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.
Similar content being viewed by others
References
A. Akansu and R. Haddard,Multiresolution Signal Decomposition (Academic, New York, 1992).
N. Boccaro,Functional Analysis: An Introduction for Physicists (Academic, New York, 1990).
N. N. Bogolubov, A. A. Logunov, A. I. Iksak, and L. T. Todorov,General Principles of Quantum Field Theory (Kluwer Academic, Dordrecht, 1990).
R. Bracewell,The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).
Y. Brychkov and A. Prudnikov,Integral Transforms of Generalized Functions (Gordon & Breach, New York, 1989).
K. Chui, ed.,An Introduction to Wavelets (Academic, New York, 1992).
K. Chui, ed.,Wavelets: A Tutorial in Theory and Applications (Academic, New York, 1992).
C. Cohen-Tannoudji, B. Diu and F. Laloë,Quantum Mechanics, Vol. II (Wiley, New York, 1977).
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.
J. F. Colombeau,Differential Calculus and Holomorphy (North-Holland Mathematical Studies, No. 64) (North-Holland, New York, 1982).
J. F. Colombeau,New Generalized Functions and Multiplication of Distributions (North-Holland Mathematical Studies, No. 84) (North-Holland, New York, 1984).
J. F. Colombeau,Elementary Introduction to New Generalized Functions (North-Holland Mathematical Studies, No. 113) (North-Holland, New York, 1985).
J. F. Colombeau, “A multiplication of distributions,”J. Math. Ann. Appl. 94, 96–115 (1983).
J. F. Colombeau and A. Y. le Roux, “Generalized functions and products appearing in equations of physics,” preprint.
F. Constantinescu,Distributions and Their Applications in Physics (Pergamon, New York, 1980).
I. Daubechies,Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Pennsylvania, 1992).
N. Despotovic and A. Takaći, “On the distributional Stieltjes transformation,”Int. J. Math. Sci. 9, 313–317 (1986).
P. A. M. Dirac,Principles of Quantum Mechanics (Oxford University Press, Oxford, 1967).
R. Gonzalez and R. Woods,Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).
A. Jain,Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, 1989).
E. Koh, “The n-th dimensional distributional Hankel transformations,”Can. J. Math. 27, 423–433 (1975).
J. LimTwo-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, 1990).
T. P. G. Liverman,Generalized Functions and Direct Operational Methods (Prentice-Hall, Englewood Cliffs, 1964).
E. G. Manovkian,Renormalization (Academic, New York, 1983).
G. Marinescu,Espaces vectorials pseudo-topolgiques et theorie des distributions (Deutscher Verlag der Wissenschaften, Berlin, 1963).
A. Messiah,Quantum Mechanics (Wiley, New York, 1958).
O. P. Misra and J. L. Lavoine,Transform Analysis of Generalized Functions (North-Holland Mathematics Studies, No. 119) (North Holland, New York, 1986).
R. Pathak and O. Gingh, “Finite Hankel transforms of distributions,”Pacific J. Math. 99 (2), 439–458 (1982).
A. Papoulis,Signal Analysis (McGraw-Hill, New York, 1977).
S. Pilipovic, “On the quasiasymptotic behavior of Stieltjes transformation of distributions,”Publ. Inst. Mathematique 40, 143–152 (1986).
S. Pilipovic, B. Stanhovic, and A Takaći,Asymptotic Behavior and Stieltjes Transformations of Distributions (Teubner, Leipzig, 1990).
W. Pratt,Digital Image Processing (Wiley, New York, 1991).
C. M. Roumieu, “Sur quelques extensions de la notion de distributions,”Ann. Scient. E. Norm. Sup. 77, 41–121 (1960).
M. Ruskai, G. Beylkin, et al., eds.,Wavelet and Their Applications (Jones & Bartlett, Boston, 1992).
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).
J. Rzewuski, “On a Hilbert space of functional power series,”Bull. Acad. Polon. Sc., Ser. Sc. Math. Astro. Phys. 18, (11), 677–685 (1970).
J. Rzewuski, “On entire functionals in quantum field theory,”Rep. Math. Phys. 1 (1), 1–27 (1970).
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).
S. Saitoh, “Analyticity in the Meyer wavelets,” preprint.
L. Schiff,Quantum Mechanics (McGraw-Hill, New York, 1968).
J. Schmeelk, “An infinite dimensional Laplacian operator,”J. Diff. Eq. 36 (1), 74–88 (1980).
J. Schmeelk, “Applications of test surfunctions,”Appl. Anal. 17 (3), 169–185 (1984).
J. Schmeelk, “Infinite-dimensional parametric distributions,”Appl. Anal. 24, 291–317 (1987).
J. Schmeelk, “Infinite-dimensional Fock spaces and associated creating and annihilation operators,”J. Math. Anal. Appl. 134 (2), 111–141 (1988).
J. Schmeelk, “A guided tour of new tempered distributions,”Found. Phys. Lett. 3 (5), 403–423 (1990).
L. Schwartz,Theorie des distributions (Hermann, Paris, 1966).
L. Schwartz, “Impossibilte de la multiplication des distributions,”C. R. Acad. Sci. (Paris) 239, 847–848 (1954).
A. Takaći, “A note on the distributional Stieltjes transformation,”Math. Proc. Camb. Soc. 94, 523–527 (1983).
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).
U. S. Vladimirov, Y. N. Drozzinov, and B. I. Zavialow,Tauberian Theorems for Generalized Functions (Kluwer Academic, Dordrecht, 1988).
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.
A. S. Wightman and R. F. Streater, PCT,Spin and Statistics and All That (Benjamin, New York, 1965).
A. H. Zemanian,Distribution Theory and Transform Analysis (McGraw-Hill, New York, 1965).
A. H. Zemanian,Realizability Theory for Continuous Linear Systems (Academic, New York, 1972).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schmeelk, J. Two-dimensional dirac delta reconsidered. Found Phys Lett 7, 315–332 (1994). https://doi.org/10.1007/BF02186682
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02186682