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Generalized functions with infinitesimals

Hans Vernaeve

TL;DR

This work develops a nonstandard-analysis framework to rigorously treat the Dirac delta and distributions in PDE contexts by representing distributions through nonstandard smooth functions. It constructs model delta objects of order $0$, uses convolution with fundamental solutions to obtain infinitesimal-accurate solutions $u$ (with $P(\partial)u\approx f$), and then passes to standard parts to recover weak and, under regularity, strong solutions. The approach extends to higher-order delta notions and delta sequences, including Fourier-representation viewpoints, and culminates in an isomorphism between the standard distribution space $\mathcal{D}'$ and the nonstandard quotient $D'/_{\approx_{D'}}$. Overall, the paper provides an elementary, intuition-driven pathway from infinitesimals to distribution theory, with concrete PDE constructions and broad conceptual parallels to Schwartz distributions.

Abstract

We give a survey of the use of infinitesimals within mathematical analysis to rigorously deal with the delta-function from physics, and more generally, with distributions in the sense of L. Schwartz. We use the framework of nonstandard analysis as introduced by A. Robinson to rigorously deal with infinitesimals. Our exposition tries to be elementary, except for familiarity with nonstandard analysis, and takes as a starting point one of the basic questions in PDE for which distribution theory was developed. No knowledge of distribution theory is required.

Generalized functions with infinitesimals

TL;DR

This work develops a nonstandard-analysis framework to rigorously treat the Dirac delta and distributions in PDE contexts by representing distributions through nonstandard smooth functions. It constructs model delta objects of order , uses convolution with fundamental solutions to obtain infinitesimal-accurate solutions (with ), and then passes to standard parts to recover weak and, under regularity, strong solutions. The approach extends to higher-order delta notions and delta sequences, including Fourier-representation viewpoints, and culminates in an isomorphism between the standard distribution space and the nonstandard quotient . Overall, the paper provides an elementary, intuition-driven pathway from infinitesimals to distribution theory, with concrete PDE constructions and broad conceptual parallels to Schwartz distributions.

Abstract

We give a survey of the use of infinitesimals within mathematical analysis to rigorously deal with the delta-function from physics, and more generally, with distributions in the sense of L. Schwartz. We use the framework of nonstandard analysis as introduced by A. Robinson to rigorously deal with infinitesimals. Our exposition tries to be elementary, except for familiarity with nonstandard analysis, and takes as a starting point one of the basic questions in PDE for which distribution theory was developed. No knowledge of distribution theory is required.
Paper Structure (8 sections, 22 theorems, 24 equations)

This paper contains 8 sections, 22 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Motivating problem
  4. Delta functions
  5. Weak solutions of linear PDE
  6. Strong solutions of linear PDE
  7. Delta functions of higher order
  8. Concluding remarks

Key Result

Theorem 1

Consider the $m$-th order linear constant coefficient PDO $P(\partial): = \sum_{\left|\alpha\right|\le m} c_\alpha \partial^\alpha$ (with $\alpha\in\mathbb N^d$, $c_\alpha\in\mathbb R$). Let $E$ be a fundamental solution for $P(\partial)$, i.e., $P(\partial) E = \delta$. Let $f\in{\mathcal{C}}^{\i

Theorems & Definitions (52)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • proof
  • Definition 4
  • Lemma 5
  • proof
  • Example 6
  • proof
  • Definition 7
  • ...and 42 more