Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Generalized function algebras as sequence space algebras


Authors: Antoine Delcroix, Maximilian F. Hasler, Stevan Pilipovic and Vincent Valmorin
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2031-2038
MSC (2000): Primary 46A45, 46F30; Secondary 46E10, 46A13, 46A50, 46E35, 46F05
DOI: https://doi.org/10.1090/S0002-9939-04-07306-X
Published electronically: February 12, 2004
MathSciNet review: 2053975
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of exponential weights. Such an algebra with embedded Dirac's delta distribution induces the discrete topology on the basic space.


References [Enhancements On Off] (What's this?)

  • 1. H. A. Biagioni, A Nonlinear Theory of Generalized Functions, Lecture Notes in Mathematics, no. 1421, Springer-Verlag, Berlin-Heidelberg-New York, 1990. MR 91g:46047
  • 2. H. A. Biagioni and J.-F. Colombeau, New generalized functions and $C^\infty$ functions with values in generalized complex numbers, J. London Math. Soc. (2) 33, 1 (1986) 169-179. MR 87m:46081
  • 3. J.-F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984. MR 86c:46042
  • 4. A. Delcroix, M. F. Hasler, S. Pilipovic, and V. Valmorin, Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces, Université Antilles-Guyane (2001) arXiv.org/abs/math.FA/0210240.
  • 5. A. Delcroix, M. F. Hasler, S. Pilipovic, and V. Valmorin, Algebras of generalized functions through sequence spaces: Functoriality and associations, Université Antilles-Guyane (2002) arXiv.org/abs/math.FA/0210249.
  • 6. A. Delcroix and D. Scarpalézos, Topology on asymptotic algebras of generalized functions and applications, Monatsh. Math. 129 (2000) 1-14. MR 2001b:46065
  • 7. Yu. V. Egorov, A contribution to the theory of new generalized functions, Russian Math. Surveys 45:5 (1990) 1-49; translated from Uspehi Mat. Nauk 45:5 (1990) 3-40.
  • 8. E. Farkas, M. Grosser, M. Kunzinger, and R. Steinbauer, On the foundations of nonlinear generalized functions I and II, Memoirs Amer. Math. Soc. 153, 2001. MR 2002f:46066
  • 9. M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Kluwer Academic Publ., Dordrecht, 2001. MR 2003d:46105
  • 10. A. H. Lightstone and A. Robinson, Nonarchimedian Fields and Asymptotic Expansions, North-Holland, Amsterdam, 1975. MR 54:2457
  • 11. J.-A. Marti, $(\mathcal C,\mathcal E, \mathcal P)$-sheaf structures and applications, Nonlinear Theory of Generalized Functions (Michael Grosser et al., eds.), Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999, 175-186. MR 2000f:46050
  • 12. M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Longman Scientific and Technical, Harlow, 1992. MR 94d:46044
  • 13. M. Oberguggenberger, The Carleman system with positive measures as initial data--generalized solutions, Transport Theory Statist. Phys. 20 (1991) 177-197. MR 92f:82055
  • 14. M. Oberguggenberger and T. Todorov, An embedding of Schwartz distributions in the algebra of asymptotic functions, Internat. J. Math. & Math. Sci., vol. 20, no. 3 (1998) 417-428. MR 99m:46097
  • 15. A. Robinson, Function theory on some nonarchimedian fields, Amer. Math. Monthly 80 (6), Part II: Papers in the Foundations of Mathematics (1973), 87-109. MR 48:8464
  • 16. E. Rosinger, Nonlinear Partial Differential Equations. An Algebraic View of Generalized Solutions, North-Holland Mathematics Studies, vol. 164, Amsterdam, 1990. MR 92d:46098
  • 17. D. Scarpalézos, Some Remarks on Functoriality of Colombeau's Construction: Topological and Microlocal Aspects and Applications, Integral Transforms and Special Functions 6 (1998), 295-307. MR 99e:46053
  • 18. L. Schwartz, Sur l'impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris 239 (1954) 847-848. MR 16:265e
  • 19. T. Todorov, An existence result for linear partial differential equations with $C^\infty$ coefficients in an algebra of generalized functions, Trans. Amer. Math. Soc. 348 (1996) 673-689. MR 96m:35004

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46A45, 46F30, 46E10, 46A13, 46A50, 46E35, 46F05

Retrieve articles in all journals with MSC (2000): 46A45, 46F30, 46E10, 46A13, 46A50, 46E35, 46F05


Additional Information

Antoine Delcroix
Affiliation: IUFM des Antilles et de la Guyane, Morne Ferret, BP 399, 97159 Pointe à Pitre cedex, Guadeloupe, F.W.I.
Email: Antoine.Delcroix@univ-ag.fr

Maximilian F. Hasler
Affiliation: Dépt. Scientifique Interfacultaire, Université des Antilles et de la Guyane, BP 7209, 97275 Schoelcher cedex, Martinique, F.W.I.
Email: Maximilian.Hasler@martinique.univ-ag.fr

Stevan Pilipovic
Affiliation: Faculty of Sciences and Mathematics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Yugoslavia
Email: pilipovic@im.ns.ac.yu

Vincent Valmorin
Affiliation: Faculté des Sciences Exactes et Naturelles, Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe à Pitre cedex, Guadeloupe, F.W.I.
Email: Vincent.Valmorin@univ-ag.fr

DOI: https://doi.org/10.1090/S0002-9939-04-07306-X
Received by editor(s): May 8, 2002
Received by editor(s) in revised form: March 19, 2003
Published electronically: February 12, 2004
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society