Multiplication of Distributions in one dimension: possible approaches and applications to $\delta$-function and its derivatives

TL;DR

This paper introduces new methods for multiplying distributions in one dimension, combining different regularizations, and applies these to products involving delta functions and their derivatives.

Contribution

The paper presents novel approaches to multiplying distributions by merging two regularizations, expanding the theoretical framework for distribution multiplication.

Findings

New multiplication methods for distributions introduced

Application to delta functions and derivatives demonstrated

Features of the new multiplication approaches analyzed in detail

Abstract

We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our theory to study a certain number of examples, involving products between Dirac delta functions and its successive derivatives.