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A New Approach to Diffeomorphism Invariant Algebras of Generalized Functions

Published online by Cambridge University Press:  13 February 2015

E. A. Nigsch
E. A. Nigsch*
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria ([email protected])
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Abstract

We develop the diffeomorphism invariant Colombeau-type algebra of nonlinear generalized functions in a modern and compact way. Using a unifying formalism for the local setting and on manifolds, the construction becomes simpler and more accessible than before.

Keywords

algebras of nonlinear generalized functionsColombeau algebradiffeomorphism invariancePrimary 46F30Secondary 58B10
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 717 - 738
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1.Biagioni, H. A., A nonlinear theory of generalized functions, 2nd edn (Springer 1990).Google Scholar
2.Bredon, G. E., Sheaf theory, 2nd edn (Springer 1997).Google Scholar
3.Colombeau, J. F., New generalized functions and multiplication of distributions (Elsevier, 1984).Google Scholar
4.Colombeau, J. F., Elementary introduction to new generalized functions (Elsevier, 1985).Google Scholar
5.Colombeau, J. F. and Langlais, M., Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Analysis Applic. 145(1 (1990), 186–196.Google Scholar
6.Colombeau, J. F. and Meril, A., Generalized functions and multiplication of distributions on C ∞ manifolds, J. Math. Analysis Applic. 186(2 (1994), 357–364.CrossRefGoogle Scholar
7.Constantine, G. M. and Savits, T. H., A multivariate Faa di Bruno formula with applications, Trans. Am. Math. Soc. 348(2 (1996), 503–520.Google Scholar
8.Dowker, C. H., Lectures on sheaf theory (Tata Institute of Fundamental Research, Bombay, 1956).Google Scholar
9.Garetto, C. and Hoermann, G., Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities, Proc. Edinb. Math. Soc. 48(3 (2005), 603–629.Google Scholar
10.Grosser, M., Farkas, E., Kunzinger, M. and Steinbauer, R., On the foundations of nonlinear generalized functions I and II, Memoirs of the American Mathematical Society, Volume 153, Issue 729 (American Mathematical Society, Providence, RI, 2001).Google Scholar
11.Grosser, M., Kunzinger, M., Oberguggenberger, M. and Steinbauer, R., Geometric theory of generalized functions with applications to general relativity (Kluwer Academic, Dordrecht, 2001).CrossRefGoogle Scholar
12.Grosser, M., Kunzinger, M., Steinbauer, R. and Vickers, J. A., A global theory of algebras of generalized functions, Adv. Math. 166(1 (2002), 50–72.Google Scholar
13.Hardy, M., Combinatorics of partial derivatives, Electron. J. Combin. 13(1) (2006).Google Scholar
14.Jelínek, J., An intrinsic definition of the Colombeau generalized functions, Commentat. Math. Univ. Carolinae 40(1 (1999), 71–95.Google Scholar
15.Jelínek, J., Equality of two diffeomorphism invariant Colombeau algebras, Commentat. Math. Univ. Carolinae 45(4 (2004), 633–662.Google Scholar
16.Kriegl, A. and Michor, P., The convenient setting of global analysis, Mathematical Surveys and Monographs, Volume 53 (American Mathematical Society, Providence, RI, 1997).Google Scholar
17.Kunzinger, M. and Nigsch, E. A., Manifold-valued generalized functions in full Colombeau spaces, Commentat. Math. Univ. Carolinae 52(4 (2011), 519–534.Google Scholar
18.Mazur, S. and Orlicz, W., Grundlegende Eigenschaften der polynomischen Operationen, Erste Mitteilung, Studia Math. 5(1 (1934), 50–68.CrossRefGoogle Scholar
19.Nigsch, E. A., Point value characterizations and related results in the full Colombeau algebras Ge(Ω) and Gd(Ω), Math. Nachr. 286(10 (2013), 1007–1021.Google Scholar
20.Oberguggenberger, M., Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, Volume 259 (Longman, Harlow, 1992).Google Scholar
21.Rosinger, E. E., Generalized solutions of nonlinear partial differential equations (Elsevier, 1987).Google Scholar
22.Schwartz, L., Sur l’impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris I 239 (1954), 847–848.Google Scholar
23.Steinbauer, R. and Vickers, J. A., The use of generalized functions and distributions in general relativity, Class. Quant. Grav. 23(10) (2006), R91–R114.Google Scholar