Logic and Analysis

, Volume 1, Issue 3, pp 205-234

First online:

Full algebra of generalized functions and non-standard asymptotic analysis

  • Todor D. TodorovAffiliated withMathematics Department, California Polytechnic State University Email author 
  • , Hans VernaeveAffiliated withUnit for Engineering Mathematics, University of Innsbruck

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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 distributions Generalized functions Colombeau algebra Multiplication of distributions Non-standard analysis Infinitesimals Ultrapower non-standard model Ultrafilter Maximal filter Robinson valuation field Ultra-metric Hahn–Banach theorem

Mathematics Subject Classification (2000)

Primary: 46F30 Secondary: 46S20 46S10 46F10 03H05 03C50