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. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, the
mathematician who introduced them.

History

Mollifiers were introduced by Kurt Otto Friedrichs in the paper , which is, according to Peter Lax in of volume 1, a watershed in the modern theory of partial differential equations. The name of the concept had a curious genesis, which is again described by Lax in of volume 1: at that time Friedrichs was a colleague of the mathematician , and since he liked to consult colleagues about english usage, he asked Flanders how to name the smoothing operator he was about to introduce in . Flanders was a puritan, his friends nicknamed him Moll after Moll Flanders in recognition of his moral qualities, and he suggested to call the new mathematical concept mollifier after himself: Friedrichs liked the idea and accepted the suggestion. Another curious fact is that the Italian verb "mollificare" has a meaning analogous to "smoothing" which is exactly what a mollifier does. However, Sergei Sobolev used mollifiers in his epoch making 1938 paper (the paper containing the proof of the Sobolev embedding theorem), as Friedrichs himself acknowledged in the later paper .

There is also a little misunderstanding in the concept of mollifier: Friedrichs defined "mollifier" the integral operator whose kernel is one of the functions nowadays called mollifiers. However, since the properties of an integral operator are completely determined by its kernel, the name mollfier was inherited by the kernel itself as a result of customary use.

Definition

Definition. If $\backslash phi$ is a smooth function on $\backslash scriptstyle\backslash mathbb\{R\}^n$ satisfying the following three requirements

where $\backslash delta(x)$ is the Dirac delta function and the limit must be undestood in the space of Schwartz distributions, then $\backslash phi$ is a mollifier. Note that when the theory of distributions was still unknown, as in the paper , the third property was formulated by saying that the convolution of this function with a function belonging to a Hilbert or Banach space converges to this last function: this clarifies why mollifiers are related to approximate identities. The function $\backslash phi$ could satisfy also further conditions: for example, if it satisfies

Originally, the following convolution operator was intended as mollifier

where $\backslash scriptstyle\backslash varphi\_\backslash epsilon(x)=\backslash epsilon^\{-n\}\backslash varphi(x/\backslash epsilon)$ and $\backslash phi$ is a smooth function satisfying the first three conditions stated above and one or more supplementary conditions as positivity and symmetry.

Concrete example

More precisely, consider the function $\backslash Psi(x)$ of the variable $\backslash scriptstyle\; x\backslash in\backslash mathbb\{R\}^n$ defined by

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 $\backslash phi$ of integral one, which can be used as mollifier as described above: it is also easy to see that $\backslash Psi$ defines a positive and symmetric mollifier.

Properties

All properties of a mollifier are related to its behaviour under the operation of convolution: we list the following ones, whose proof can be found in every text on distribution theory, as for example

Smoothing property

For any distribution $T$, the following sequence of convolutions indexed by the real number $\backslash epsilon$

where $\backslash ast$ denotes convolution, is a sequence of smooth functions.

Approximation of identity

For any distribution $T$, the following sequence of convolutions indexed by the real number $\backslash epsilon$ converges to $T$

Support of convolution

For any distribution $T$,

where $\backslash mathrm\{supp\}$ indicates the support in the sense of distributions, and $+$ indicates their Minkowski addition.

Applications

The basic applications of mollifiers is to prove properties valid for smooth functions also in nonsmooth situations:

Product of distributions

In some theories of generalized functions, mollifiers are used to define the multiplication of distributions: precisely, given two distributions $S$ and $T$, the limit of the product of a smooth function and a distribution

defines (if it exists) their product in various theories of generalized functions.

"Weak=Strong" theorems

Very informally, mollifiers are used to prove the identity of two different kind of extension of differential operators: the strong extension and the weak extension. The paper illustrates quite well this concept: however the high quantity of technical details needed to show what this really means prevent us from being formally detailled in this short description.

Smooth cutoff functions

By convolution of the characteristic function of the unit ball with $\backslash phi\_2$ (defined as in with $\backslash epsilon\; =\; 2)$, one obtains the function

\chi_{B_1,2}(x)=\chi_{B_1}\ast\varphi_2(x)=\int_{\mathbb{R}^n}\!\!\!\chi_{B_1}(y)\varphi_2(x-y)\mathrm{d}y=\int_{B_1}\!\!\!\varphi_2(x-y)\mathrm{d}y

which is a smooth function equal to $1$ on , with support contained in . It is easy to see how this construction can be generalized to obtain a smooth function identical to one on a neighbourhood of a given compact set, and equal to zero in every point whose distance from this set is greater than a given $\backslash epsilon$: a proof of this fact can be found in , Theorem 1.4.1. Such a function is called a (smooth) cutoff function: those functions are used to eliminate singularities of a given (generalized) function by multiplication. They leave unchanged the value of the (generalized) function they multiply only on a given set, thus modifying its support: also cutoff functions are the basic parts of smooth partitions of unity.

See also