Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-03T10:18:37.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Full and Special Colombeau Algebras

Part of: Distributions, generalized functions, distribution spaces

Published online by Cambridge University Press:  13 June 2018

M. Grosser  and
E. A. Nigsch
M. Grosser
Affiliation:
Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria (michael.grosser@univie.ac.at)
E. A. Nigsch
Affiliation:
Wolfgang-Pauli-Institut, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria (eduard.nigsch@univie.ac.at)
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce full diffeomorphism-invariant Colombeau algebras with added ε-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.

Keywords

Colombeau algebrageneralized functionfullspecialsharp topologypoint valueslocalitysheaf property

MSC classification

Primary: 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 961 - 994
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aragona, J., Fernandez, R. and Juriaans, S. O., Natural topologies on Colombeau algebras, Topol. Methods Nonlinear Anal. 34(1) (2009), 161–180.Google Scholar
2.Biagioni, H. A., A nonlinear theory of generalized functions, 2nd edn (Springer-Verlag, Berlin, 1990).Google Scholar
3.Bourbaki, N., General topology, chapters 1–4, The Elements of Mathematics, Volume III (Springer, Berlin, 1995).Google Scholar
4.Catuogno, P. and Olivera, C., Tempered generalized functions and Hermite expansions, Nonlinear Anal. 74(2) (2011), 479–493.Google Scholar
5.Colombeau, J. F., New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, Volume 84 (North-Holland Publishing Co., Amsterdam, 1984).Google Scholar
6.Colombeau, J. F., Elementary introduction to new generalized functions, North-Holland Mathematics Studies, Volume 113 (North-Holland Publishing Co., Amsterdam, 1985).Google Scholar
7.Colombeau, J. and Galé, J., Holomorphic generalized functions, J. Math. Anal. Appl. 103 (1984), 117–133.Google Scholar
8.Colombeau, J. F. and Le Roux, A. Y., Multiplications of distributions in elasticity and hydrodynamics, J. Math. Phys. 29 (1988), 315–319.Google Scholar
9.Dave, S., Geometrical embeddings of distributions into algebras of generalized functions, Math. Nachr. 283(11) (2010), 1575–1588.Google Scholar
10.Dave, S., Hörmann, G. and Kunzinger, M., Optimal regularization processes on complete Riemannian manifolds, Tokyo J. Math. 36(1) (2013), 25–47.Google Scholar
11.Egorov, Yu. V., A contribution to the theory of generalized functions, Russ. Math. Surv. 45(5) (1990), 1–49.Google Scholar
12.Godement, R., Topologie algébrique et théorie des faisceaux, 3rd edn. Actualités Scientifiques et Industrielles 1252. Publications de l'Institut de Mathématique de l'Université de Strasbourg, XIII (Hermann, Paris, 1973).Google Scholar
13.Grosser, M., Farkas, E., Kunzinger, M. and Steinbauer, R., On the foundations of nonlinear generalized functions I, II, Mem. Am. Math. Soc. 729 (2001).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 Publishers, Dordrecht, 2001).Google Scholar
15.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
16.Hasler, M. F. and Marti, J.-A., Towards pointvalue characterizations in multi-parameter algebras, Novi Sad J. Math. 41(1) (2011), 21–31.Google Scholar
17.Jelínek, J., An intrinsic definition of the Colombeau generalized functions, Commentat. Math. Univ. Carol. 40(1) (1999), 71–95.Google Scholar
18.König, H., Multiplikation von Distributionen I, Math. Ann. 128 (1955), 420–452.Google Scholar
19.Kriegl, A. and Michor, P. W., The convenient setting of global analysis, Mathematical Surveys and Monographs, Volume 53 (American Mathematical Society, Providence, RI, 1997).Google Scholar
20.Kunzinger, M. and Nigsch, E. A., Manifold-valued generalized functions in full Colombeau spaces, Commentat. Math. Univ. Carol. 52(4) (2011), 519–534.Google Scholar
21.Kunzinger, M. and Oberguggenberger, M., Characterization of Colombeau generalized functions by their pointvalues, Math. Nachr. 203(1) (1999), 147–157.Google Scholar
22.Kunzinger, M. and Steinbauer, R., A rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves, J. Math. Phys. 40(3) (1999), 1479–1489.Google Scholar
23.Lojasiewicz, S., Sur la valeur et la limite d'une distribution en un point, Studia Math. 16 (1957), 1–36.Google Scholar
24.Mayerhofer, E., On the characterization of p-adic Colombeau–Egorov generalized functions by their point values, Math. Nachr. 280(11) (2007), 1297–1301.Google Scholar
25.Mikusiński, J., Sur la méthode de généralisation de Laurent Schwartz et sur la convergence faible, Fundam. Math. 35 (1948).Google Scholar
26.Mikusiński, J., Antosik, P. and Sikorski, R., Theory of distributions. The sequential approach (Elsevier, Amsterdam, 1973).Google Scholar
27.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
28.Nigsch, E. A., The functional analytic foundation of Colombeau algebras, J. Math. Anal. Appl. 421(1) (2015), 415–435.Google Scholar
29.Nigsch, E. A., Nonlinear generalized sections of vector bundles, J. Math. Anal. Appl. 440 (2016), 183–219.Google Scholar
30.Nigsch, E. A., On a nonlinear Peetre theorem in full Colombeau algebras, Commentat. Math. Univ. Carol. 58(1) (2017), 69–77.Google Scholar
31.Nigsch, E. A., Nonlinear generalized sections and vector bundle homomorphisms in Colombeau spaces of generalized functions, Math. Nachr. 290(13) (2017), 1991–2008.Google Scholar
32.Oberguggenberger, M., Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics, Volume 259 (Longman, Harlow, UK, 1992).Google Scholar
33.Rosinger, E. E., Distributions and nonlinear partial differential equations, Lecture Notes in Mathematics, Volume 684 (Springer-Verlag, Berlin, 1978).Google Scholar
34.Scarpalézos, D., Colombeau's generalized functions: topological structures; microlocal properties. A simplified point of view I, Bull. Cl. Sci. Math. Nat., Sci. Math. 121(25) (2000), 89–114.Google Scholar
35.Schwartz, L., Sur l'impossibilité de la multiplication des distributions, Comptes Rendus de l'Académie des Sciences 239 (1954), 847–848.Google Scholar
36.Schwartz, L., Espaces de fonctions différentiables à valeurs vectorielles, J. Anal. Math. 4(1) (1955), 88–148.Google Scholar
37.Steinbauer, R., Geodesics and geodesic deviation for impulsive gravitational waves, J. Math. Phys. 39(4) (1998), 2201–2212.Google Scholar