Cutoff Function
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, mollifiers (also known as ''approximations to the identity'') are
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s 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 operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
. 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 Kurt Otto Friedrichs (September 28, 1901 – December 31, 1982) was a noted German-American mathematician. He was the co-founder of the Courant Institute at New York University, and a recipient of the National Medal of Science. Biography Friedri ...
, who introduced them.


Historical notes

Mollifiers were introduced by
Kurt Otto Friedrichs Kurt Otto Friedrichs (September 28, 1901 – December 31, 1982) was a noted German-American mathematician. He was the co-founder of the Courant Institute at New York University, and a recipient of the National Medal of Science. Biography Friedri ...
in his paper , which is considered a watershed in the modern theory of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
.See the commentary of
Peter Lax Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dyn ...
on the paper in .
The name of this mathematical object had a curious genesis, and
Peter Lax Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dyn ...
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 Catholic Church, Roman Catholic practices, maintaining that the Church of England had not been fully reformed and should become m ...
, nicknamed by his friends Moll after
Moll Flanders ''Moll Flanders'' is a novel by Daniel Defoe, first published in 1722. It purports to be the true account of the life of the eponymous Moll, detailing her exploits from birth until old age. By 1721, Defoe had become a recognised novelist, wit ...
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 Prof Sergei Lvovich Sobolev (russian: Серге́й Льво́вич Со́болев) H FRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduc ...
used mollifiers in his epoch making 1938 paper, which contains the proof of the
Sobolev embedding theorem In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...
: 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 Two types of language change can be characterized as natural language, linguistic drift: a wikt:unidirectional, unidirectional short-term and Period of time, cyclic long-term drift. Short-term unidirectional drift According to Edward Sapir, Sapir, ...
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 learnin ...
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 ''\varphi'' is a
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
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 ...
:\int_\!\varphi(x)\mathrmx=1 :\lim_\varphi_\epsilon(x) = \lim_\epsilon^\varphi(x / \epsilon)=\delta(x) where \delta(x) is the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
and the limit must be understood in the space of Schwartz
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
s, then ''\varphi'' is a mollifier. The function ''\varphi'' could also satisfy further conditions: for example, if it satisfies :''\varphi''(x) ≥ 0 for all ''x'' ∈ ℝ''n'', then it is called a positive mollifier :''\varphi''(x)=''\mu''(, x, ) 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 i ...
''\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, property above was formulated by saying that the
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of the function ''\scriptstyle\varphi_\epsilon'' 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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 Friedrichs may refer to: Names * Bob Friedrichs, American baseball pitcher * Carl Joseph Friedrichs, German printer, author and gold prospector * Friedrich Friedrichs, German World War I fighter ace * Fritz Walter Paul Friedrichs, German chemist ...
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 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'' ...
: :\Phi_\epsilon(f)(x)=\int_\varphi_\epsilon(x-y) f(y)\mathrmy where \varphi_\epsilon(x)=\epsilon^\varphi(x/\epsilon) and ''\varphi'' is a
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
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 ...
''\varphi''(x) of a
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
in ℝ''n'' defined by \varphi(x) = \begin e^/I_n& \text , x, < 1\\ 0& \text , x, \geq 1 \end where the numerical constant I_n 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. F ...
for . ''\varphi'' can be therefore used as mollifier as described above: one can see that ''\varphi''(x) 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 operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
: we list the following ones, whose proofs can be found in every text on distribution theory.


Smoothing property

For any distribution T, 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 real ...
\epsilon :T_\epsilon = T\ast\varphi_\epsilon where \ast denotes
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
, is a family of
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s.


Approximation of identity

For any distribution T, 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 real ...
\epsilon converges to T :\lim_T_\epsilon = \lim_T\ast\varphi_\epsilon=T\in D^\prime(\mathbb^n)


Support of convolution

For any distribution T, :\mathrmT_\epsilon=\mathrm(T\ast\varphi_\epsilon)\subset\mathrmT+\mathrm\varphi_\epsilon where \mathrm indicates the
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
in the sense of distributions, and + indicates their
Minkowski addition In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...
.


Applications

The basic application of mollifiers is to prove that properties valid for
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s 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 S and T, 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 In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
and a
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
:\lim_S_\epsilon\cdot T=\lim_S\cdot T_\epsilon\oversetS\cdot T 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 Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
B_1 = \ with the
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
''\varphi_'' (defined as in with \epsilon = 1/2), one obtains the function : \chi_(x)=\chi_\ast\varphi_(x)=\int_\!\!\!\chi_(x-y)\varphi_(y)\mathrmy=\int_\!\!\! \chi_(x-y) \varphi_(y)\mathrmy \ \ \ (\because\ \mathrm(\varphi_)=B_) which is a
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
equal to 1 on B_ = \, with support contained in B_=\. This can be seen easily by observing that if , x, 1/2 and , y, 1/2 then , x-y, 1. Hence for , x, 1/2, : \int_\!\!\!\chi_(x-y) \varphi_(y)\mathrmy= \int_\!\!\! \varphi_(y)\mathrmy=1 . 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; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
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 \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 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 Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
. 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, thus modifying its
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
: also cutoff functions are the basic parts of smooth partitions of unity.


See also

*
Approximate identity In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate ...
*
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 bump f ...
*
Convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
*
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 Kurt Otto Friedrichs (September 28, 1901 – December 31, 1982) was a noted German-American mathematician. He was the co-founder of the Courant Institute at New York University, and a recipient of the National Medal of Science. Biography Friedri ...
*
Non-analytic smooth function In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is no ...
*
Sergei Sobolev Prof Sergei Lvovich Sobolev (russian: Серге́й Льво́вич Со́болев) H FRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduc ...
*
Weierstrass transform In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''. Specifically, it is the function defined ...


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 its ...
of solutions of
elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, wher ...
s is investigated by using mollifiers. *. A selection from Friedrichs' works with a biography and commentaries of
David Isaacson David T. Isaacson is a United States Army major general who serves as the director of manpower and personnel of the Joint Staff since June 2022. He most recently served as the Chief of Staff of the United States Cyber Command United State ...
,
Fritz John Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a 1 ...
,
Tosio Kato was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis. Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of Tokyo. After disruption o ...
,
Peter Lax Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dyn ...
,
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 Harold Weitzner is an American applied mathematician and physicist whose primary research is plasma physics. He is Professor Emeritus of Mathematics at the Courant Institute of Mathematical Sciences and has served as Director of the Magneto-Fluid ...
. *. *. *{{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