  In analogy to the classical isomorphism between
$\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}) ,\mathcal{S}^{\prime}(\mathbb{R}^{m})
) $ and $\mathcal{S}^{\prime}(\mathbb{R}^{n+m}) $, we show that a large class
of moderate linear mappings acting between the space
$\mathcal{G}\_{\mathcal{S}}(\mathbb{R}^{n}) $ of Colombeau rapidly decreasing
generalized functions and the space $\mathcal{G}\_{\tau}(\mathbb{R}^{n}) $ of
temperate ones admits generalized integral representations, with kernels
belonging to $\mathcal{G}\_{\tau}(\mathbb{R}^{n+m}) $. Furthermore, this result
contains the classical one in the sense of the generalized distribution
equality.
