On the radial derivative of the delta distribution

Abstract

Possibilities for defining the radial derivative of the delta distribution $\delta(\underline{x})$ in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar to but different from the standard distributions. The radial derivative of $\delta(\underline{x})$ then belongs to that new class of so-called signumdistributions. It is shown that these signumdistributions obey easy-to-handle calculus rules which are in accordance with those for the standard distributions in $\mathbb{R}^m$.