A comparison on the commutative neutrix convolution of distributions and the exchange formula

Title:

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Creator:

Kilicman, Adem

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:76485c2e-92bb-49af-baf3-153b02556cb9
uuid:76485c2e-92bb-49af-baf3-153b02556cb9

Subject:

distributions, ultradistributions, delta-function, neutrix limit, neutrix product, neutrix convolution, and exchange formula

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

463-471

Source:

Czechoslovak Mathematical Journal | 2001 Volume:51 | Number:3

Harvested from:

CDK

Metadata only:

false