| Title:
|
A comparison on the commutative neutrix convolution of distributions and the exchange formula (English) |
| Author:
|
Kiliçman, Adem |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
463-471 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
distributions |
| Keyword:
|
ultradistributions |
| Keyword:
|
delta-function |
| Keyword:
|
neutrix limit |
| Keyword:
|
neutrix product |
| Keyword:
|
neutrix convolution |
| Keyword:
|
exchange formula |
| MSC:
|
46F10 |
| idZBL:
|
Zbl 1079.46514 |
| idMR:
|
MR1851540 |
| . |
| Date available:
|
2009-09-24T10:44:16Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127662 |
| . |
| Reference:
|
[kn:cor] J.G. van der Corput: Introduction to the neutrix calculus.J. Analyse Math. 7 (1959–60), 291–398. MR 0124678 |
| Reference:
|
[kn:fi] B. Fisher: Neutrices and the convolution of distributions.Zb. Rad. Prirod.-Mat. Fak., Ser. Mat., Novi Sad 17 (1987), 119–135. MR 0939303 |
| Reference:
|
[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 1319771 |
| Reference:
|
[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 1222286 |
| Reference:
|
[kn:gel] I.M. Gel’fand and G.E. Shilov: Generalized functions, Vol. I.Academic Press, 1964. MR 0166596 |
| Reference:
|
[kn:jon] D.S. Jones: The convolution of generalized functions.Quart. J. Math. Oxford Ser. (2) 24 (1973), 145–163. Zbl 0256.46054, MR 0336325, 10.1093/qmath/24.1.145 |
| Reference:
|
[kn:tre] F. Treves: Topological vector spaces, distributions and kernels.Academic Press, 1970. MR 0225131 |
| . |
