Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-17T20:05:14.585Z Has data issue: false hasContentIssue false
  • English
  • Français

A Convenient Notion of Compact Set for Generalized Functions

Part of: Topological linear spaces and related structures Distributions, generalized functions, distribution spaces Topological rings and modules

Published online by Cambridge University Press:  03 April 2017

Paolo Giordano  and
Michael Kunzinger
Paolo Giordano*
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria (paolo.giordano@univie.ac.at; michael.kunzinger@univie.ac.at)
Michael Kunzinger*
Affiliation:
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria (paolo.giordano@univie.ac.at; michael.kunzinger@univie.ac.at)
*
*Corresponding author.
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

Keywords

functionally compact setsColombeau generalized functionsgeneralized smooth functionslocally convex modules

MSC classification

Primary: 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) 13J99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 57 - 92
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., Colombeau, J. F. and Juriaans, S. O., Locally convex topological algebras of generalized functions: compactness and nuclearity in a nonlinear context, Trans. Amer. Math. Soc., to appear.Google Scholar
2. Aragona, J., Fernandez, R. and Juriaans, S. O., A Discontinuous Colombeau Differential Calculus. Monatsh. Math. 144 (2005), 13–29.CrossRefGoogle Scholar
3. Aragona, J., Fernandez, R. and Juriaans, S. O., Natural topologies on Colombeau algebras, Topol. Methods Nonlinear Anal. 34(1) (2009), 161–180.Google Scholar
4. Aragona, J. and Juriaans, S. O., Some structural properties of the topological ring of Colombeau's generalized numbersx, Vernaeve, Comm. Algebra 29(5) (2001), 2201–2230.CrossRefGoogle Scholar
5. Aragona, J., Juriaans, S. O., Oliveira, O. R. B. and Scarpalɉzos, D., Algebraic and geometric theory of the topological ring of Colombeau generalized functions, Proc. Edinb. Math. Soc. (2) 51(3) (2008), 545564.Google Scholar
6. Aragona, J., Fernandez, R. and Juriaans, S. O., Oberguggenberger, M., Differential calculus and integration of generalized functions over membranes. Monatsh. Math. 166 (2012), 1–18.CrossRefGoogle Scholar
7. Biagioni, H. A., A Nonlinear Theory of Generalized Functions, Lecture Notes in Mathematics 1421 , Springer, Berlin, 1990.Google Scholar
8. Colombeau, J. F., New generalized functions and multiplication of distributions. North-Holland, Amsterdam, 1984.Google Scholar
9. Colombeau, J. F., Elementary introduction to new generalized functions (North-Holland, Amsterdam, 1985).Google Scholar
10. Delcroix, A., Hasler, M. F., Pilipovic, S. and Valmorin, V., Sequence spaces with exponent weights. Realizations of Colombeau type algebras, Dissertationes Mathematicae 447 (2007), 1–73.Google Scholar
11. Garetto, C., Topological structures in Colombeau algebras: Topological C-modules and duality theory, Acta Appl. Math. 88(1) (2005), 81–123.Google Scholar
12. Garetto, C., Topological structures in Colombeau algebras: investigation of the duals of 𝒢 c (Ω), 𝒢(Ω) and 𝒢 𝒮 (ℝ n ), Monatsh. Math. 146(3) (2005), 203–226.Google Scholar
13. Garetto, C., Fundamental solutions in the Colombeau framework: applications to solvability and regularity theory, Acta. Appl. Math. 102 (2008), 281–318.Google Scholar
14. Garetto, C., Closed graph and open mapping theorems for topological ℂ̃-modules and applications, Math. Nachr. 282(8) (2009), 1159–1188.Google Scholar
15. Garetto, C. and Hrmann, G., On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets, Bull. Acad. Serbe Cl. Sci. 31 (2006), 115–136.Google Scholar
16. Garetto, C. and Vernaeve, H., Hilbert ℂ̃-modules: structural properties and applications to variational problems, Trans. Amer. Math. Soc. 363(4) (2011), 2047–2090.CrossRefGoogle Scholar
17. Giordano, P. and Kunzinger, M., New topologies on Colombeau generalized numbers and the Fermat-Reyes theorem, J. Math. Anal. and Appl. 399 (2013), 229–238.CrossRefGoogle Scholar
18. Giordano, P., Kunzinger, M. and Vernaeve, H., Strongly internal sets and generalized smooth functions, J. Math. Anal. Appl. 422 (2015), 56–71.Google Scholar
19. Giordano, P. and Baglini, L. Luperi, Asymptotic gauges: Generalization of Colombeau type algebras, Math. Nachr. (2015), 1–28.Google Scholar
20. Giordano, P. and Nigsch, E., Unifying order structures for Colombeau algebras, Math. Nachr. 288(11-12) (2015), 1286–1302,.CrossRefGoogle Scholar
21. Giordano, P. and Wu, E., Categorical frameworks for generalized functions, Arab. J. Math., 4(4) (2015), 301–328.Google Scholar
22. Grosser, M., Kunzinger, M., Oberguggenberger, M. and Steinbauer, R., Geometric theory of generalized functions (Kluwer, Dordrecht, 2001).Google Scholar
23. Mayerhofer, E., Spherical completeness of the non-Archimedean ring of Colombeau generalized numbers, Bull. Inst. Math. Acad. Sin. (N.S.) 2(3) (2007), 769–783.Google Scholar
24. Mayerhofer, E., On Lorentz geometry in algebras of generalized functions, Proc. Roy. Soc. Edinburgh Sect. A 138(4) (2008), 843–871.Google Scholar
25. Oberguggenberger, M., Multiplication of Distributions and Applications to Partial Differential Equations, volume 259 of Pitman Research Notes in Mathematics (Longman, Harlow, 1992).Google Scholar
26. Oberguggenberger, M. and Kunzinger, M., Characterization of Colombeau generalized functions by their pointvalues, Math. Nachr. 203 (1999), 147–157.Google Scholar
27. Oberguggenberger, M. and Vernaeve, H., Internal sets and internal functions in Colombeau theory, J. Math. Anal. Appl. 341 (2008), 649–659.CrossRefGoogle Scholar
28. Scarpalɉzos, D., Some remarks on functoriality of Colombeau's construction; topological and microlocal aspects and applications, Int. Transf. Spec. Fct. 6(1–4) (1998), 295–307.Google Scholar
29. Scarpalɉzos, 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
30. Vernaeve, H., Ideals in the ring of Colombeau generalized numbers, Comm. Algebra 38(6) (2010), 2199–2228.Google Scholar
31. Vernaeve, H., Nonstandard principles for generalized functions, J. Math. Anal. Appl., 384(2) (2011), 536–548.Google Scholar