Mollifier

dimension

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

In

They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them.^[2]

Historical notes

Mollifiers were introduced by

Previously,

It must be pointed out that the term "mollifier" has undergone

is one of the functions nowadays called mollifiers. However, since the properties of a linear integral operator are completely determined by its kernel, the name mollifier was inherited by the kernel itself as a result of common usage.

Definition

Modern (distribution based) definition

Definition 1. If ${\displaystyle \varphi }$ is a

on ℝⁿ, n ≥ 1, satisfying the following three requirements

(1)   it is compactly supported^[8]

(2)  ${\displaystyle \int _{\mathbb {R} ^{n}}\!\varphi (x)\mathrm {d} x=1}$

(3)  ${\displaystyle \lim _{\epsilon \to 0}\varphi _{\epsilon }(x)=\lim _{\epsilon \to 0}\epsilon ^{-n}\varphi (x/\epsilon )=\delta (x)}$

where ${\displaystyle \delta (x)}$ is the Dirac delta function and the limit must be understood in the space of Schwartz distributions, then ${\displaystyle \varphi }$ is a mollifier. The function ${\displaystyle \varphi }$ could also satisfy further conditions:^[9] for example, if it satisfies

(4)  ${\displaystyle \varphi }$${\displaystyle (x)}$ ≥ 0 for all x ∈ ℝⁿ, then it is called a positive mollifier

(5)  ${\displaystyle \varphi }$${\displaystyle (x)}$=${\displaystyle \mu }$${\displaystyle (|x|)}$ for some

infinitely differentiable function

${\displaystyle \mu }$ : ℝ⁺ → ℝ, then it is called a symmetric mollifier

Notes on Friedrichs' definition

Note 1. When the theory of distributions was still not widely known nor used,^[10] property (3) above was formulated by saying that the convolution of the function ${\displaystyle \scriptstyle \varphi _{\epsilon }}$ with a given function belonging to a proper

Note 2. As briefly pointed out in the "Historical notes" section of this entry, originally, the term "mollifier" identified the following convolution operator:^[13][14]

${\displaystyle \Phi _{\epsilon }(f)(x)=\int _{\mathbb {R} ^{n}}\varphi _{\epsilon }(x-y)f(y)\mathrm {d} y}$

where ${\displaystyle \varphi _{\epsilon }(x)=\epsilon ^{-n}\varphi (x/\epsilon )}$ and ${\displaystyle \varphi }$ is a

satisfying the first three conditions stated above and one or more supplementary conditions as positivity and symmetry.

Concrete example

Consider the bump function ${\displaystyle \varphi }$${\displaystyle (x)}$ of a variable in ℝⁿ defined by

${\displaystyle \varphi (x)={\begin{cases}e^{-1/(1-|x|^{2})}/I_{n}&{\text{ if }}|x|<1\\0&{\text{ if }}|x|\geq 1\end{cases}}}$

where the numerical constant ${\displaystyle I_{n}}$ ensures normalization. This function is infinitely differentiable, non analytic with vanishing derivative for |x| = 1. ${\displaystyle \varphi }$ can be therefore used as mollifier as described above: one can see that ${\displaystyle \varphi }$${\displaystyle (x)}$ defines a positive and symmetric mollifier.^[15]

Properties

All properties of a mollifier are related to its behaviour under the operation of convolution: we list the following ones, whose proofs can be found in every text on distribution theory.^[16]

Smoothing property

For any distribution ${\displaystyle T}$, the following family of convolutions indexed by the real number ${\displaystyle \epsilon }$

${\displaystyle T_{\epsilon }=T\ast \varphi _{\epsilon }}$

where ${\displaystyle \ast }$ denotes

.

Approximation of identity

For any distribution ${\displaystyle T}$, the following family of convolutions indexed by the real number ${\displaystyle \epsilon }$ converges to ${\displaystyle T}$

${\displaystyle \lim _{\epsilon \to 0}T_{\epsilon }=\lim _{\epsilon \to 0}T\ast \varphi _{\epsilon }=T\in D^{\prime }(\mathbb {R} ^{n})}$

Support of convolution

For any distribution ${\displaystyle T}$,

${\displaystyle \operatorname {supp} T_{\epsilon }=\operatorname {supp} (T\ast \varphi _{\epsilon })\subset \operatorname {supp} T+\operatorname {supp} \varphi _{\epsilon }}$,

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

Applications

The basic application of mollifiers is to prove that properties valid for

are also valid 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 ${\displaystyle S}$ and ${\displaystyle T}$, the limit of the

${\displaystyle \lim _{\epsilon \to 0}S_{\epsilon }\cdot T=\lim _{\epsilon \to 0}S\cdot T_{\epsilon }{\overset {\mathrm {def} }{=}}S\cdot T}$

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 (Friedrichs 1944) illustrates this concept quite well: however the high number of technical details needed to show what this really means prevent them from being formally detailed in this short description.

Smooth cutoff functions

By convolution of the

${\displaystyle B_{1}=\{x:|x|<1\}}$ with the ${\displaystyle \varphi _{1/2}}$ (defined as in (3) with ${\displaystyle \epsilon =1/2}$), one obtains the function

${\displaystyle \chi _{B_{1},1/2}(x)=\chi _{B_{1}}\ast \varphi _{1/2}(x)=\int _{\mathbb {R} ^{n}}\!\!\!\chi _{B_{1}}(x-y)\varphi _{1/2}(y)\mathrm {d} y=\int _{B_{1/2}}\!\!\!\chi _{B_{1}}(x-y)\varphi _{1/2}(y)\mathrm {d} y\ \ \ (\because \ \mathrm {supp} (\varphi _{1/2})=B_{1/2})}$

which is a

equal to ${\displaystyle 1}$ on ${\displaystyle B_{1/2}=\{x:|x|<1/2\}}$, with support contained in ${\displaystyle B_{3/2}=\{x:|x|<3/2\}}$. This can be seen easily by observing that if ${\displaystyle |x|}$ ≤ ${\displaystyle 1/2}$ and ${\displaystyle |y|}$ ≤ ${\displaystyle 1/2}$ then ${\displaystyle |x-y|}$ ≤ ${\displaystyle 1}$. Hence for ${\displaystyle |x|}$ ≤ ${\displaystyle 1/2}$,

${\displaystyle \int _{B_{1/2}}\!\!\!\chi _{B_{1}}(x-y)\varphi _{1/2}(y)\mathrm {d} y=\int _{B_{1/2}}\!\!\!\varphi _{1/2}(y)\mathrm {d} y=1}$.

One can see how this construction can be generalized to obtain a smooth function identical to one on a

from this set is greater than a given ${\displaystyle \epsilon }$.

smooth partitions of unity

.

See also

• Approximate identity
• Bump function
• Convolution
• Distribution (mathematics)
• Generalized function
• Kurt Otto Friedrichs
• Non-analytic smooth function
• Sergei Sobolev
• Weierstrass transform

Notes

References

• Zbl 0061.26201
  . The first paper where mollifiers were introduced.
• Zbl 0051.32703, archived from the original on 2013-01-05. A paper where the differentiability of solutions of elliptic partial differential equations
  is investigated by using mollifiers.
• Zbl 0613.01020. A selection from Friedrichs' works with a biography and commentaries of David Isaacson, Fritz John, Tosio Kato, Peter Lax, Louis Nirenberg, Wolfgag Wasow, Harold Weitzner
  .
• Zbl 0545.49018
  .
• Zbl 0712.35001
  .
• embedding theorem, introducing and using integral operators
  very similar to mollifiers, without naming them.