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.

Read more about this topic:  Mollifier

Famous quotes containing the word concrete:

    “freeways fifty lanes wide
    on a concrete continent
    spaced with bland billboards
    illustrating imbecile illusions of happiness”
    —Lawrence Ferlinghetti (b. 1919)

    “The clock of communism has stopped striking. But its concrete building has not yet come crashing down. For that reason, instead of freeing ourselves, we must try to save ourselves from being crushed by its rubble.”
    —Alexander Solzhenitsyn (b. 1918)