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Distributions pp 137-152 | Cite as

Fundamental Solutions

  • J. J. Duistermaat
  • J. A. C. Kolk
Chapter
Part of the Cornerstones book series (COR)

Abstract

Let \(P = P(\partial ) = \sum\nolimits_{|a| \le m} {C_\alpha \partial ^\alpha }\) be a linear partial differential operator in R n with constant coefficients, as introduced in (7.5). A fundamental solution of P is a distribution \(E\ \in\ \mathcal {D}^\prime({\rm \mathbf R}^n )\ {\rm such\ that}\ P\,E - \delta,\) the Dirac measure at the origin.

Keywords

Harmonic Function Open Subset Fundamental Solution Wave Operator Linear Partial Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atiyah, M.F.: Resolution of singularities and division of distributions. Comm. Pure Appl. Math. 23, 145-150 (1970).MATHMathSciNetGoogle Scholar
  2. 2.
    Bernshtein, I.N., Gel’fand, S.I.: Meromorphic property of the function P. Funct. Anal. Appl. 3,68-69 (1969).CrossRefGoogle Scholar
  3. 3.
    Bochner, S.: Sur les fonctions presques p ériodiques de Bohr. C. R. Acad. Sci. Paris 180, 1156-1158 (1925).Google Scholar
  4. 4.
    Bourbaki, N.: El éments de Math ématique, Livre V: Espaces Vectoriels Topologiques. Fasc. XVII: Chap. I, II, Fasc. XVIII: Chap. III-V. Hermann, Paris (1964).Google Scholar
  5. 5.
    Duistermaat, J.J.: Selfsimilarity of “Riemann’s nondifferentiable function.” Nieuw Arch. Wisk. (4)9, 303-337 (1991).MATHMathSciNetGoogle Scholar
  6. 6.
    Duistermaat, J.J.: Fourier Integral Operators. Birkh äuser, Boston (1996).MATHGoogle Scholar
  7. 7.
    Duistermaat, J.J., Kolk, J.A.C.: Multidimensional Real Analysis, Vols. I and II. Cambridge University Press, Cambridge (2004).Google Scholar
  8. 8.
    Ehrenpreis, L.: Solutions of some problems of division I. Amer. J. Math. 76, 883-903 (1954).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989).MATHGoogle Scholar
  10. 10.
    Harish-Chandra: Collected Papers, Vols. I-IV. Springer-Verlag, Berlin (1984).Google Scholar
  11. 11.
    H örmander, L.: The Analysis of Linear Partial Differential Operators, Vols. I-IV. Springer-Verlag, Berlin (1983-85).Google Scholar
  12. 12.
    H örmander, L.: [11, Vol. I]. Second edition. Springer-Verlag, Berlin (1990).Google Scholar
  13. 13.
    Knapp, A.W.: Basic Real Analysis. Birkh äuser, Boston (2005).Google Scholar
  14. 14.
    Knapp, A.W.: Advanced Real Analysis. Birkh äuser, Boston (2005).Google Scholar
  15. 15.
    Kolk, J.A.C., Varadarajan, V.S.: Riesz distributions. Math. Scand. 68, 273-291 (1991).MATHMathSciNetGoogle Scholar
  16. 16.
    Malgrange, B.: Existence et approximation des solutions des équations aux d ériv ées partielles et des équations de convolution. Ann. Inst. Fourier (Grenoble) 6, 271-355 (1955-56).MathSciNetGoogle Scholar
  17. 17.
    Peetre, J.: R éctification a l’article “Une caract érisation abstraite des op érateurs diff érentiels”. Math. Scand. 8, 116-120 (1960).MATHMathSciNetGoogle Scholar
  18. 18.
    Riesz, M.: L’int égrale de Riemann-Liouville et le probl ème de Cauchy. Acta Math. 81, 1-223 (1949).CrossRefMathSciNetGoogle Scholar
  19. 19.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973).MATHGoogle Scholar
  20. 20.
    Schwartz, L.: Th éorie des Distributions, Tome I-II. Hermann, Paris (1950-51).Google Scholar
  21. 21.
    Stroock, D.W.: A Concise Introduction to the Theory of Integration. Third edition. Birkh äuser, Boston (1999).MATHGoogle Scholar
  22. 22.
    Varadarajan, V.S.: Euler Through Time: A New Look at Old Themes. Amer. Math. Soc., Providence (2006).MATHGoogle Scholar
  23. 23.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Second edition. Cambridge University Press, Cambridge (1944).MATHGoogle Scholar
  24. 24.
    Weyl, H.: The method of orthogonal projection in potential theory. Duke Math. J. 7, 411-444 (1940).CrossRefMathSciNetGoogle Scholar
  25. 25.
    Whitney, H.: Elementary structure of real algebraic varieties. Ann. of Math. (2) 66, 545-556 (1957).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematical InstituteUtrecht UniversityUtrechtThe Netherlands

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