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

Kiliçman, Adem. "A comparison on the commutative neutrix convolution of distributions and the exchange formula." Czechoslovak Mathematical Journal 51.3 (2001): 463-471. <http://eudml.org/doc/30648>.

@article{Kiliçman2001,
abstract = {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.},
author = {Kiliçman, Adem},
journal = {Czechoslovak Mathematical Journal},
keywords = {distributions; ultradistributions; delta-function; neutrix limit; neutrix product; neutrix convolution; exchange formula; distributions; ultradistributions; delta-function; neutrix limit; neutrix product; neutrix convolution; exchange formula},
language = {eng},
number = {3},
pages = {463-471},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A comparison on the commutative neutrix convolution of distributions and the exchange formula},
url = {http://eudml.org/doc/30648},
volume = {51},
year = {2001},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 463
EP - 471
AB - 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.
LA - eng
KW - distributions; ultradistributions; delta-function; neutrix limit; neutrix product; neutrix convolution; exchange formula; distributions; ultradistributions; delta-function; neutrix limit; neutrix product; neutrix convolution; exchange formula
UR - http://eudml.org/doc/30648
ER -