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Full algebra of generalized functions and non-standard asymptotic analysis

(2008) Logic And Analysis. 1(3-4). p.205-234
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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.
Keywords
ultrafilter, ultrapower non-standard model, infinitesimals, non-standard analysis, multiplication of distributions, Colombeau algebra, Schwartz distributions, generalized functions, maximal filter, Robinson valuation field, ultra-metric, Hahn-Banach theorem

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MLA
Todorov, Todor, and Hans Vernaeve. “Full Algebra of Generalized Functions and Non-standard Asymptotic Analysis.” Logic And Analysis 1.3-4 (2008): 205–234. Print.
APA
Todorov, T., & Vernaeve, H. (2008). Full algebra of generalized functions and non-standard asymptotic analysis. Logic And Analysis, 1(3-4), 205–234.
Chicago author-date
Todorov, Todor, and Hans Vernaeve. 2008. “Full Algebra of Generalized Functions and Non-standard Asymptotic Analysis.” Logic And Analysis 1 (3-4): 205–234.
Chicago author-date (all authors)
Todorov, Todor, and Hans Vernaeve. 2008. “Full Algebra of Generalized Functions and Non-standard Asymptotic Analysis.” Logic And Analysis 1 (3-4): 205–234.
Vancouver
1.
Todorov T, Vernaeve H. Full algebra of generalized functions and non-standard asymptotic analysis. Logic And Analysis. Springer; 2008;1(3-4):205–34.
IEEE
[1]
T. Todorov and H. Vernaeve, “Full algebra of generalized functions and non-standard asymptotic analysis,” Logic And Analysis, vol. 1, no. 3–4, pp. 205–234, 2008.
@article{675590,
  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.},
  author       = {Todorov, Todor and Vernaeve, Hans},
  issn         = {1863-3617},
  journal      = {Logic And Analysis},
  keywords     = {ultrafilter,ultrapower non-standard model,infinitesimals,non-standard analysis,multiplication of distributions,Colombeau algebra,Schwartz distributions,generalized functions,maximal filter,Robinson valuation field,ultra-metric,Hahn-Banach theorem},
  language     = {eng},
  number       = {3-4},
  pages        = {205--234},
  publisher    = {Springer},
  title        = {Full algebra of generalized functions and non-standard asymptotic analysis},
  url          = {http://dx.doi.org/10.1007/s11813-008-0008-y},
  volume       = {1},
  year         = {2008},
}

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