Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T13:57:42.898Z Has data issue: false hasContentIssue false
  • English
  • Français

Vanishing Theorems in Colombeau Algebras of Generalized Functions

Published online by Cambridge University Press:  20 November 2018

V. Valmorin
V. Valmorin*
Affiliation:
Département Math-Info, Université des Antilles et de la Guyane, Campus de Fouillole: 97159 Pointe à Pitre Cedex, France. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using a canonical linear embedding of the algebra ${{G}^{\infty }}\left( \Omega \right)$ of Colombeau generalized functions in the space of $\overline{\mathbb{C}}$ -valued $\mathbb{C}$-linear maps on the space $D\left( \Omega \right)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${{G}^{\infty }}$ . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

Keywords

32A6045P0546F30Colombeau generalized functionslinear integral operatorsgeneralized holomorphic functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 4 , 01 December 2008 , pp. 618 - 626
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Aragona, J., On existence theorems for the operator on generalized differential forms. Proc. London Math. Soc. 53(1986), no. 3, 474–488.Google Scholar
[2] Bernard, S., Colombeau, J-F., and Delcroix, A., Generalized integral operators and applications. Math. Proc. Cambridge Philos. Soc. 141(2006), no. 3, 521–546. Soc.Google Scholar
[3] Colombeau, J-F., New Generalized Functions and Multiplication of Distributions. North-Holland Math. Stud. 84, North-Holland, Amsterdam, 1984.Google Scholar
[4] Colombeau, J-F., Elementary Introduction to New Generalized Functions. North-Holland Math. Stud. 113, North-Holland, Amsterdam, 1985.Google Scholar
[5] Colombeau, J-F. and Galé, J.E., Holomorphic generalized functions. J. Math. Anal. Appl. 103(1984), no. 1, 117–133.Google Scholar
[6] Colombeau, J-F. and Galé, J.E., Analytic continuation of generalized functions. Acta Math. Hung. 52(1988), no. 1-2, 57–60.Google Scholar
[7] Garetto, C., Topological structures in Colombeau algebras: investigation of the duals of . Monatshf. Math. 146(2005), no. 3, 203–226.Google Scholar
[8] Grosser, M., Kunzinger, M., Oberguggenberger, M., and Steinbauer, R., Geometric Theory of Generalized Functions with Applications to General relativity. Mathematics and Its Applications 537, Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[9] Khelif, A. and Scarpalezos, D., Zeros of generalized holomorphic functions. Monatsh. Math 149(2006), no. 4, 323–335.Google Scholar
[10] Oberguggenberger, M., Multiplication of distributions and application to partial differential equations. Pitman Res. Notes Math. Ser. 259, Longman, Harlow, New York, 1992.Google Scholar
[11] Oberguggenberger, M. and Kunzinger, M., Characterization of Colombeau generalized functions by their pointvalues. Math. Nachr. 203(1999), 147–157.Google Scholar
[12] Oberguggenberger, M., Pilipovic, S., and Valmorin, V., Global representatives of Colombeau holomorphic generalized functions. Monatsh. Math. 151(2007), no. 1, 67–74.Google Scholar
[13] Pilipović, S., Scarpalezos, D., and Valmorin, V., Equalities in algebras of generalized functions. Forum Math. 18(2006), no. 5, 789–801.Google Scholar