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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], and let $C^\\infty(\\Omega)$ be the algebra of [[smooth functions|0V33BNKZ9S]] on $\\Omega$.\nLet $\\mathcal{E}(\\Omega)$ be the algebra of families $(u_\\varepsilon)_{\\varepsilon \\in (0,1]}$ in $C^\\infty(\\Omega)$, with pointwise operations.\nFor a [[compact set|0R6EWFKRCF]] $K \\subseteq \\Omega$ and a multi-index $\\alpha$, let $p_{K,\\alpha}(u) = \\sup_{x \\in K} \\abs{\\partial^\\alpha u(x)}$.\n\n1. A family $(u_\\varepsilon)$ is *moderate* if for each $K$ and $\\alpha$ there is $N \\ge 0$ with $p_{K,\\alpha}(u_\\varepsilon) = O(\\varepsilon^{-N})$ as $\\varepsilon \\to 0$.\n2. A family $(u_\\varepsilon)$ is *negligible* if for each $K$, $\\alpha$ and $m \\ge 0$, $p_{K,\\alpha}(u_\\varepsilon) = O(\\varepsilon^{m})$ as $\\varepsilon \\to 0$.\n\nThe moderate families form a subalgebra $\\mathcal{E}_M(\\Omega)$, and the negligible families form an [[ideal|096Z9MP2ME]] $\\mathcal{N}(\\Omega)$ of it.\nThe *special Colombeau algebra* on $\\Omega$ is the [[quotient algebra|0076HWHGHQ]] $\\mathcal{G}(\\Omega) = \\mathcal{E}_M(\\Omega) / \\mathcal{N}(\\Omega)$.\n\n## Remarks\n\n$\\mathcal{G}(\\Omega)$ 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.\nA smooth function $f$ embeds as the class of the constant family $u_\\varepsilon = f$, and this embedding keeps products.\nOn $\\R^n$, 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 $0$.\nThis embedding depends on the choice of $\\rho$, and the moment condition makes it agree with the embedding of smooth functions.\nThe [[Schwartz impossibility theorem|Q0C1H111EN]] shows that the product of continuous functions cannot also be kept; in $\\mathcal{G}$ it is kept only up to association.\n\nExample: on $\\R$, let $\\rho$ be smooth with compact support and $\\int \\rho = 1$, and let $u_\\varepsilon(x) = \\varepsilon^{-1} \\rho(x / \\varepsilon)$.\nThen $\\sup \\abs{\\partial^\\alpha u_\\varepsilon} = \\varepsilon^{-1-\\alpha} \\sup \\abs{\\rho^{(\\alpha)}}$, so the family is moderate.\nIts class is associated with the delta: $\\int u_\\varepsilon \\varphi \\to \\varphi(0)$ for every test function $\\varphi$.\nIts square is the class of $(u_\\varepsilon^2)$.\n\nNon-example: the constant family $u_\\varepsilon = e^{1/\\varepsilon}$ is not moderate, since $e^{1/\\varepsilon}$ grows faster than every power $\\varepsilon^{-N}$.\nSo it defines no element of $\\mathcal{G}(\\R)$.\n",
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