Colombeau algebra

Remarks

$\mathcal{G}(\Omega)$ is a commutative associative algebra, and the partial derivatives $\partial_j$, applied to each $u_\varepsilon$, pass to the quotient and satisfy the Leibniz rule. A smooth function $f$ embeds as the class of the constant family $u_\varepsilon = f$, and this embedding keeps products. On $\R^n$, distributions embed by convolution with a rescaled mollifier $\rho_\varepsilon(x) = \varepsilon^{-n} \rho(x / \varepsilon)$, where $\rho$ is a Schwartz function with $\int \rho = 1$ and all higher moments $0$. This embedding depends on the choice of $\rho$, and the moment condition makes it agree with the embedding of smooth functions. The Schwartz impossibility theorem shows that the product of continuous functions cannot also be kept; in $\mathcal{G}$ it is kept only up to association.

Example: on $\R$, let $\rho$ be smooth with compact support and $\int \rho = 1$, and let $u_\varepsilon(x) = \varepsilon^{-1} \rho(x / \varepsilon)$. Then $\sup \abs{\partial^\alpha u_\varepsilon} = \varepsilon^{-1-\alpha} \sup \abs{\rho^{(\alpha)}}$, so the family is moderate. Its class is associated with the delta: $\int u_\varepsilon \varphi \to \varphi(0)$ for every test function $\varphi$. Its square is the class of $(u_\varepsilon^2)$.

Non-example: the constant family $u_\varepsilon = e^{1/\varepsilon}$ is not moderate, since $e^{1/\varepsilon}$ grows faster than every power $\varepsilon^{-N}$. So it defines no element of $\mathcal{G}(\R)$.