In mathematics, mollifiers (also known as approximations to the identity) are particular smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a (generalized) function, convolving it with a mollifier "mollifies" it, that is, its sharp features are smoothed, while still remaining close to the original. [1]
They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them. [2]
Mollifiers were introduced by Kurt Otto Friedrichs in his paper ( Friedrichs 1944 , pp. 136–139), which is considered a watershed in the modern theory of partial differential equations. [3] 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". [4] 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. [3] 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. [5]
Previously, Sergei Sobolev had used mollifiers in his epoch making 1938 paper, [6] 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...". [7]
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.
Definition 1. Let be a smooth function on , , and put for . Then is a mollifier if it satisfies the following three requirements:
where is the Dirac delta function, and the limit must be understood as taking place in the space of Schwartz distributions. The function may also satisfy further conditions of interest; [9] for example, if it satisfies
then it is called a positive mollifier, and if it satisfies
then it is called a symmetric mollifier.
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 with a given function belonging to a proper Hilbert or Banach space converges as ε → 0 to that function: [11] this is exactly what Friedrichs did. [12] This also clarifies why mollifiers are related to approximate identities. [13]
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]
where and 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 of a variable in defined by
where the numerical constant ensures normalization. This function is infinitely differentiable, non analytic with vanishing derivative for |x| = 1. can be therefore used as mollifier as described above: one can see that defines a positive and symmetric mollifier. [15]
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]
For any distribution , the following family of convolutions indexed by the real number
where denotes convolution, is a family of smooth functions.
For any distribution , the following family of convolutions indexed by the real number converges to
For any distribution ,
where 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 and , 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:
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 ( Friedrichs 1944 ) illustrates this approach.
By convolution of the characteristic function of the unit ball with the smooth function (defined as in (3) 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 of a given compact set, and equal to zero in every point whose distance from this set is greater than a given . [17] 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.
In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.
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 do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function, but can be every intersection of the graph itself with a hyperplane parallel to a fixed x-axis and to the y-axis.
In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this.
In mathematics, the support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. If the domain of is a topological space, then the support of is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc.
In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra over the real or complex numbers that at the same time is also a Fréchet space. The multiplication operation for is required to be jointly continuous. If is an increasing family of seminorms for the topology of , the joint continuity of multiplication is equivalent to there being a constant and integer for each such that for all . Fréchet algebras are also called B0-algebras.
In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function is called refinable with respect to the mask if
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.
In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension.
In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case of elliptic or hypoelliptic regularity.
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking,
An approach to nonlinear congruential methods of generating uniform pseudorandom numbers in the interval [0,1) is the Inversive congruential generator with prime modulus. A generalization for arbitrary composite moduli with arbitrary distinct primes will be present here.
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on Lp spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.
Computational anatomy (CA) is the study of shape and form in medical imaging. The study of deformable shapes in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form . In CA, this orbit is in general considered a smooth Riemannian manifold since at every point of the manifold there is an inner product inducing the norm on the tangent space that varies smoothly from point to point in the manifold of shapes . This is generated by viewing the group of diffeomorphisms as a Riemannian manifold with , associated to the tangent space at . This induces the norm and metric on the orbit under the action from the group of diffeomorphisms.
In probability theory, a branch of mathematics, white noise analysis, otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener process. It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes.
In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.
In optimal transport, a branch of mathematics, polar factorization of vector fields is a basic result due to Brenier (1987), with antecedents of Knott-Smith (1984) and Rachev (1985), that generalizes many existing results among which are the polar decomposition of real matrices, and the rearrangement of real-valued functions.