Full algebra of generalized functions and non-standard asymptotic analysis

Logic and Analysis 1 (3-4):205-234 (2008)
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
Keywords 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
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DOI 10.1007/s11813-008-0008-y
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Martin Davis (1978). Applied Nonstandard Analysis. Journal of Symbolic Logic 43 (2):383-384.
John L. Kelley (1962). General Topology. Journal of Symbolic Logic 27 (2):235-235.

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