Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis

Details Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis

2007-12-17

arxiv.org/abs/0712.2603v1

Logic and Analysis, Vol. 1, Issue 3, 2008.

Mathematics

Functional Analysis

29 pages

Scientific paper

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau's solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn-Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by J.F. Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.

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