin.
Paper Structure (33 sections, 14 theorems, 509 equations, 2 tables)

This paper contains 33 sections, 14 theorems, 509 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Signumdistributions
  3. Cartesian operators
  4. Signum--pairs and cross--pairs of operators
  5. Spherical operators
  6. Action uniqueness of some operators
  7. Cartesian derivatives of signumdistributions
  8. Actions on $T_\lambda$ and $U_\lambda$
  9. The operators $\underline{x}$, $r$ and $r^2$
  10. The operators $\underline{\partial}$ and $\Delta$
  11. The operators $\mathbb{E}$, $\Gamma$ and $\partial_{\underline{\omega}}$
  12. The operators $\partial_{\underline{\omega}}\, \underline{\omega}$, $\underline{\omega}\, \partial_{\underline{\omega}}\, \underline{\omega}$, $\Delta^*$ and ${\bf Z}^*$
  13. The operators $\frac{1}{\underline{x}}$, $\frac{1}{r}$ and $\frac{1}{r^2}$
  14. The operators $\omega_j, j=1,\ldots,m$
  15. The operators $\underline{\omega}\, \partial_r$, $\frac{1}{r}\, \partial_{\underline{\omega}}$ and $\partial_r$
  16. ...and 18 more sections

Key Result

Lemma 1

For a scalar distribution $T$ it holds that for any distribution $\underline{S}$ such that $\underline{x}\, \underline{S} = T$.

Theorems & Definitions (43)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Proposition 3
  • Definition 3
  • Definition 4
  • ...and 33 more