Dirac delta and singular distributions: The general non-good function case

Abstract

A rigorous approach for defining Dirac type singular distributions is detailed for the case where the distribution is to be applied to functions that are normally to be considered in physical applications and which are outside formal distribution theory. Four principles that underpin singular distributions are detailed. The approach is applied to examples in the recent literature and the limitations of several conclusions is made clear. A non-linear distribution which extracts the number of zero crossings of a function is defined.