An elementary purely algebraic approach to generalized functions and operational calculus

TL;DR

This paper introduces an algebraic framework for generalized functions and operational calculus by representing Schwartz distributions as Mikusinski functions, enabling multiplication by Laurent series and simplifying Heaviside's calculus.

Contribution

It presents a purely algebraic approach to generalized functions using Mikusinski functions, offering a new basis for operational calculus.

Findings

Representation of Schwartz distributions as Mikusinski functions

Mikusinski functions admit multiplication by Laurent series

Simplification of Heaviside's operational calculus

Abstract

The space of Schwartz distributions of finite order is represented as a factor space of the space of, what we call, Mikusinski functions. The point of Mikusinski functions is that they admit a multiplication by convergent Laurent series. It is shown that this multiplication provides a natural simple basis for Heaviside's operational calculus.