Mollifier#Smooth cutoff functions

Mollifiers were introduced by Kurt Otto Friedrichs in his paper {{Harv|Friedrichs|1944|pp=136–139}}, which is considered a watershed in the modern theory of partial differential equations.See the commentary of Peter Lax on the paper {{Harv|Friedrichs|1944}} in {{Harv|Friedrichs|1986|loc=volume 1, p. 117}}. The name of this mathematical object has a curious genesis, and Peter Lax tells the story in his commentary on that paper published in Friedrichs' "Selecta".{{Harv|Friedrichs|1986|loc=volume 1, p. 117}} According to him, at that time, the mathematician Donald Alexander Flanders was a colleague of Friedrichs; since he liked to consult colleagues about English usage, he asked Flanders for advice on naming the smoothing operator he was using. Flanders was a modern-day puritan, nicknamed by his friends Moll after Moll Flanders in recognition of his moral qualities: he suggested calling the new mathematical concept a "mollifier" as a pun incorporating both Flanders' nickname and the verb 'to mollify', meaning 'to smooth over' in a figurative sense.In {{Harv|Friedrichs|1986|loc=volume 1, p. 117}} Lax writes "On English usage Friedrichs liked to consult his friend and colleague, Donald Flanders, a descendant of puritans and a puritan himself, with the highest standard of his own conduct, noncensorious towards others. In recognition of his moral qualities he was called Moll by his friends. When asked by Friedrichs what to name the smoothing operator, Flanders remarked that they could be named "mollifier" after himself; Friedrichs was delighted, as on other occasions, to carry this joke into print."

Previously, Sergei Sobolev had used mollifiers in his epoch making 1938 paper,See {{Harv|Sobolev|1938}}. which contains the proof of the Sobolev embedding theorem: Friedrichs himself acknowledged Sobolev's work on mollifiers, stating "These mollifiers were introduced by Sobolev and the author...".{{Harvtxt|Friedrichs|1953|p=196}}.

It must be pointed out that the term "mollifier" has undergone linguistic drift since the time of these foundational works: Friedrichs defined as "mollifier" the integral operator whose kernel 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.

Image:Heat eqn.gif

{{EquationRef|1|Definition 1.}} Let $\backslash varphi$ be a smooth function on $\backslash R^n$, $n\; \backslash ge\; 1$, and put $\backslash varphi\_\backslash epsilon(x)\; :=\; \backslash epsilon^\{-n\}\backslash varphi(x\; /\; \backslash epsilon)$ for $\backslash epsilon\; >\; 0\; \backslash in\backslash R$.

Then $\backslash varphi$ is a mollifier if it satisfies the following three requirements:

:{{EquationRef|2|(1){{spaces|2}}}} it is compactly supported,This is satisfied if, for instance, $\backslash varphi(x)$ is a bump function.

:{{EquationRef|3|(2){{spaces|2}}}}$\backslash int\_\{\backslash R^n\}\backslash !\backslash varphi(x)\backslash mathrm\{d\}x=1$,

:{{EquationRef|4|(3){{spaces|2}}}}$\backslash lim\_\{\backslash epsilon\backslash to\; 0\}\backslash varphi\_\backslash epsilon(x)\; =\; \backslash lim\_\{\backslash epsilon\backslash to\; 0\}\backslash epsilon^\{-n\}\backslash varphi(x\; /\; \backslash epsilon)=\backslash delta(x)$,

where $\backslash delta(x)$ is the Dirac delta function, and the limit must be understood as taking place in the space of Schwartz distributions. The function $\backslash varphi$ may also satisfy further conditions of interest;See {{harv|Giusti|1984|p=11}}. for example, if it satisfies

:{{EquationRef|5|(4){{spaces|2}}}}$\backslash varphi(x)\backslash ge\; 0$ for all $x\; \backslash in\; \backslash R^n$,

then it is called a positive mollifier, and if it satisfies

:{{EquationRef|6|(5){{spaces|2}}}}$\backslash varphi(x)=\backslash mu(|x|)$ for some infinitely differentiable function $\backslash mu:\backslash R^+\backslash to\backslash R$'',

then it is called a symmetric mollifier.

Note 1. When the theory of distributions was still not widely known nor used,As when the paper {{Harv|Friedrichs|1944}} was published, few years before Laurent Schwartz widespread his work. property {{EquationNote|4|(3)}} above was formulated by saying that the convolution of the function $\backslash scriptstyle\backslash varphi\_\backslash epsilon$ with a given function belonging to a proper Hilbert or Banach space converges as ε → 0 to that function:Obviously the topology with respect to convergence occurs is the one of the Hilbert or Banach space considered. this is exactly what Friedrichs did.See {{harv|Friedrichs|1944|pp=136–138}}, properties PI, PII, PIII and their consequence PIII₀. This also clarifies why mollifiers are related to approximate identities.Also, in this respect, {{harvtxt|Friedrichs|1944|pp=132}} says:-"The main tool for the proof is a certain class of smoothing operators approximating unity, the "mollifiers".

Note 2. As briefly pointed out in the "Historical notes" section of this entry, originally, the term "mollifier" identified the following convolution operator:See {{harv|Friedrichs|1944|p=137}}, paragraph 2, "Integral operators".

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

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

Consider the bump function $\backslash varphi$$(x)$ of a variable in $\backslash mathbb\{R\}^n$ defined by

0& \text{ if } |x|\geq 1

\end{cases}

where the numerical constant $I\_n$ ensures normalization. This function is infinitely differentiable, non analytic with vanishing derivative for {{math||x| {{=}} 1}}. $\backslash varphi$ can be therefore used as mollifier as described above: one can see that $\backslash varphi$$(x)$ defines a positive and symmetric mollifier.See {{Harv|Hörmander|1990|p=14}}, lemma 1.2.3.: the example is stated in implicit form by first defining

:$f(t)=\backslash exp(\{-1/t\})$ for $t\backslash in\backslash mathbb\{R\}\_+$,

and then considering

:$f(x)=f\backslash big(1-|x|^2\backslash big)=\backslash exp\backslash big(-1/(1-|x|^2)\backslash big)$ for $x\backslash in\backslash mathbb\{R\}^n$.

File:Mollifier Illustration.svg one]]

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.See for example {{Harv|Hörmander|1990}}.

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

:$T\_\backslash epsilon\; =\; T\backslash ast\backslash varphi\_\backslash epsilon$

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

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

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

For any distribution $T$,

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

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

The basic application of mollifiers is to prove that properties valid for smooth functions are also valid in nonsmooth situations.

In some theories of generalized functions, mollifiers are used to define the multiplication of distributions. Given two distributions $S$ and $T$, the limit of the product of the smooth function obtained from one operand via mollification, with the other operand defines, when it exists, their product in various theories of generalized functions:

:$S\backslash cdot\; T\; :=\; \backslash lim\_\{\backslash epsilon\backslash to\; 0\}S\_\backslash epsilon\backslash cdot\; T=\backslash lim\_\{\backslash epsilon\backslash to\; 0\}S\backslash cdot\; T\_\backslash epsilon$.

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 by Friedrichs which introduces mollifiers {{Harv|Friedrichs|1944}} illustrates this approach.

By convolution of the characteristic function of the unit ball with the smooth function $\backslash varphi\_\{1/2\}$ (defined as in {{EquationNote|4|(3)}} with $\backslash epsilon\; =\; 1/2$), one obtains the function

:$$

\begin{align}

\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})

\end{align}

which is a smooth function equal to $1$ on , with support contained in . This can be seen easily by observing that if $|x|\; \backslash le\; 1/2$ and $|y|\; \backslash le\; 1/2$ then $|x-y|\; \backslash le\; 1$. Hence for $|x|\; \backslash le\; 1/2$,

:$$

\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 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 {{Harv|Hörmander|1990|p=25}}, Theorem 1.4.1. Such a function is called a (smooth) cutoff function; these are used to eliminate singularities of a given (generalized) function via multiplication. They leave unchanged the value of the multiplicand on a given set, but modify its support. Cutoff functions are used to construct smooth partitions of unity.

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

{{reflist|29em}}

• {{Citation

| last = Friedrichs

| first = Kurt Otto

| author-link = Kurt Otto Friedrichs

| title = The identity of weak and strong extensions of differential operators

| journal = Transactions of the American Mathematical Society

| volume = 55

| issue = 1

| pages = 132–151

| date = January 1944

| doi = 10.1090/S0002-9947-1944-0009701-0

| jstor = 1990143

| mr = 0009701

| zbl = 0061.26201

| doi-access = free

}}. The first paper where mollifiers were introduced.

• {{Citation

| last = Friedrichs

| first = Kurt Otto

| author-link = Kurt Otto Friedrichs

| title = On the differentiability of the solutions of linear elliptic differential equations

| journal = Communications on Pure and Applied Mathematics

| volume = VI

| issue = 3

| pages = 299–326

| year = 1953

| url = http://www3.interscience.wiley.com/journal/113395283/abstract

| archive-url = https://archive.today/20130105122505/http://www3.interscience.wiley.com/journal/113395283/abstract

| url-status = dead

| archive-date = 2013-01-05

| doi = 10.1002/cpa.3160060301

| mr = 0058828

| zbl = 0051.32703

}}. A paper where the differentiability of solutions of elliptic partial differential equations is investigated by using mollifiers.

• {{Citation

| last = Friedrichs

| first = Kurt Otto

| author-link = Kurt Otto Friedrichs

| editor-last = Morawetz

| editor-first = Cathleen S.

| editor-link = Cathleen Synge Morawetz

| title = Selecta

| place = Boston-Basel-Stuttgart

| publisher = Birkhäuser Verlag

| year = 1986

| series = Contemporary Mathematicians

| volume =

| pages = 427 (Vol. 1); pp. 608 (Vol. 2)

| url = https://books.google.com/books?id=l1Z_yHVjor4C&q=Kurt+Otto+Friedrichs+selecta&pg=PP1

| doi =

| zbl = 0613.01020

| isbn = 0-8176-3270-0}}. 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.

• {{Citation

| last = Giusti

| first = Enrico

| author-link = Enrico Giusti

| title = Minimal surfaces and functions of bounded variations

| place = Basel-Boston-Stuttgart

| publisher = Birkhäuser Verlag

| year = 1984

| series = Monographs in Mathematics

| volume = 80

| pages = xii+240

| url = https://books.google.com/books?id=dNgsmArDoeQC&q=Minimal+surfaces+and+functions+of+bounded+variations

| mr = 0775682

| zbl = 0545.49018

| isbn = 0-8176-3153-4

}}.

• {{Citation

| last = Hörmander

| first = Lars

| author-link = Lars Hörmander

| title = The analysis of linear partial differential operators I

| place = Berlin-Heidelberg-New York

| publisher = Springer-Verlag

| year = 1990

| series = Grundlehren der Mathematischen Wissenschaft

| volume = 256

| edition = 2nd

| url =

| doi =

| mr = 1065136

| zbl= 0712.35001

| isbn = 0-387-52343-X

}}.

• {{Citation

| last = Sobolev

| first = Sergei L.

| author-link = Sergei Sobolev

| title = Sur un théorème d'analyse fonctionnelle

| journal = Recueil Mathématique (Matematicheskii Sbornik)

| volume = 4(46)

| issue = 3

| pages = 471–497

| year = 1938

| language = Russian, French

| url = http://mi.mathnet.ru/eng/msb/v46/i3/p471

| zbl = 0022.14803

}}. The paper where Sergei Sobolev proved his embedding theorem, introducing and using integral operators very similar to mollifiers, without naming them.

Category:Functional analysis

Category:Smooth functions

Category:Schwartz distributions