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Definition D138
Standard mollifier

Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers such that
 (i) $$\phi : \mathbb{R}^N \to \mathbb{R}, \quad \phi(x) = e^{- \pi |x|^2}$$ (ii) $C^{\infty}_c$ is a D392: Set of test functions
A D18: Map $\eta : (0, \infty) \to C^{\infty}_c$ is the standard mollifier with respect to $\mathbb{R}^N$ if and only if $$\forall \, \varepsilon > 0 : \forall \, x \in \mathbb{R}^N : \eta_{\varepsilon}(x) = \frac{1}{\varepsilon^N} \phi \left( \frac{x}{\varepsilon} \right)$$
Results
 ▶ The standard mollifier integrates to one