Mollifier

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A mollifier (top). At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version.
A mollifier (top). At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version.

In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a function which is rather irregular, by convolving it with a mollifier the function gets "mollified", that is, its sharp features are smoothed, while still remaining close to the original nonsmooth function.

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[edit] Definition

A function undergoing progressively more mollification.
A function undergoing progressively more mollification.

If φ is a compactly supported smooth function (i.e. a bump function) on Rs of integral equal to one, then the sequence

φn(x) = ns φ(n x)       (1)

converges to the Dirac delta function in the space of Schwartz distributions.

This means, that for any distribution T, the sequence

Tn = T ∗ φn,

where ∗ denotes convolution, converges to T. On the other hand, this is a sequence of smooth functions.

[edit] Concrete example

More precisely, consider the function ψ defined by

\Psi(x) = e^{-1/(1-x^2)} \mbox{ for } |x| < 1

and zero elsewhere. It is easily seen that this function is infinitely differentiable, with vanishing derivative for |x| = 1. Divide this function by its integral over the whole space to get a function φ of integral one, which can be used as mollifier as described above.

The function Ψ(x).
The function Ψ(x).

[edit] Smooth cutoff function

By convolution of the characteristic function of the unit ball B = { x | |x| < 1 } with φ2 (defined as in (1) with n = 2), one obtains a smooth function equal to 1 on { x | |x| < 1/2 }, with support contained in { x | |x| < 3/2 }.

It is easy to see how this can be generalized to obtain a smooth function identical to one on a given compact set, and equal to zero in every point of distance greater than a given ε to this set. Such a function is called a (smooth) cutoff function.

[edit] See also