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:

    ““The city’s grotesque iron skeletons
    Would knock their drunken penthouse heads together
    And cake their concrete dirt off in the streets.””
    —Robert Frost (1874–1963)

    “Love wants to be confirmed with concrete symbols, but recklessness loves instability.”
    —Franz Grillparzer (1791–1872)