ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Ringoid
Semiring
Ring
Left ring action
Module
Linear combination
Linear map
Linear form
Distribution
Distributional derivative
Weak derivative
Real matrix function derivative
Euclidean real function derivative
Differentiable euclidean real function
Euclidean real function slope
Euclidean real function slope function
N-times differentiable function
Infinitely differentiable function
Test function
Mollifier
Definition D138
Standard mollifier
Formulation 1
Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers such that
(i) \begin{equation} \phi : \mathbb{R}^N \to \mathbb{R}, \quad \phi(x) = e^{- \pi |x|^2} \end{equation}
(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 \begin{equation} \forall \, \varepsilon > 0 : \forall \, x \in \mathbb{R}^N : \eta_{\varepsilon}(x) = \frac{1}{\varepsilon^N} \phi \left( \frac{x}{\varepsilon} \right) \end{equation}
Results
The standard mollifier integrates to one