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Logic and Analysis

, 1:205 | Cite as

Full algebra of generalized functions and non-standard asymptotic analysis

  • Todor D. Todorov
  • Hans Vernaeve
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.

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 

References

  1. 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 ObispoGoogle Scholar
  2. 2.
    Biagioni HA (1990) A Nonlinear theory of generalized functions. Lecture Notes in Mathematics, vol 1421, (XII). Springer, HeidelbergGoogle Scholar
  3. 3.
    Bremermann H (1965) Distributions, complex variables, and Fourier transforms. Addison-Wesley, Palo AltoMATHGoogle Scholar
  4. 4.
    Capiński M, Cutland NJ (1995) Nonstandard methods for stochastic fluid mechanics. World Scientific, Singapore-New Jersey-London-Hong KongMATHGoogle Scholar
  5. 5.
    Chang CC, Keisler HJ (1998) Model theory studies in logic and the foundations of mathematics, vol 73. Elsevier, AmsterdamGoogle Scholar
  6. 6.
    Colombeau JF (1984) New generalized functions and multiplication of distributions, Math Studies, vol 84. Amsterdam, North-HollandGoogle Scholar
  7. 7.
    Colombeau JF (1985) Elementary introduction to new generalized functions. North Holland, AmsterdamMATHGoogle Scholar
  8. 8.
    Colombeau JF, Le Roux AY (1988) Multiplication of distributions in elasticity and hydrodynamics. J Math Phys 29: 315–319MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Colombeau JF (1990) Multiplication of distributions. Bull AMS 23(2): 251–268MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Colombeau JF (1992) Multiplication of distributions: a tool in mathematics, numerical engineering and theoretical physics. Lecture Notes in Mathematics, vol 1532. Springer, BerlinGoogle Scholar
  11. 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]Google Scholar
  12. 12.
    Davis M (2005) Applied nonstandard analysis. Dover Publications, Inc., New YorkGoogle Scholar
  13. 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 YorkGoogle Scholar
  14. 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, DordrechtGoogle Scholar
  15. 15.
    Ingleton AW (1952) The Hahn-Banach theorem for non-archimedean valued fields. Proc Cambridge Phil Soc 48: 41–45MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kaneko A (1988) Introduction to hyperfunctions. Kluwer, DordrechtMATHGoogle Scholar
  17. 17.
    Kelley JL (1975) General topology. Prindle, Springer, New YorkMATHGoogle Scholar
  18. 18.
    Levi-Civita T (1954) Sugli Infiniti ed Infinitesimi Attuali Quali Elementi Analitici (1892–1893), Opere Mathematiche, vol 1. Bologna, p1–39Google Scholar
  19. 19.
    Bang-He L (1978) Non-standard analysis and multiplication of disributions. Sci Sinica 21: 561–585MathSciNetGoogle Scholar
  20. 20.
    Lightstone AH, Robinson A (1975) Nonarchimedean fields and asymptotic expansions. North-Holland, AmsterdamMATHGoogle Scholar
  21. 21.
    Lindstrøm T (1988) An invitation to nonstandard analysis. In: Cutland N(eds) Nonstandard analysis and its applications. Cambridge University Press, London, pp 1–105Google Scholar
  22. 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)Google Scholar
  23. 23.
    Luxemburg WAJ (1976) On a class of valuation fields introduced by Robinson. Israel J Math 25: 189–201MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mayerhofer E (2006) The wave equation on singular space-times. PhD Thesis, Faculty of Mathematics, University of Vienna, ViennaGoogle Scholar
  25. 25.
    Mayerhofer E (2007) Spherical completeness of the non-archimedean ring of Colombeau generalized numbers. Bull Inst Math Acad Sin (New Series) 2(3): 769–783MATHMathSciNetGoogle Scholar
  26. 26.
    Oberguggenberger M (1992) Multiplication of distributions and applications to partial differential equations. Pitman Research Notes Math., vol 259. Longman, HarlowGoogle Scholar
  27. 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 130–150Google Scholar
  28. 28.
    Oberguggenberger M, Todorov T (1998) An embedding of Schwartz distributions in the algebra of asymptotic functions. Int J Math Math Sci 21: 417–428MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Oberguggenberger M, Vernaeve H (2008) Internal sets and internal functions in Colombeau theory. J Math Anal Appl (to appear)Google Scholar
  30. 30.
    Pestov V (1991) On a valuation field invented by A. Robinson and certain structures connected with it. Israel J Math 74: 65–79MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Ribenboim P (1999) The theory of classical valuations. Springer Monographs in Mathematics. Springer, HeidelbergGoogle Scholar
  32. 32.
    Robinson A (1966) Nonstandard analysis. North Holland, AmsterdamGoogle Scholar
  33. 33.
    Robinson A (1973) Function theory on some nonarchimedean fields. Amer Math Monthly 80(6), Part II: Papers in the Foundations of Mathematics, pp 87–109Google Scholar
  34. 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. 35.
    Schwartz L (1954) Sur l’impossibilité de la multiplication des distributions. C R Acad Sci Paris 239: 847–848MATHMathSciNetGoogle Scholar
  36. 36.
    Stroyan KD, Luxemburg WAJ (1976) Introduction to the theory of infinitesimals. Academic Press, New YorkMATHGoogle Scholar
  37. 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 369–383Google Scholar
  38. 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/0601722Google Scholar
  39. 39.
    VanDer Waerden BL (1964) Modern algebra, 3rd printing. Ungar Publishing, New YorkGoogle Scholar
  40. 40.
    Vernaeve H (2007) Ideals in the ring of Colombeau generalized numbers, preprint, arXiv:0707.0698Google Scholar
  41. 41.
    Vladimirov V (1979) Generalized functions in mathematical physics. Mir-Publisher, MoscowGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  2. 2.Unit for Engineering MathematicsUniversity of InnsbruckInnsbruckAustria

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