The Space \(\mathcal{D}^{\prime}(\Omega )\) of Distributions

  • Dorina Mitrea
Chapter
Part of the Universitext book series (UTX)

Abstract

In this chapter the space of distributions is introduced and studied from the perspective of a topological vector space with various other additional features, such as the concept of support, multiplication with a smooth function, distributional derivatives, tensor product, and a partially defined convolution product. Here the nature of distributions with higher-order gradients continuous or bounded is also discussed.

References

  1. [8]
    J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and applications, Translated from the Dutch by J. P. van Braam Houckgeest, Cornerstones, Birkhäuser Boston, Inc., Boston, MA, 2010.Google Scholar
  2. [15]
    G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press, 2nd edition, 1998.Google Scholar
  3. [17]
    I. M. Gel’fand and G. E. Šilov, Generalized Functions, Vol. 1: Properties and Operations, Academic Press, New York and London, 1964.Google Scholar
  4. [20]
    I. M. Gel’fand, M. I. Graev, and N. Y. Vilenkin, Generalized Functions, Vol. 5: Integral Geometry and Representation theory, Academic Press, 1966.Google Scholar
  5. [24]
    G. Grubb, Distributions and Operators, Springer-Verlag, 2009.Google Scholar
  6. [25]
    M. A. Al-Gwaiz, Theory of Distributions, Pure and Applied Mathematics, CRC Press, 1992.Google Scholar
  7. [30]
    L. Hörmander, Linear Partial Differential Operators, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 116, Springer, 1969.Google Scholar
  8. [31]
    L. Hörmander, The Analysis of Linear Partial Differential Operators I, Distribution Theory and Fourier Analysis, Springer-Verlag, 2003.Google Scholar
  9. [53]
    D. Mitrea, I. Mitrea, M. Mitrea, and S. Monniaux, Groupoid Metrization Theory With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis, Birkhäuser, Springer New York, Heidelberg, Dordrecht, London, 2013.Google Scholar
  10. [59]
    W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., 1991.Google Scholar
  11. [60]
    L. Schwartz, Théorie des Distributions, I, II, Hermann, Paris, 1950–51.Google Scholar
  12. [65]
    R. Strichartz, A guide to Distribution Theory and Fourier Transforms, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.Google Scholar
  13. [68]
    M. E. Taylor, Partial Differential Equations. I. Basic Theory, Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996.Google Scholar
  14. [70]
    F. Tréves, Topological Vector Spaces, Distributions and Kernels, Dover Publications, 2006.Google Scholar
  15. [72]
    V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Dorina Mitrea
    • 1
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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