On regularizations of the integral representation of Dirac delta function -- elementary approach

TL;DR

This paper explores various elementary regularizations of the integral representation of the Dirac delta function, connecting Schwartz's, Mikusiński and Sikorski's, and Sato's approaches with mathematical clarity.

Contribution

It provides a clear, elementary exposition of multiple regularizations of the Dirac delta's integral representation, bridging different distribution theories.

Findings

Unified framework for regularizations of the Dirac delta

Connections between Schwartz, Mikusiński, Sikorski, and Sato approaches

Enhanced intuitive understanding of distribution regularizations

Abstract

The article presents, in an elementary way, but with mathematical precision and without harm to the intuition, the path from the integral representation to the Dirac delta, starting with Schwartz's functional approach. Next, the considered representation is presented in a more intuitive sequential approach, formulated by Mikusi\'nski and Sikorski. Finally, we present the third regularization that can be related to Sato's approach to distributions.