In
mathematics, mollifiers (also known as ''approximations to the identity'') are
smooth functions with special properties, used for example in
distribution theory to create
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of smooth functions approximating nonsmooth
(generalized) functions, via
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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 (generalized) function.
They are also known as Friedrichs mollifiers after
Kurt Otto Friedrichs, who introduced them.
Historical notes
Mollifiers were introduced by
Kurt Otto Friedrichs in his paper , which is considered a watershed in the modern theory of
partial differential equations.
[See the commentary of Peter Lax on the paper in .] The name of this mathematical object had a curious genesis, and
Peter Lax tells the whole story in his commentary on that paper published in Friedrichs' "''Selecta''".
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 an advice on how to name the smoothing operator he was using.
Flanders was a
puritan
The Puritans were English Protestants in the 16th and 17th centuries who sought to purify the Church of England of Roman Catholic practices, maintaining that the Church of England had not been fully reformed and should become more Protestant. ...
, nicknamed by his friends Moll after
Moll Flanders in recognition of his moral qualities: he suggested to call 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.
Previously,
Sergei Sobolev used mollifiers in his epoch making 1938 paper, which contains the proof of the
Sobolev embedding theorem: Friedrichs himself acknowledged Sobolev's work on mollifiers stating that:-"''These mollifiers were introduced by Sobolev and the author...''".
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
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms invol ...
whose
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
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
If ''
'' is a
smooth function on ℝ''
n'', ''n'' ≥ 1, satisfying the following three requirements
: it is
compactly supported
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
:
:
where
is the
Dirac delta function and the limit must be understood in the space of Schwartz
distributions, then ''
'' is a mollifier. The function ''
'' could also satisfy further conditions: for example, if it satisfies
:''
''
≥ 0 for all ''x'' ∈ ℝ''
n'', then it is called a positive mollifier
:''
''
=''
''
for some
infinitely differentiable function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
''
'' : 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, property above was formulated by saying that the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the function ''
'' with a given function belonging to a proper
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
or
Banach space converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
s as ''ε'' → 0 to that function: this is exactly what
Friedrichs did. This also clarifies why mollifiers are related to
approximate identities.
[Also, in this respect, 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:
:
where
and ''
'' is a
smooth function satisfying the first three conditions stated above and one or more supplementary conditions as positivity and symmetry.
Concrete example
Consider the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''
''
of a
variable in ℝ''
n'' defined by
where the numerical constant
ensures normalization. This function is
infinitely differentiable, non analytic with vanishing
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
for . ''
'' can be therefore used as mollifier as described above: one can see that ''
''
defines a ''positive and symmetric mollifier''.
Properties
All properties of a mollifier are related to its behaviour under the operation of
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
: we list the following ones, whose proofs can be found in every text on
distribution theory.
Smoothing property
For any distribution
, the following family of convolutions indexed by the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
:
where
denotes
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
, is a family of
smooth functions.
Approximation of identity
For any distribution
, the following family of convolutions indexed by the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
converges to
:
Support of convolution
For any distribution
,
:
where
indicates the
support in the sense of distributions, and
indicates their
Minkowski addition.
Applications
The basic application of mollifiers is to prove that properties valid for
smooth functions are also valid in nonsmooth situations:
Product of distributions
In some theories of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s, mollifiers are used to define the
multiplication of distributions: precisely, given two distributions
and
, the limit of the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of a
smooth function and a
distribution
:
defines (if it exists) their product in various theories of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s.
"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 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
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of the
unit ball with the
smooth function ''
'' (defined as in with
), one obtains the function
:
which is a
smooth function equal to
on
, with support contained in
. This can be seen easily by observing that if
≤
and
≤
then
≤
. Hence for
≤
,
:
.
One can see how this construction can be generalized to obtain a smooth function identical to one on a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of a given
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
, and equal to zero in every point whose
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from this set is greater than a given
.
[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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
by
multiplication. They leave unchanged the value of the (
generalized)
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
*
Approximate identity
*
Bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bum ...
*
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
*
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
*
Generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
*
Kurt Otto Friedrichs
*
Non-analytic smooth function
*
Sergei Sobolev
*
Weierstrass transform
Notes
References
*. The first paper where mollifiers were introduced.
*. A paper where the
differentiability
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
of solutions of
elliptic partial differential equations is investigated by using mollifiers.
*. A selection from Friedrichs' works with a biography and commentaries of
David Isaacson,
Fritz John,
Tosio Kato,
Peter Lax,
Louis Nirenberg
Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Nearly all of his work was in the field of partial differential equat ...
,
Wolfgag Wasow,
Harold Weitzner.
*.
*.
*{{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 operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms invol ...
s very similar to mollifiers, without naming them.
Functional analysis
Smooth functions