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Czechoslovak Mathematical Journal

, Volume 51, Issue 3, pp 463–471 | Cite as

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

  • Adem Kilicman
Article

Abstract

Let \({\tilde f}\), \({\tilde g}\) be ultradistributions in \(\mathcal{Z}{\text{'}}\) and let \(\tilde fn = \tilde f*\delta n\) and \(\tilde gn = \tilde g*\sigma n\) where \({\text{\{ }}\delta _n \} \) is a sequence in \(\mathcal{Z}\) which converges to the Dirac-delta function \(\delta \). Then the neutrix product \(\tilde f\tilde g\) is defined on the space of ultradistributions \(\mathcal{Z}{\text{'}}\) as the neutrix limit of the sequence \(\left\{ {\frac{1}{2}\left( {\tilde fn\tilde g + \tilde f\tilde gn} \right)} \right\}\) provided the limit \({\tilde h}\) exist in the sense that
$$\mathop {{\text{N - lim}}}\limits_{n \to \infty } \frac{1}{2}\left\langle {\tilde f_n \tilde g + \tilde f\tilde g_n ,\psi } \right\rangle = \left\langle {\tilde h,\psi } \right\rangle $$
for all Ψ in \(\mathcal{Z}\). We also prove that the neutrix convolution product \(fg\) exist in \(\mathcal{D}'\), if and only if the neutrix product \(\tilde f\tilde g\) exist in \(\mathcal{Z}{\text{'}}\) and the exchange formula \(F(fg) = \tilde f\tilde g\) is then satisfied.
distributions ultradistributions delta-function neutrix limit neutrix product neutrix convolution exchange formula 

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Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2001

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  • Adem Kilicman

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