Foundations of Physics Letters

, Volume 7, Issue 4, pp 315–332 | Cite as

Two-dimensional dirac delta reconsidered

  • John Schmeelk
Article

Abstract

Distribution theory continues to be of significant importance in many branches of applied mathematics and especially within the research activities of theoretical and applied physicists. It is the belief of the author that the Dirac delta functional offers enormous impact in fostering advances within distribution theory together with its applications. Whenever one requires an example of a singular, ultra or new generalized function, a version of the Dirac delta satisfies that need. In this paper we have collected several very recent and important results for the Dirac delta and formulated them within a two-dimensional domain. We then go on and graph a three-dimensional version of the result implementing the software, Pro-Matlab. Within many branches of signal analysis the geometrical aspects of a particular mathematical concept are of paramount importance to the user. For example, when one implements a transform as a filter, the geometrical considerations give strong evidence of the utility of the filter for the particular application. We have also included a preliminary beginning for considering wavelet transforms applied to distributions.

Key words

rapid-descent test functions tempered distributions Fourier-transforms convolutions Stieltjes transforms Hankel transforms wavelet transforms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Akansu and R. Haddard,Multiresolution Signal Decomposition (Academic, New York, 1992).Google Scholar
  2. 2.
    N. Boccaro,Functional Analysis: An Introduction for Physicists (Academic, New York, 1990).Google Scholar
  3. 3.
    N. N. Bogolubov, A. A. Logunov, A. I. Iksak, and L. T. Todorov,General Principles of Quantum Field Theory (Kluwer Academic, Dordrecht, 1990).Google Scholar
  4. 4.
    R. Bracewell,The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).Google Scholar
  5. 5.
    Y. Brychkov and A. Prudnikov,Integral Transforms of Generalized Functions (Gordon & Breach, New York, 1989).Google Scholar
  6. 6.
    K. Chui, ed.,An Introduction to Wavelets (Academic, New York, 1992).Google Scholar
  7. 7.
    K. Chui, ed.,Wavelets: A Tutorial in Theory and Applications (Academic, New York, 1992).Google Scholar
  8. 8.
    C. Cohen-Tannoudji, B. Diu and F. Laloë,Quantum Mechanics, Vol. II (Wiley, New York, 1977).Google Scholar
  9. 9.
    J. F. Colombeau, “Some aspects of infinite-dimensional holomorphy in mathematical physics,” inAspects of Mathematics and its Applications, J. A. Barroso, ed. (Elsevier, Amsterdam, 1986), pp. 253–263.Google Scholar
  10. 10.
    J. F. Colombeau,Differential Calculus and Holomorphy (North-Holland Mathematical Studies, No. 64) (North-Holland, New York, 1982).Google Scholar
  11. 11.
    J. F. Colombeau,New Generalized Functions and Multiplication of Distributions (North-Holland Mathematical Studies, No. 84) (North-Holland, New York, 1984).Google Scholar
  12. 12.
    J. F. Colombeau,Elementary Introduction to New Generalized Functions (North-Holland Mathematical Studies, No. 113) (North-Holland, New York, 1985).Google Scholar
  13. 13.
    J. F. Colombeau, “A multiplication of distributions,”J. Math. Ann. Appl. 94, 96–115 (1983).CrossRefGoogle Scholar
  14. 14.
    J. F. Colombeau and A. Y. le Roux, “Generalized functions and products appearing in equations of physics,” preprint.Google Scholar
  15. 15.
    F. Constantinescu,Distributions and Their Applications in Physics (Pergamon, New York, 1980).Google Scholar
  16. 16.
    I. Daubechies,Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Pennsylvania, 1992).Google Scholar
  17. 17.
    N. Despotovic and A. Takaći, “On the distributional Stieltjes transformation,”Int. J. Math. Sci. 9, 313–317 (1986).CrossRefGoogle Scholar
  18. 18.
    P. A. M. Dirac,Principles of Quantum Mechanics (Oxford University Press, Oxford, 1967).Google Scholar
  19. 19.
    R. Gonzalez and R. Woods,Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).Google Scholar
  20. 20.
    A. Jain,Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, 1989).Google Scholar
  21. 21.
    E. Koh, “The n-th dimensional distributional Hankel transformations,”Can. J. Math. 27, 423–433 (1975).Google Scholar
  22. 22.
    J. LimTwo-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, 1990).Google Scholar
  23. 23.
    T. P. G. Liverman,Generalized Functions and Direct Operational Methods (Prentice-Hall, Englewood Cliffs, 1964).Google Scholar
  24. 24.
    E. G. Manovkian,Renormalization (Academic, New York, 1983).Google Scholar
  25. 25.
    G. Marinescu,Espaces vectorials pseudo-topolgiques et theorie des distributions (Deutscher Verlag der Wissenschaften, Berlin, 1963).Google Scholar
  26. 26.
    A. Messiah,Quantum Mechanics (Wiley, New York, 1958).Google Scholar
  27. 27.
    O. P. Misra and J. L. Lavoine,Transform Analysis of Generalized Functions (North-Holland Mathematics Studies, No. 119) (North Holland, New York, 1986).Google Scholar
  28. 28.
    R. Pathak and O. Gingh, “Finite Hankel transforms of distributions,”Pacific J. Math. 99 (2), 439–458 (1982).Google Scholar
  29. 29.
    A. Papoulis,Signal Analysis (McGraw-Hill, New York, 1977).Google Scholar
  30. 30.
    S. Pilipovic, “On the quasiasymptotic behavior of Stieltjes transformation of distributions,”Publ. Inst. Mathematique 40, 143–152 (1986).Google Scholar
  31. 31.
    S. Pilipovic, B. Stanhovic, and A Takaći,Asymptotic Behavior and Stieltjes Transformations of Distributions (Teubner, Leipzig, 1990).Google Scholar
  32. 32.
    W. Pratt,Digital Image Processing (Wiley, New York, 1991).Google Scholar
  33. 33.
    C. M. Roumieu, “Sur quelques extensions de la notion de distributions,”Ann. Scient. E. Norm. Sup. 77, 41–121 (1960).Google Scholar
  34. 34.
    M. Ruskai, G. Beylkin, et al., eds.,Wavelet and Their Applications (Jones & Bartlett, Boston, 1992).Google Scholar
  35. 35.
    J. Rzewuski, “On a triplet including the Hilbert space of entire functionals,”Bull. Acad. Polon. Sc., Ser. Sc. Math. Astro. Phys. 17 (7), 459–466 (1969).Google Scholar
  36. 36.
    J. Rzewuski, “On a Hilbert space of functional power series,”Bull. Acad. Polon. Sc., Ser. Sc. Math. Astro. Phys. 18, (11), 677–685 (1970).Google Scholar
  37. 37.
    J. Rzewuski, “On entire functionals in quantum field theory,”Rep. Math. Phys. 1 (1), 1–27 (1970).CrossRefGoogle Scholar
  38. 38.
    J. Rzewuski, “Some estimates for generating functionals with an application to quantum field theory,”Bull. Acad. Polon. Sc., Ser. Sc. Math. Astro. Phys. 19 (3), 235–249 (1971).Google Scholar
  39. 39.
    S. Saitoh, “Analyticity in the Meyer wavelets,” preprint.Google Scholar
  40. 40.
    L. Schiff,Quantum Mechanics (McGraw-Hill, New York, 1968).Google Scholar
  41. 41.
    J. Schmeelk, “An infinite dimensional Laplacian operator,”J. Diff. Eq. 36 (1), 74–88 (1980).CrossRefGoogle Scholar
  42. 42.
    J. Schmeelk, “Applications of test surfunctions,”Appl. Anal. 17 (3), 169–185 (1984).Google Scholar
  43. 43.
    J. Schmeelk, “Infinite-dimensional parametric distributions,”Appl. Anal. 24, 291–317 (1987).Google Scholar
  44. 44.
    J. Schmeelk, “Infinite-dimensional Fock spaces and associated creating and annihilation operators,”J. Math. Anal. Appl. 134 (2), 111–141 (1988).Google Scholar
  45. 45.
    J. Schmeelk, “A guided tour of new tempered distributions,”Found. Phys. Lett. 3 (5), 403–423 (1990).CrossRefGoogle Scholar
  46. 46.
    L. Schwartz,Theorie des distributions (Hermann, Paris, 1966).Google Scholar
  47. 47.
    L. Schwartz, “Impossibilte de la multiplication des distributions,”C. R. Acad. Sci. (Paris) 239, 847–848 (1954).Google Scholar
  48. 48.
    A. Takaći, “A note on the distributional Stieltjes transformation,”Math. Proc. Camb. Soc. 94, 523–527 (1983).Google Scholar
  49. 49.
    G. Velo and A. S. Wightman, eds.,Renormalization Theory (Proceedings, NATO Advanced Study Institute, International School of Mathematical Physics, Sicily, Italy, August 1975) (Reidel, Dordrecht, 1976).Google Scholar
  50. 50.
    U. S. Vladimirov, Y. N. Drozzinov, and B. I. Zavialow,Tauberian Theorems for Generalized Functions (Kluwer Academic, Dordrecht, 1988).Google Scholar
  51. 51.
    A. S. Wightman and K. O. Friedrich, “Differential equations of mathematical physics,” an Air Force Office of Scientific Research Scientific Report, American University, October 1, 1966.Google Scholar
  52. 52.
    A. S. Wightman and R. F. Streater, PCT,Spin and Statistics and All That (Benjamin, New York, 1965).Google Scholar
  53. 53.
    A. H. Zemanian,Distribution Theory and Transform Analysis (McGraw-Hill, New York, 1965).Google Scholar
  54. 54.
    A. H. Zemanian,Realizability Theory for Continuous Linear Systems (Academic, New York, 1972).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • John Schmeelk
    • 1
  1. 1.Department of Mathematical SciencesVirginia Commonwealth UniversityRichmond

Personalised recommendations