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Algebras of generalized functions with smooth parameter dependence

Part of: Topological rings and modules Distributions, generalized functions, distribution spaces Nonlinear functional analysis

Published online by Cambridge University Press:  04 January 2012

Annegret Burtscher  and
Michael Kunzinger
Annegret Burtscher
Affiliation:
Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria ([email protected]; [email protected])
Michael Kunzinger
Affiliation:
Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria ([email protected]; [email protected])
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Abstract

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We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of and establish some properties of its ideals.

Keywords

algebras of generalized functionsColombeau algebrassmooth parametrizationalgebraic properties

MSC classification

Secondary: 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 13J25: Ordered rings 46T30: Distributions and generalized functions on nonlinear spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 105 - 124
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Aragona, J. and Juriaans, S. O, Some structural properties of the topological ring of Colombeau's generalized numbers, Commun. Alg. 29 (2001), 2201–2230.CrossRefGoogle Scholar
2.Aragona, J., Juriaans, S. O, Oliveira, O. R. B. and Scarpalézos, D., Algebraic and geometric theory of the topological ring of Colombeau generalized functions, Proc. Edinb. Math. Soc. 51 (2008), 545–564.CrossRefGoogle Scholar
3.Banaschewski, B., Ring theory and pointfree topology, Topol. Applic. 137 (2004), 21–37.CrossRefGoogle Scholar
4.Biagioni, H. A., A nonlinear theory of generalized functions, 2nd edition, Lecture Notes in Mathematics, Volume 1421 (Springer, 1990).CrossRefGoogle Scholar
5.Bigard, A., Keimel, K. and Wolfenstein, S., Groupes et anneaux réticulés, Lecture Notes in Mathematics, Volume 608 (Springer, 1977).CrossRefGoogle Scholar
6.Burtscher, A., Isomorphisms of algebras of smooth and generalized functions, Master's Thesis, Universität Wien (available at http://othes.univie.ac.at/4988/1/2009-05-22_0308854.pdf).Google Scholar
7.Burtscher, A., An algebraic approach to manifold-valued generalized functions, Monatsh. Math., doi:10.1007/s00605-011-0317-1 (in press).CrossRefGoogle Scholar
8.Colombeau, J.-F., New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, Volume 84 (North-Holland, Amsterdam, 1984).Google Scholar
9.Colombeau, J.-F., Elementary introduction to new generalized functions, North-Holland Mathematics Studies, Volume 113 (North-Holland, Amsterdam, 1985).Google Scholar
10.Garetto, C., Topological structures in Colombeau algebras: topological -modules and duality theory, Acta Appl. Math. 88 (2005), 81–123.CrossRefGoogle Scholar
11.Garetto, C., Topological structures in Colombeau algebras: investigation of the duals of c(Ω), (Ω) and S(ℝn), Monatsh. Math. 146 (2005), 203–226.CrossRefGoogle Scholar
12.Garetto, C. and Vernaeve, H., Hilbert -modules: structural properties and applications to variational problems, Trans. Am. Math. Soc. 363 (2011), 2047–2090.CrossRefGoogle Scholar
13.Gillman, L. and Jerison, M., Rings of continuous functions, University Series in Higher Mathematics (Van Nostrand, Princeton, NJ, 1960).Google Scholar
14.Grosser, M., Kunzinger, M., Oberguggenberger, M. and Steinbauer, R., Geometric theory of generalized functions with applications to general relativity, Mathematics and Its Applications, Volume 537 (Kluwer Academic, Dordrecht, 2001).Google Scholar
15.Hirsch, M. W., Differential topology, Graduate Texts in Mathematics, Volume 33 (Springer, 1994).Google Scholar
16.Kunzinger, M., Generalized functions valued in a smooth manifold, Monatsh. Math. 137 (2002), 31–49.CrossRefGoogle Scholar
17.Kunzinger, M. and Steinbauer, R., Generalized pseudo-Riemannian geometry, Trans. Am. Math. Soc. 354 (2002), 4179–4199.CrossRefGoogle Scholar
18.Kunzinger, M., Steinbauer, R. and Vickers, J. A., Intrinsic characterization of manifold-valued generalized functions, Proc. Lond. Math. Soc. (3) 87 (2003), 451–470.CrossRefGoogle Scholar
19.Kunzinger, M., Steinbauer, R. and Vickers, J. A, Sheaves of nonlinear generalized functions and manifold-valued distributions, Trans. Am. Math. Soc. 361 (2009), 5177–5192.CrossRefGoogle Scholar
20.Lee, J. M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Volume 218 (Springer, 2003).Google Scholar
21.Madsen, I. and Tornehave, J., From calculus to cohomology (Cambridge University Press, 1997).Google Scholar
22.Marzouk, Al. and Perrot, B., Regularity results for generalized solutions of algebraic equations and algebraic differential equations, Preprint.Google Scholar
23.Mayerhofer, E., On Lorentz geometry in algebras of generalized functions, Proc. R. Soc. Edinb. A138 (2008), 843–871.CrossRefGoogle Scholar
24.Nedeljkov, M., Pilipović, S. and Scarpalézos, D., The linear theory of Colombeau generalized functions, Pitman Research Notes in Mathematics Series, Volume 385 (Longman, Harlow, 1998).Google Scholar
25.Oberguggenberger, M., Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, Volume 259 (Longman, Harlow, 1992).Google Scholar
26.Oberguggenberger, M., Pilipović, S. and Scarpalézos, D., Positivity and positive definiteness in generalized function algebras, J. Math. Analysis Applic. 328 (2007), 1321–1335.CrossRefGoogle Scholar
27.Scarpalézos, D., Colombeau's generalized functions: topological structures; microlocal properties – a simplified point of view, I, Bull. Cl. Sci. Math. Nat. Sci. Math. 25 (2000), 89–114.Google Scholar
28.Scarpalézos, D., Colombeau's generalized functions: topological structures; microlocal properties – a simplified point of view, II, Publ. Inst. Math. (Beograd) (N.S.) 76 (2004), 111–125.CrossRefGoogle Scholar
29.Vernaeve, H., Ideals in the ring of Colombeau generalized numbers, Commun. Alg. 38 (2010), 2199–2228.CrossRefGoogle Scholar
30.Vernaeve, H., Isomorphisms of algebras of generalized functions, Monatsh. Math. 162 (2011), 225–237.CrossRefGoogle Scholar