Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Standard Version of Egorov Algebra of Generalized Functions

Todor D. Todorov

Abstract

We consider a non-standard version of Egorov's algebra of generalized functions, with improved properties of the generalized scalars and embedding of the Schwartz distributions compared with the original standard Egorov's version. The embedding of distributions is similar to, but different from author's works in the past and independently done by Hans Vernaeve.

Non-Standard Version of Egorov Algebra of Generalized Functions

Abstract

We consider a non-standard version of Egorov's algebra of generalized functions, with improved properties of the generalized scalars and embedding of the Schwartz distributions compared with the original standard Egorov's version. The embedding of distributions is similar to, but different from author's works in the past and independently done by Hans Vernaeve.
Paper Structure (6 sections, 6 theorems, 1 equation)

This paper contains 6 sections, 6 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Notations and Set-Theoretical Framework
  3. Non-Standard Version of Egorov Algebra $\widehat{^*\mathcal{E}}(\Omega)$
  4. Non-Standard Delta-Function
  5. Embedding of Distributions into $\widehat{^*\mathcal{E}}(\Omega)$
  6. Regular Algebra

Key Result

Theorem 3.2

Theorems & Definitions (13)

  • Definition 3.1: Non-Standard Version of Egorov Algebra
  • Theorem 3.2: Basic Properties of $\widehat{^*\mathcal{E}}(\Omega)$
  • proof
  • Lemma 4.1: Non-Standard Delta-Function
  • Theorem 4.2: Regularization in $^*\mathcal{E}(\Omega)$
  • proof
  • Definition 5.1: Embedding of Distributions in $\widehat{^*\mathcal{E}}(\Omega)$
  • Theorem 5.2: Properties of the Embedding
  • proof
  • Corollary 5.4: Multiplication of Classical Functions
  • ...and 3 more