Distributions: a spherical co-ordinates approach

TL;DR

This paper introduces signumdistributions to handle derivatives of distributions in spherical coordinates, addressing the complexities and non-uniqueness issues in defining radial derivatives.

Contribution

It proposes a new class of distributions, signumdistributions, to facilitate derivatives in spherical coordinates, extending the theoretical framework.

Findings

Signumdistributions are suitable for radial derivatives of distributions.

Radial derivatives of distributions are not unique, leading to an equivalence class.

The framework extends the understanding of distribution derivatives in spherical coordinates.

Abstract

When expressing a distribution in Euclidean space in spherical co-ordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the delta-distribution (its angular derivatives being zero) led to the introduction of a new kind of distributions, the so-called signumdistributions, as continuous linear functionals on a space of test functions showing a singularity at the origin. In this paper we search for a definition of the radial and angular derivatives of a general standard distribution and again, as expected, we are inevitably led to consider signumdistributions. Although these signumdistributions provide an adequate framework for the actions on distributions aimed at, it turns out that the derivation with respect to the radial distance of a general (signum)distribution is still not yet unique, but gives rise to an equivalent class of (signum)distributions.