A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

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

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.

Keywords:

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

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