University Putra Malaysia, Department of Mathematics, 43400 UPM, Serdang, Selangor, Malaysia, e-mail: akilic@fsas.upm.edu.my
Abstract: Let $\tilde f$, $\tilde g$ be ultradistributions in $\cal Z'$ and let $\tilde f_n = \tilde f * \delta_n$ and $\tilde g_n = \tilde g * \sigma_n$ where $\{\delta_n \}$ is a sequence in $\cal Z$ which converges to the Dirac-delta function $\delta$. Then the neutrix product $\tilde f \diamond\tilde g$ is defined on the space of ultradistributions $\cal Z'$ as the neutrix limit of the sequence $\{{1 \over2}(\tilde f_n \tilde g + \tilde f \tilde g_n)\}$ provided the limit $\tilde h$ exist in the sense that
\Nlim_{n\to\infty}{1 \over2} \langle\tilde f_n \tilde g +\tilde f \tilde g_n, \psi\rangle= \langle\tilde h, \psi\rangle
for all $\psi$ in $\cal Z$. We also prove that the neutrix convolution product $f \dast g$ exist in $\cal D'$, if and only if the neutrix product $\tilde f \diamond\tilde g$ exist in $\cal Z'$ and the exchange formula
F(f \dast g) = \tilde f \diamond\tilde g
is then satisfied.
Keywords: distributions, ultradistributions, delta-function, neutrix limit, neutrix product, neutrix convolution, exchange formula
Classification (MSC 2000): 46F10
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