Skip to main content
SpringerLink
Log in

Topological properties of regular generalized function algebras

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We investigate the topological density of various subalgebras of regular generalized functions in the Colombeau algebra \(\mathcal{G }(\varOmega )\) of generalized functions with its natural (so-called sharp) topology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aragona, J., Fernandez, R., Juriaans, S.O.: A discontinuous Colombeau differential calculus. Monatsh. Math. 144, 13–29 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biagioni, H.: A Nonlinear Theory of Generalized Functions. Lecture Notes Mathematics 1421, Springer, Berlin (1990)

  3. Delcroix, A.: Generalized integral operators and Schwartz kernel type theorems. J. Math. Anal. Appl. 306(2), 481–501 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Delcroix, A.: Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity. J. Math. Anal. Appl. 327(1), 564–584 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dapić, N., Pilipović, S., Scarpalézos, D.: Microlocal analysis of Colombeau’s generalized functions: propagation of singularities. J. Anal. Math. 75, 51–66 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Garetto, C.: Topological structures in Colombeau algebras: topological \(\widetilde{{\mathbb{C}}}\)-modules and duality theory. Acta Appl. Math. 88(1), 81–123 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Garetto, C., Hörmann, G.: Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proc. Edinb. Math. Soc. 48, 603–629 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric theory of generalized functions with applications to general relativity. Kluwer, Dordrecht (2001)

  9. Hörmann, G., Kunzinger, M.: Microlocal properties of basic operations in Colombeau algebras. J. Math. Anal. Appl. 261, 254–270 (2001)

    Google Scholar 

  10. Hörmann, G., Oberguggenberger, M., Pilipović, S.: Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients. Trans. Am. Math. Soc. 358, 3363–3383 (2006)

    Article  MATH  Google Scholar 

  11. Landau, E.: Einige Ungleichungen für zweimal differentiierbare Funktionen. Proc. Lond. Math. Soc. Ser. 2(13), 43–49 (1913–1914)

  12. Nedeljkov, M., Pilipović, S., Scarpalézos, D.: The linear theory of Colombeau generalized functions. Pitman Res. Not. Math. 385, Longman Sci. Techn. (1998)

  13. Oberguggenberger, M.: Multiplication of distributions and applications to partial differential equations. Pitman Res. Not. Math. 259, Longman Sci. Techn. (1992)

  14. Oberguggenberger, M.: Regularity theory in Colombeau algebras. Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.) T CXXXIII 31, 147–162 (2006)

    Google Scholar 

  15. Pilipović, S., Scarpalézos, D., Valmorin, V.: Real analytic generalized functions. Monatsh. Math. 156(1), 85–102 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Scarpalézos, D.: Topologies dans les espaces de nouvelles fonctions généralisées de Colombeau. \(\widetilde{{\mathbb{C}}}\)-modules topologiques. Université Paris 7, (1992)

  17. Vernaeve, H.: Pointwise characterizations in generalized function algebras. Monatsh. Math. 158(2), 195–213 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We are grateful to D. Scarpalézos for very useful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ghent University, Krijgslaan 281, 9000 , Ghent, Belgium

    H. Vernaeve

Authors
  1. H. Vernaeve
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Vernaeve.

Additional information

Communicated by A. Constantin.

Supported by research grant 1.5.138.13N of the Research Foundation—Flanders (FWO).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vernaeve, H. Topological properties of regular generalized function algebras. Monatsh Math 173, 433–439 (2014). https://doi.org/10.1007/s00605-013-0495-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-013-0495-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation