Skip to main content

Full algebra of generalized functions and non-standard asymptotic analysis

Logic and Analysis volume 1, pages 205–234 (2008)Cite this article

Abstract

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (Canada)

References

  1. Berger G (2005) Non-standard analysis, multiplication of Schwartz distributions, and delta-like solution of Hopf’s Equation. Master Thesis # LD729. 6. S52 M3 B47 in R.E. Kennedy Library of the California Polytechnic State University, San Luis Obispo

  2. Biagioni HA (1990) A Nonlinear theory of generalized functions. Lecture Notes in Mathematics, vol 1421, (XII). Springer, Heidelberg

    Google Scholar 

  3. Bremermann H (1965) Distributions, complex variables, and Fourier transforms. Addison-Wesley, Palo Alto

    MATH  Google Scholar 

  4. Capiński M, Cutland NJ (1995) Nonstandard methods for stochastic fluid mechanics. World Scientific, Singapore-New Jersey-London-Hong Kong

    MATH  Google Scholar 

  5. Chang CC, Keisler HJ (1998) Model theory studies in logic and the foundations of mathematics, vol 73. Elsevier, Amsterdam

    Google Scholar 

  6. Colombeau JF (1984) New generalized functions and multiplication of distributions, Math Studies, vol 84. Amsterdam, North-Holland

    Google Scholar 

  7. Colombeau JF (1985) Elementary introduction to new generalized functions. North Holland, Amsterdam

    MATH  Google Scholar 

  8. Colombeau JF, Le Roux AY (1988) Multiplication of distributions in elasticity and hydrodynamics. J Math Phys 29: 315–319

    Article  MATH  MathSciNet  Google Scholar 

  9. Colombeau JF (1990) Multiplication of distributions. Bull AMS 23(2): 251–268

    Article  MATH  MathSciNet  Google Scholar 

  10. Colombeau JF (1992) Multiplication of distributions: a tool in mathematics, numerical engineering and theoretical physics. Lecture Notes in Mathematics, vol 1532. Springer, Berlin

  11. Colombeau JF, Gsponer A, Perrot B (2007) Nonlinear generalized functions and Heisenberg-Pauli foundations of quantum field theory. a preprint, arXiv: 0705.2396v1 [math-ph]

  12. Davis M (2005) Applied nonstandard analysis. Dover Publications, Inc., New York

    Google Scholar 

  13. Grosser M, Farkas E, Kunzinger M, Steinbauer R (2001) On the foundations of nonlinear generalized functions I and II. Memoirs of the AMS, vol 153(729). American Mathematical Society, New York

  14. Grosser M, Kunzinger M, Oberguggenberger M, Steinbauer R (2001) Geometric theory of generalized functions withy applications to general relativity. Mathematics and its applications, vol 537. Kluwer, Dordrecht

  15. Ingleton AW (1952) The Hahn-Banach theorem for non-archimedean valued fields. Proc Cambridge Phil Soc 48: 41–45

    Article  MATH  MathSciNet  Google Scholar 

  16. Kaneko A (1988) Introduction to hyperfunctions. Kluwer, Dordrecht

    MATH  Google Scholar 

  17. Kelley JL (1975) General topology. Prindle, Springer, New York

    MATH  Google Scholar 

  18. Levi-Civita T (1954) Sugli Infiniti ed Infinitesimi Attuali Quali Elementi Analitici (1892–1893), Opere Mathematiche, vol 1. Bologna, p1–39

  19. Bang-He L (1978) Non-standard analysis and multiplication of disributions. Sci Sinica 21: 561–585

    MathSciNet  Google Scholar 

  20. Lightstone AH, Robinson A (1975) Nonarchimedean fields and asymptotic expansions. North-Holland, Amsterdam

    MATH  Google Scholar 

  21. Lindstrøm T (1988) An invitation to nonstandard analysis. In: Cutland N(eds) Nonstandard analysis and its applications. Cambridge University Press, London, pp 1–105

    Google Scholar 

  22. Luxemburg WAJ (1962) Non-standard analysis: lectures on A. Robinson’s theory of infinitesimals and infinitely large numbers. California Institute of Technology, Pasadena, California (Second edition 1973)

  23. Luxemburg WAJ (1976) On a class of valuation fields introduced by Robinson. Israel J Math 25: 189–201

    Article  MATH  MathSciNet  Google Scholar 

  24. Mayerhofer E (2006) The wave equation on singular space-times. PhD Thesis, Faculty of Mathematics, University of Vienna, Vienna

  25. Mayerhofer E (2007) Spherical completeness of the non-archimedean ring of Colombeau generalized numbers. Bull Inst Math Acad Sin (New Series) 2(3): 769–783

    MATH  MathSciNet  Google Scholar 

  26. Oberguggenberger M (1992) Multiplication of distributions and applications to partial differential equations. Pitman Research Notes Math., vol 259. Longman, Harlow

  27. Oberguggenberger M (1995) Contributions of nonstandard analysis to partial differential equations. In: Cutland NJ, Neves V, Oliveira F, Sousa-Pinto J (eds). Developments in nonstandard mathematics. Longman Press, Harlow, pp 130–150

    Google Scholar 

  28. Oberguggenberger M, Todorov T (1998) An embedding of Schwartz distributions in the algebra of asymptotic functions. Int J Math Math Sci 21: 417–428

    Article  MATH  MathSciNet  Google Scholar 

  29. Oberguggenberger M, Vernaeve H (2008) Internal sets and internal functions in Colombeau theory. J Math Anal Appl (to appear)

  30. Pestov V (1991) On a valuation field invented by A. Robinson and certain structures connected with it. Israel J Math 74: 65–79

    Article  MATH  MathSciNet  Google Scholar 

  31. Ribenboim P (1999) The theory of classical valuations. Springer Monographs in Mathematics. Springer, Heidelberg

    Google Scholar 

  32. Robinson A (1966) Nonstandard analysis. North Holland, Amsterdam

    Google Scholar 

  33. Robinson A (1973) Function theory on some nonarchimedean fields. Amer Math Monthly 80(6), Part II: Papers in the Foundations of Mathematics, pp 87–109

  34. Salbany S, Todorov T (1998) Nonstandard analysis in point-set topology, Lecture Notes, vol 666, (52 pp) of Erwing Schrödinger Institute for Mathematical Physics, Vienna (ftp at http://ftp.esi.ac.at, URL: http://www.esi.ac.at)

  35. Schwartz L (1954) Sur l’impossibilité de la multiplication des distributions. C R Acad Sci Paris 239: 847–848

    MATH  MathSciNet  Google Scholar 

  36. Stroyan KD, Luxemburg WAJ (1976) Introduction to the theory of infinitesimals. Academic Press, New York

    MATH  Google Scholar 

  37. Todorov T (1999) Pointwise values and fundamental theorem in the algebra of asymptotic functions. In: Grosser M, Hörmann G, Kunzinger M, Oberguggenberger M (eds) Non-linear theory of generalized functions. CRC Research Notes in Mathematics, vol 401. Chapman & Hall, Boca Raton, pp 369–383

  38. Todorov TD, Wolf RS (2004) Hahn field representation of A. Robinson’s asymptotic numbers. In: Delcroix A, Hasler M, Marti J-A, Valmorin V (eds). Nonlinear algebraic analysis and applications, proceedings of the ICGF 2000. Cambridge Scientific Publishers, Cambridge, pp 357–374 arXiv:math.AC/0601722

    Google Scholar 

  39. VanDer Waerden BL (1964) Modern algebra, 3rd printing. Ungar Publishing, New York

    Google Scholar 

  40. Vernaeve H (2007) Ideals in the ring of Colombeau generalized numbers, preprint, arXiv:0707.0698

  41. Vladimirov V (1979) Generalized functions in mathematical physics. Mir-Publisher, Moscow

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, California Polytechnic State University, San Luis Obispo, CA, 93407, USA

    Todor D. Todorov

  2. Unit for Engineering Mathematics, University of Innsbruck, 6020, Innsbruck, Austria

    Hans Vernaeve

Authors
  1. Todor D. Todorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Vernaeve
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Todor D. Todorov.

Additional information

T. D. Todorov and H. Vernaeve were partly supported by START-project Y237 of the Austrian Science Fund. The second author was also supported by research grant M949 of Austrian Science Fund.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Todorov, T.D., Vernaeve, H. Full algebra of generalized functions and non-standard asymptotic analysis. Log Anal 1, 205–234 (2008). https://doi.org/10.1007/s11813-008-0008-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11813-008-0008-y

Keywords

  • Schwartz distributions
  • Generalized functions
  • Colombeau algebra
  • Multiplication of distributions
  • Non-standard analysis
  • Infinitesimals
  • Ultrapower non-standard model
  • Ultrafilter
  • Maximal filter
  • Robinson valuation field
  • Ultra-metric
  • Hahn–Banach theorem

Mathematics Subject Classification (2000)

  • Primary: 46F30
  • Secondary: 46S20
  • 46S10
  • 46F10
  • 03H05
  • 03C50