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:

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

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