Title

Full algebra of generalized functions and non-standard asymptotic analysis

Author Info

Todor D. Todorov, California Polytechnic State University - San Luis ObispoFollow
Hans Vernaeve, University of Innsbruck

Recommended Citation

Postprint version. Published in Logic and Analysis, Volume 1, Issue 3, June 18, 2008, pages 205-234.

The definitive version is available at https://doi.org/10.1007/s11813-008-0008-y.

Abstract

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 Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.

Disciplines

Mathematics

Copyright

2008 Springer.