Mollifier

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A mollifier (top) in dimension one. At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version.

In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) 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 (generalized) function.[1]

They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them.[2]

Historical notes

Mollifiers were introduced by Kurt Otto Friedrichs in his paper (Friedrichs 1944), which is considered a watershed in the modern theory of partial differential equations.[3] 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".[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 an advice on how to name the smoothing operator he was using.[3] 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 'to mollify', meaning 'to smooth over' in a figurative sense.[5]

Previously, Sergei Sobolev 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 that:-"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

A function undergoing progressive mollification.

Modern (distribution based) definition

Definition 1. If [math]\displaystyle{ \varphi }[/math] is a smooth function on ℝn, n ≥ 1, satisfying the following three requirements

(1)   it is compactly supported[8]
(2)  [math]\displaystyle{ \int_{\mathbb{R}^n}\!\varphi(x)\mathrm{d}x=1 }[/math]
(3)  [math]\displaystyle{ \lim_{\epsilon\to 0}\varphi_\epsilon(x) = \lim_{\epsilon\to 0}\epsilon^{-n}\varphi(x / \epsilon)=\delta(x) }[/math]

where [math]\displaystyle{ \delta(x) }[/math] is the Dirac delta function and the limit must be understood in the space of Schwartz distributions, then [math]\displaystyle{ \varphi }[/math] is a mollifier. The function [math]\displaystyle{ \varphi }[/math] could also satisfy further conditions:[9] for example, if it satisfies

(4)  [math]\displaystyle{ \varphi }[/math][math]\displaystyle{ (x) }[/math] ≥ 0 for all x ∈ ℝn, then it is called a positive mollifier
(5)  [math]\displaystyle{ \varphi }[/math][math]\displaystyle{ (x) }[/math]=[math]\displaystyle{ \mu }[/math][math]\displaystyle{ (|x|) }[/math] for some infinitely differentiable function [math]\displaystyle{ \mu }[/math] : 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,[10] property (3) above was formulated by saying that the convolution of the function [math]\displaystyle{ \scriptstyle\varphi_\epsilon }[/math] 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]

[math]\displaystyle{ \Phi_\epsilon(f)(x)=\int_{\mathbb{R}^n}\varphi_\epsilon(x-y) f(y)\mathrm{d}y }[/math]

where [math]\displaystyle{ \varphi_\epsilon(x)=\epsilon^{-n}\varphi(x/\epsilon) }[/math] and [math]\displaystyle{ \varphi }[/math] 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 bump function [math]\displaystyle{ \varphi }[/math][math]\displaystyle{ (x) }[/math] of a variable in ℝn defined by

[math]\displaystyle{ \varphi(x) = \begin{cases} e^{-1/(1-|x|^2)}/I_n& \text{ if } |x| \lt 1\\ 0& \text{ if } |x|\geq 1 \end{cases} }[/math]

where the numerical constant [math]\displaystyle{ I_n }[/math] ensures normalization. This function is infinitely differentiable, non analytic with vanishing derivative for |x| = 1. [math]\displaystyle{ \varphi }[/math] can be therefore used as mollifier as described above: one can see that [math]\displaystyle{ \varphi }[/math][math]\displaystyle{ (x) }[/math] defines a positive and symmetric mollifier.[15]

The function [math]\displaystyle{ \varphi }[/math][math]\displaystyle{ (x) }[/math] in dimension one

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.[16]

Smoothing property

For any distribution [math]\displaystyle{ T }[/math], the following family of convolutions indexed by the real number [math]\displaystyle{ \epsilon }[/math]

[math]\displaystyle{ T_\epsilon = T\ast\varphi_\epsilon }[/math]

where [math]\displaystyle{ \ast }[/math] denotes convolution, is a family of smooth functions.

Approximation of identity

For any distribution [math]\displaystyle{ T }[/math], the following family of convolutions indexed by the real number [math]\displaystyle{ \epsilon }[/math] converges to [math]\displaystyle{ T }[/math]

[math]\displaystyle{ \lim_{\epsilon\to 0}T_\epsilon = \lim_{\epsilon\to 0}T\ast\varphi_\epsilon=T\in D^\prime(\mathbb{R}^n) }[/math]

Support of convolution

For any distribution [math]\displaystyle{ T }[/math],

[math]\displaystyle{ \operatorname{supp}T_\epsilon=\operatorname{supp}(T\ast\varphi_\epsilon)\subset\operatorname{supp}T+\operatorname{supp}\varphi_\epsilon }[/math],

where [math]\displaystyle{ \operatorname{supp} }[/math] indicates the support in the sense of distributions, and [math]\displaystyle{ + }[/math] 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 functions, mollifiers are used to define the multiplication of distributions: precisely, given two distributions [math]\displaystyle{ S }[/math] and [math]\displaystyle{ T }[/math], the limit of the product of a smooth function and a distribution

[math]\displaystyle{ \lim_{\epsilon\to 0}S_\epsilon\cdot T=\lim_{\epsilon\to 0}S\cdot T_\epsilon\overset{\mathrm{def}}{=}S\cdot T }[/math]

defines (if it exists) 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 (Friedrichs 1944) 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 of the unit ball [math]\displaystyle{ B_1 = \{x : |x|\lt 1\} }[/math] with the smooth function [math]\displaystyle{ \varphi_{1/2} }[/math] (defined as in (3) with [math]\displaystyle{ \epsilon = 1/2 }[/math]), one obtains the function

[math]\displaystyle{ \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}) }[/math]

which is a smooth function equal to [math]\displaystyle{ 1 }[/math] on [math]\displaystyle{ B_{1/2} = \{ x: |x| \lt 1/2 \} }[/math], with support contained in [math]\displaystyle{ B_{3/2}=\{ x: |x| \lt 3/2 \} }[/math]. This can be seen easily by observing that if [math]\displaystyle{ |x| }[/math] ≤ [math]\displaystyle{ 1/2 }[/math] and [math]\displaystyle{ |y| }[/math] ≤ [math]\displaystyle{ 1/2 }[/math] then [math]\displaystyle{ |x-y| }[/math] ≤ [math]\displaystyle{ 1 }[/math]. Hence for [math]\displaystyle{ |x| }[/math] ≤ [math]\displaystyle{ 1/2 }[/math],

[math]\displaystyle{ \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 }[/math].

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 [math]\displaystyle{ \epsilon }[/math].[17] Such a function is called a (smooth) cutoff function: those functions are used to eliminate singularities of a given (generalized) function by multiplication. They leave unchanged the value of the (generalized) function 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

Notes

  1. ↑ Respect to the topology of the given space of generalized functions.
  2. ↑ See (Friedrichs 1944).
  3. ↑ 3.0 3.1 See the commentary of Peter Lax on the paper (Friedrichs 1944) in (Friedrichs 1986).
  4. ↑ (Friedrichs 1986)
  5. ↑ In (Friedrichs 1986) Lax writes precisely that:-"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, Flander sremarked that they could be named mollifier after himself; Friedrichs was delighted, as on other occasions, to carry this joke into print."
  6. ↑ See (Sobolev 1938).
  7. ↑ (Friedrichs 1953).
  8. ↑ Such as a bump function
  9. ↑ See (Giusti 1984).
  10. ↑ As when the paper (Friedrichs 1944) was published, few years before Laurent Schwartz widespread his work.
  11. ↑ Obviously the topology with respect to convergence occurs is the one of the Hilbert or Banach space considered.
  12. ↑ See (Friedrichs 1944), properties PI, PII, PIII and their consequence PIII0.
  13. ↑ 13.0 13.1 Also, in this respect, (Friedrichs 1944) says:-"The main tool for the proof is a certain class of smoothing operators approximating unity, the "mollifiers".
  14. ↑ See (Friedrichs 1944), paragraph 2, "Integral operators".
  15. ↑ See (Hörmander 1990), lemma 1.2.3.: the example is stated in implicit form by first defining
    [math]\displaystyle{ f(t)=\exp({-1/t}) }[/math] for [math]\displaystyle{ t\in\mathbb{R}_+ }[/math],
    and then considering
    [math]\displaystyle{ f(x)=f\big(1-|x|^2\big)=\exp\big(1-|x|^2\big) }[/math] for [math]\displaystyle{ x\in\mathbb{R}^n }[/math].
  16. ↑ See for example (Hörmander 1990).
  17. ↑ A proof of this fact can be found in (Hörmander 1990), Theorem 1.4.1.

References



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