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

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.