Skip to main content
SpringerLink
Log in

Analytic regularizations, finite part prescriptions and products of distributions

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

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

References

  1. Schwartz, L.: Sur l'impossibilité de la multiplication des distributions. C. R. A. Sc.239, 847–848 (1954)

    Google Scholar 

  2. Walter, H.F.: Über die Multiplikation von Distributionen in einem Folgenmodell. Math. Ann.189, 211–221 (1970)

    Google Scholar 

  3. Brauns, G., Liese, R.: Canonical products of distributions and causal solutions of nonlinear wave equations. J. Diff. Equ.16, 399–412 (1974)

    Google Scholar 

  4. Güttinger, W.: Products of improper operators and the renormalization problem of quantum field theory. Progress of Theoret. Phys.13, 612–626 (1955)

    Google Scholar 

  5. Taylor, J.G.: The renormalization constants in perturbation theory. Nuovo Cimento17, 695–702 (1960)

    Google Scholar 

  6. de Jager, E.M.: Application of distributions in mathematical physics. Mathematical Centre Tracts 10, 1969

  7. Mikusiński, J.: Irregular operations on distributions. Stud. Math.20, 163–169 (1961)

    Google Scholar 

  8. Itano, M.: On the theory of the multiplicative products of distributions. J. Sc. Hirosh. Univ.30, 151–181 (1966)

    Google Scholar 

  9. Shiraishi, R.: On the value of distributions at a point and the multiplicative products. J. Sc. Hirosh. Univ.31, 89–104 (1967)

    Google Scholar 

  10. Fisher, B.: The product of distributions. Quart. J. Math. Oxford22, 291–298 (1971)

    Google Scholar 

  11. Fisher, B.: Singular products of distributions. Math. Ann.203, 103–116 (1973)

    Google Scholar 

  12. Hirata, Y., Ogata, H.: On the exchange formula for distributions. J. Sc. Hirosh. Univ.22, 147–152 (1958)

    Google Scholar 

  13. Jones, D.S.: The convolution of generalized functions. Quart. J. Math. Oxford24, 145–163 (1973)

    Google Scholar 

  14. Bremermann, H.J., Durand, L.: On analytic continuation, multiplication, and Fourier transformations of Schwartz distributions. J. Math. Phys.2, 240–258 (1961)

    Google Scholar 

  15. Hörmander, L.: Fourier integral operators. I. Acta Math.127, 79–183 (1971)

    Google Scholar 

  16. König, H.: Multiplikation von Distributionen. Math. Ann.128, 420–452 (1955)

    Google Scholar 

  17. Rosinger, E.: An associative, commutative distribution multiplication. Techn. Rep. No. 80, March 1976, Technion, Haifa

  18. Thurber, J., Katz, J.: Application of fractional powers of delta functions. In: Victoria Symposium on Nonstandard Analysis. Lecture Notes in Mathematics 369. Berlin, Heidelberg, New York: Springer 1974

    Google Scholar 

  19. Bogoliubov, N.N., Parasiuk, O.S.: Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder. Acta Math.97, 227–266 (1957)

    Google Scholar 

  20. Hepp, K.: Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. math. Phys.2, 301–326 (1966)

    Google Scholar 

  21. Bollini, C.G., Giambiagi, J.J., González Dominguez, A.: Analytic regularization and the divergences of quantum field theory. Nuovo Cimento31, 550–561 (1964)

    Google Scholar 

  22. Speer, E.R.: Generalized Feynman amplitudes. Princeton: Princeton University Press 1969

    Google Scholar 

  23. Bollini, C.G., Giambiagi, J.J.: Dimensional renormalization: the number of dimensions as a regularizing parameter. Nuovo Cimento12B, 20–25 (1972)

    Google Scholar 

  24. t'Hooft, G., Veltman, M.: Diagrammar. CERN Preprint 73-9

  25. Keller, K.: Konstruktion von Produkten in einer für Feldtheorien wichtigen Klasse von Distributionen. Thesis, Aachen, July 1974

  26. Keller, K.: On the multiplication of distributions. I-V. Techn. Rep., TH Aachen, May 1975–January 1977

  27. Keller, K.: Irregular operations in quantum field theory. I. Multiplication of distributions. February 1977. Rep. on Math. Phys. (to appear)

  28. Keller, K.: Irregular operations in quantum field theory. II. Evaluation functionals and products, convolutions, integrals of distributions. April 1977. Rep. on Math. Phys. (to appear)

  29. Schwartz, L.: Théorie des distributions. Paris: Hermann 1966

    Google Scholar 

  30. Gel'fand, I.M., Shilov, G.E.: Generalized Functions (Distributions), Vol. 1. New York: Academic Press 1964

    Google Scholar 

  31. Itano, M., Hatano, S.: Note on the multiplicative products of 84-1 and 84-2. Mem. Fac. Gen. Ed. Hirosh. Univ. III8, 159–174 (1974)

    Google Scholar 

  32. Fisher, B.: Neutrices and the product of distributions. Stud. Math.57, 263–274 (1976)

    Google Scholar 

  33. Shiraishi, R., Itano, M.: On the multiplicative products of distributions. J. Sc. Hirosh. Univ.28, 223–235 (1964)

    Google Scholar 

  34. Bernshtein, I.N.: The analytic continuation of generalized functions with respect to a parameter. Funkts. Analiz.6, 26–40 (1972)

    Google Scholar 

  35. Horváth, J.: Finite parts of distributions. In: Linear Operators and Approximation, Proceedings of the Oberwolfach Conference 1971 (ed. P.L. Butzer, J.P. Kahane, B. Szökefalvi-Nagy). Basel, Stuttgart: Birkhäuser 1972

    Google Scholar 

  36. Riesz, M.: L'integrale de Riemann-Liouville et le problème de Cauchy. Acta Math.81, 1–223 (1949)

    Google Scholar 

  37. Edwards, R.E.: Functional analysis. New York: Holt, Rinehart, and Winston 1965

    Google Scholar 

  38. Łojasiewicz, S.: Sur la valeur et la limite d'une distribution en un point. Stud. Math.16, 1–36 (1957)

    Google Scholar 

  39. Fuchssteiner, B.: Eine assoziative Algebra über einem Unterraum der Distributionen. Math. Ann.178, 302–314 (1968)

    Google Scholar 

  40. Brédimas, B.: La différentiation d'ordre complexe et les produits canonique et de convolution généralisé. C.R. Acad. Sc. Paris282, Sér. A, 37–40 (1976);283, Sér. A. 1095–1098 (1976)

    Google Scholar 

  41. Tillmann, H.G.: Darstellung der Schwartzschen Distributionen durch analytische Funktionen. Math. Z.77, 106–124 (1961)

    Google Scholar 

  42. Vladimirov, V.S.: Methods of the theory of functions of many complex variables. Cambridge: M.I.T. Press 1966

    Google Scholar 

  43. Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. 2. Fourier analysis, self-adjointness. New York: Academic Press 1975

    Google Scholar 

Download references

Author information

Author notes

  1. Klaus Keller

    Present address: Mathematisches Institut, Lehrstuhl I, Universität Dortmund, D-4600, Dortmund, Federal Republic of Germany

Authors and Affiliations

  1. Institut für Theoretische Physik, Lehrstuhl E, Technische Hochschule Aachen, D-5100, Aachen, Federal Republic of Germany

    Klaus Keller

Authors
  1. Klaus Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work supported in part by the Deutsche Forschungsgemeinschaft

Rights and permissions

Reprints and permissions

About this article

Cite this article

Keller, K. Analytic regularizations, finite part prescriptions and products of distributions. Math. Ann. 236, 49–84 (1978). https://doi.org/10.1007/BF01420256

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01420256

Keywords

Search

Navigation