On the Noncommutative Neutrix Product of Distributions

Özçaḡ, Emin; Ege, İnci; Gürçay, Haşmet; Jolevska-Tuneska, Biljana

doi:https://doi.org/10.1155/2007/81907

Abstract and Applied Analysis

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Volume 2007 | Article ID 081907 | https://doi.org/10.1155/2007/81907

On the Noncommutative Neutrix Product of Distributions

Emin Özçaḡ,¹İnci Ege,¹Haşmet Gürçay,¹and Biljana Jolevska-Tuneska²

Academic Editor: Agacik Zafer

Received21 Aug 2007

Accepted05 Nov 2007

Published26 Nov 2007

Abstract

Let f and g be distributions and let gn=(g*δn)(x), where δn(x) is a certain sequence converging to the Dirac-delta function δ(x). The noncommutative neutrix product f∘g of f and g is defined to be the neutrix limit of the sequence {fgn}, provided the limit h exists in the sense that N‐limn→∞〈f(x)gn(x),φ(x)〉=〈h(x),φ(x)〉, for all test functions in 𝒟. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products x+rlnx+∘x−−r−1lnx− and x−−r−1lnx−∘x+rlnx+ are proved to exist and are evaluated for r=1,2,…. It is consequently seen that these two products are in fact equal.

References

J. G. van der Corput, “Introduction to the neutrix calculus,” Journal d'Analyse Mathématique, vol. 7, pp. 281–399, 1960.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. Hoskins and J. S. Pinto, Distributions Ultradistributions and Other Generalized Functions, Ellis Horwood, Chichester, UK, 1994.
B. Fisher, “Neutrices and the product of distributions,” Studia Mathematica, vol. 57, no. 3, pp. 263–274, 1976.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
B. Fisher, “A non-commutative neutrix product of distributions,” Mathematische Nachrichten, vol. 108, no. 1, pp. 117–127, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
B. Fisher and K. Taş, “On the non-commutative neutrix product of the distributions $x_{+}^{λ}$ and $x_{+}^{μ}$ ,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1639–1644, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Fisher and K. Taş, “On the non-commutative neutrix product of the distributions $x_{+}^{- r} {ln}^{p} x_{+}$ and $x_{+}^{μ} {ln}^{q} x$ ,” Integral Transforms and Special Functions, vol. 17, no. 7, pp. 513–519, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Özçaḡ, B. Fisher, and H. Gürçay, “Some results on the non-commutative neutrix product of distributions,” Integral Transforms and Special Functions, vol. 11, no. 1, pp. 49–60, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. J. Ng and H. van Dam, “Neutrix calculus and finite quantum field theory,” Journal of Physics A, vol. 38, no. 18, pp. L317–L323, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Jack Ng and H. van Dam, “An application of neutrix calculus to quantum theory,” International Journal of Modern Physics A, vol. 21, no. 2, pp. 297–312, 2006.
View at: Google Scholar | Zentralblatt MATH
B. Fisher, “The product of distributions,” The Quarterly Journal of Mathematics, vol. 22, no. 2, pp. 291–298, 1971.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Fisher, “On defining the product of distributions,” Mathematische Nachrichten, vol. 99, no. 1, pp. 239–249, 1980.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Fisher and A. Kiliçman, “Neutrices and the non-commutative neutrix product of distributions,” Boletín Sociedad Matemática Mexicana, vol. 3, no. 2, pp. 299–307, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. I: Properties and Operations, Academic Press, New York, NY, USA, 1964.
View at: Zentralblatt MATH | MathSciNet
E. Özçaḡ and B. Fisher, “On defining the distribution $x_{+}^{- r} ln x_{+}$ ,” Rostocker Mathematisches Kolloquium, no. 42, pp. 25–30, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2007 Emin Özçaḡ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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