Logic and Analysis

, 1:205

Full algebra of generalized functions and non-standard asymptotic analysis


DOI: 10.1007/s11813-008-0008-y

Cite this article as:
Todorov, T.D. & Vernaeve, H. Log Anal (2008) 1: 205. doi:10.1007/s11813-008-0008-y


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.


Schwartz distributionsGeneralized functionsColombeau algebraMultiplication of distributionsNon-standard analysisInfinitesimalsUltrapower non-standard modelUltrafilterMaximal filterRobinson valuation fieldUltra-metricHahn–Banach theorem

Mathematics Subject Classification (2000)

Primary: 46F30Secondary: 46S2046S1046F1003H0503C50

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  2. 2.Unit for Engineering MathematicsUniversity of InnsbruckInnsbruckAustria