Mollifier - Concrete Example

Concrete Example

Consider the function of the variable ℝn defined by

\varphi(x) = \begin{cases} e^{-1/(1-|x|^2)}& \text{ if } |x| < 1\\ 0& \text{ if } |x|\geq 1 \end{cases}

It is easily seen that this function is infinitely differentiable, non analytic with vanishing derivative for |x| = 1. Divide this function by its integral over the whole space to get a function of integral one, which can be used as mollifier as described above: it is also easy to see that defines a positive and symmetric mollifier.

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