In mathematics, mollifiers are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth functions, via 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 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 ofpartial differential equations. 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 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, 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 , 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 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 mollfier was inherited by the kernel itself as a result of common usage.
Definition
Modern (distribution based) definition
If ' is a smooth function on ℝn, n ≥ 1, satisfying the following three requirements 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
Concrete example
Consider the function ' of a variable in ℝn defined by where the numerical constant ensures normalization. It is easily seen that this function is infinitely differentiable, non analytic with vanishing derivative for. ' can be therefore used as mollifier as described above: it is also easy to see that defines a positive and symmetric mollifier.
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.
Smoothing property
For any distribution, the following family of convolutions indexed by the real number where denotes convolution, is a family of smooth functions.
Approximation of identity
For any distribution, the following family of convolutions indexed by the real number 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 applications of mollifiers is to prove properties valid for smooth functions also 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 and, the limit of the product of a smooth function and a distribution defines 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 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.
By convolution of the characteristic function of the unit ball with the smooth function ', 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 ≤, It is easy to 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. Such a function is called a cutoff function''': those functions are used to eliminate singularities of a given function by multiplication. They leave unchanged the value of the function they multiply only on a given set, thus modifying its support: also cutoff functions are the basic parts of smooth partitions of unity.
