Czechoslovak Mathematical Journal

, Volume 51, Issue 3, pp 463–471

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Article

Abstract

Let $${\tilde f}$$, $${\tilde g}$$ be ultradistributions in $$\mathcal{Z}{\text{'}}$$ and let $$\tilde fn = \tilde f*\delta n$$ and $$\tilde gn = \tilde g*\sigma n$$ where $${\text{\{ }}\delta _n \}$$ is a sequence in $$\mathcal{Z}$$ which converges to the Dirac-delta function $$\delta$$. Then the neutrix product $$\tilde f\tilde g$$ is defined on the space of ultradistributions $$\mathcal{Z}{\text{'}}$$ as the neutrix limit of the sequence $$\left\{ {\frac{1}{2}\left( {\tilde fn\tilde g + \tilde f\tilde gn} \right)} \right\}$$ provided the limit $${\tilde h}$$ exist in the sense that
$$\mathop {{\text{N - lim}}}\limits_{n \to \infty } \frac{1}{2}\left\langle {\tilde f_n \tilde g + \tilde f\tilde g_n ,\psi } \right\rangle = \left\langle {\tilde h,\psi } \right\rangle$$
for all Ψ in $$\mathcal{Z}$$. We also prove that the neutrix convolution product $$fg$$ exist in $$\mathcal{D}'$$, if and only if the neutrix product $$\tilde f\tilde g$$ exist in $$\mathcal{Z}{\text{'}}$$ and the exchange formula $$F(fg) = \tilde f\tilde g$$ is then satisfied.
distributions ultradistributions delta-function neutrix limit neutrix product neutrix convolution exchange formula

References

1. 
J.G. van der Corput: Introduction to the neutrix calculus. J. Analyse Math. 7 (1959–60), 291–398.Google Scholar
2. 
B. Fisher: Neutrices and the convolution of distributions. Zb. Rad. Prirod.-Mat. Fak., Ser. Mat., Novi Sad 17 (1987), 119–135.Google Scholar
3. 
B. Fisher and Li Chen Kuan: A commutative neutrix convolution product of distributions. Zb. Rad. Prirod.-Mat. Fak., Ser. Mat., Novi Sad (1) 23 (1993), 13–27.Google Scholar
4. 
B. Fisher, E. Özçag and L.C. Kuan: A commutative neutrix convolution of distributions and exchange formula. Arch. Math. 28 (1992), 187–197.Google Scholar
5. 
I.M. Gel'fand and G.E. Shilov: Generalized functions, Vol. I. Academic Press, 1964.Google Scholar
6. 
D.S. Jones: The convolution of generalized functions. Quart. J. Math. Oxford Ser. (2) 24 (1973), 145–163.Google Scholar
7. 
F. Treves: Topological vector spaces, distributions and kernels. Academic Press, 1970.Google Scholar