| Title:
|
The product of distributions on $R^m$ (English) |
| Author:
|
Lin-Zhi, Cheng |
| Author:
|
Fisher, Brian |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
4 |
| Year:
|
1992 |
| Pages:
|
605-614 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The fixed infinitely differentiable function $\rho (x)$ is such that $\{n\rho (n x)\}$ is a re\-gular sequence converging to the Dirac delta function $\delta $. The function $\delta _{\bold n}(\bold x)$, with $\bold x=(x_1, \dots , x_m)$ is defined by $$ \delta _{\bold n}(\bold x)=n_1 \rho (n_1 x_1)\dots n_m \rho (n_m x_m). $$ The product $f \circ g$ of two distributions $f$ and $g$ in $\mathcal D'_m$ is the distribution $h$ defined by $$ \operatornamewithlimits{N\mbox{--}\lim}\limits _{n_1\rightarrow \infty } \dots \operatornamewithlimits{N\mbox{--}\lim}\limits _{n_m\rightarrow \infty } \langle f_{\bold n} g_{\bold n}, \phi \rangle = \langle h, \phi \rangle, $$ provided this neutrix limit exists for all $\phi (\bold x)=\phi _1(x_1)\dots \phi _m(x_m)$, where $f_{\bold n}=f \ast \delta _{\bold n}$ and $g_{\bold n}=g\ast \delta _{\bold n}$. (English) |
| Keyword:
|
distribution |
| Keyword:
|
neutrix limit |
| Keyword:
|
neutrix product |
| MSC:
|
46F10 |
| idZBL:
|
Zbl 0818.46035 |
| idMR:
|
MR1240181 |
| . |
| Date available:
|
2009-01-08T17:58:53Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118531 |
| . |
| Reference:
|
[1] Cheng L.Z., Fisher B.: Several products of distributions on $R^m$.Proc. R. Soc. Lond. A 426 (1989), 425-439. MR 1030468 |
| Reference:
|
[2] van der Corput J.G.: Introduction to the neutrix calculus.J. Analyse Math. 7 (1959-60), 291-398. Zbl 0097.10503, MR 0124678 |
| Reference:
|
[3] Fisher B.: The product of distributions.Quart. J. Math. (2) 22 (1971), 291-298. Zbl 0213.13104, MR 0287308 |
| Reference:
|
[4] Fisher B.: The product of the distributions $x_+^{-r-1/2}$ and $x_-^{-r-1/2}$.Proc. Camb. Phil. Soc. 71 (1972), 123-130. Zbl 0239.46031, MR 0296690 |
| Reference:
|
[5] Fisher B.: The neutrix distribution product $x_+^{-r}\delta ^{(r-1)}(x)$.Studia Sci. Math. Hungar. 9 (1974), 439-441. MR 0412805 |
| Reference:
|
[6] Fisher B., Li C.K.: On the product of distributions in $m$ variables.Jiangsu Coll. Jnl. 11 (1990), 1-10. MR 1069541 |
| Reference:
|
[7] Schwartz L.: Théorie des distributions.Vol. I, II, Herman, 1957. Zbl 0962.46025, MR 0107812 |
| . |
