Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in $\mathcal Z^{\prime }$ and let $\tilde{f}_n = \tilde{f} * \delta _n$ and $\tilde{g}_n = \tilde{g} * \sigma _n$ where $\lbrace \delta _n \rbrace $ is a sequence in $\mathcal Z$ which converges to the Dirac-delta function $\delta $. Then the neutrix product $\tilde{f} \diamond \tilde{g}$ is defined on the space of ultradistributions $\mathcal Z^{\prime }$ as the neutrix limit of the sequence $\lbrace {1 \over 2}(\tilde{f}_n \tilde{g} + \tilde{f} \tilde{g}_n)\rbrace $ provided the limit $\tilde{h}$ exist in the sense that \[ \mathop {\mathrm N\text{-}lim}_{n\rightarrow \infty }{1 \over 2} \langle \tilde{f}_n \tilde{g} +\tilde{f} \tilde{g}_n, \psi \rangle = \langle \tilde{h}, \psi \rangle \] for all $\psi $ in $\mathcal Z$. We also prove that the neutrix convolution product $f \mathbin {\diamondsuit \!\!\!\!*\,}g$ exist in $\mathcal D^{\prime }$, if and only if the neutrix product $\tilde{f} \diamond \tilde{g}$ exist in $\mathcal Z^{\prime }$ and the exchange formula \[ F(f \mathbin {\diamondsuit \!\!\!\!*\,}g) = \tilde{f} \diamond \tilde{g} \] is then satisfied.
Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in $\mathcal Z^{\prime }$ and let $\tilde{f}_n = \tilde{f} * \delta _n$ and $\tilde{g}_n = \tilde{g} * \sigma _n$ where $\lbrace \delta _n \rbrace $ is a sequence in $\mathcal Z$ which converges to the Dirac-delta function $\delta $. Then the neutrix product $\tilde{f} \diamond \tilde{g}$ is defined on the space of ultradistributions $\mathcal Z^{\prime }$ as the neutrix limit of the sequence $\lbrace {1 \over 2}(\tilde{f}_n \tilde{g} + \tilde{f} \tilde{g}_n)\rbrace $ provided the limit $\tilde{h}$ exist in the sense that \[ \mathop {\mathrm N\text{-}lim}_{n\rightarrow \infty }{1 \over 2} \langle \tilde{f}_n \tilde{g} +\tilde{f} \tilde{g}_n, \psi \rangle = \langle \tilde{h}, \psi \rangle \] for all $\psi $ in $\mathcal Z$. We also prove that the neutrix convolution product $f \mathbin {\diamondsuit \!\!\!\!*\,}g$ exist in $\mathcal D^{\prime }$, if and only if the neutrix product $\tilde{f} \diamond \tilde{g}$ exist in $\mathcal Z^{\prime }$ and the exchange formula \[ F(f \mathbin {\diamondsuit \!\!\!\!*\,}g) = \tilde{f} \diamond \tilde{g} \] is then satisfied.
@article{CMJ_2001_51_3_a1,
author = {Kili\c{c}man, Adem},
title = {A comparison on the commutative neutrix convolution of distributions and the exchange formula},
journal = {Czechoslovak Mathematical Journal},
pages = {463--471},
year = {2001},
volume = {51},
number = {3},
mrnumber = {1851540},
zbl = {1079.46514},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a1/}
}
TY - JOUR
AU - Kiliçman, Adem
TI - A comparison on the commutative neutrix convolution of distributions and the exchange formula
JO - Czechoslovak Mathematical Journal
PY - 2001
SP - 463
EP - 471
VL - 51
IS - 3
UR - http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a1/
LA - en
ID - CMJ_2001_51_3_a1
ER -
%0 Journal Article
%A Kiliçman, Adem
%T A comparison on the commutative neutrix convolution of distributions and the exchange formula
%J Czechoslovak Mathematical Journal
%D 2001
%P 463-471
%V 51
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a1/
%G en
%F CMJ_2001_51_3_a1
Kiliçman, Adem. A comparison on the commutative neutrix convolution of distributions and the exchange formula. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 463-471. http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a1/
[kn:cor] J.G. van der Corput: Introduction to the neutrix calculus. J. Analyse Math. 7 (1959–60), 291–398. | MR
[kn:fi] B. Fisher: Neutrices and the convolution of distributions. Zb. Rad. Prirod.-Mat. Fak., Ser. Mat., Novi Sad 17 (1987), 119–135. | MR
[kn:li] 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. | MR
[kn:ozli] B. Fisher, E. Özçaḡ and L. C. Kuan: A commutative neutrix convolution of distributions and exchange formula. Arch. Math. 28 (1992), 187–197. | MR
[kn:gel] I.M. Gel’fand and G.E. Shilov: Generalized functions, Vol. I. Academic Press, 1964. | MR
[kn:jon] D.S. Jones: The convolution of generalized functions. Quart. J. Math. Oxford Ser. (2) 24 (1973), 145–163. | DOI | MR | Zbl
[kn:tre] F. Treves: Topological vector spaces, distributions and kernels. Academic Press, 1970. | MR
