Mollifier

In mathematics, mollifiers are smooth functions with special properties, used in distribution theory to create a sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

General idea

If φ is a compactly supported smooth function on R^s of integral equal to one, then the sequence

φ_n(x) = n^s 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 T_n = T * φ_n, where * denotes convolution, converges to T. On the other hand, this is a sequence of smooth functions.

Concrete example

More precisely, consider the function ψ defined by ψ(x)=exp(1/(x ²-1)) 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.

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 set, and equal to zero in every point of distance greater than a given ε to this set. Such a function is called (smooth) cutoff function .