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. 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 (Friedrichs 1944, pp. 136–139), which is considered a watershed in the modern theory of partial differential equations. 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". 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. 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. Previously, Sergei Sobolev had 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 "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 mollifier was inherited by the kernel itself as a result of common usage.
Definition
Modern (distribution based) definition Definition 1. Let φ {\displaystyle \varphi } be a smooth function on R n {\displaystyle \mathbb {R} ^{n}} , n ≥ 1 {\displaystyle n\geq 1} , and put φ ϵ ( x ) := ϵ − n φ ( x / ϵ ) {\displaystyle \varphi _{\epsilon }(x):=\epsilon ^{-n}\varphi (x/\epsilon )} for ϵ > 0 ∈ R {\displaystyle \epsilon >0\in \mathbb {R} } . Then φ {\displaystyle \varphi } is a mollifier if it satisfies the following three requirements:
(1) it is compactly supported, (2) ∫ R n φ ( x ) d x = 1 {\displaystyle \int _{\mathbb {R} ^{n}}\!\varphi (x)\mathrm {d} x=1} , (3) lim ϵ → 0 φ ϵ ( x ) = lim ϵ → 0 ϵ − n φ ( x / ϵ ) = δ ( x ) {\displaystyle \lim _{\epsilon \to 0}\varphi _{\epsilon }(x)=\lim _{\epsilon \to 0}\epsilon ^{-n}\varphi (x/\epsilon )=\delta (x)} , where δ ( x ) {\displaystyle \delta (x)} is the Dirac delta function, and the limit must be understood as taking place in the space of Schwartz distributions. The function φ {\displaystyle \varphi } may also satisfy further conditions of interest; for example, if it satisfies
(4) φ ( x ) ≥ 0 {\displaystyle \varphi (x)\geq 0} for all x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , then it is called a positive mollifier, and if it satisfies
(5) φ ( x ) = μ ( | x | ) {\displaystyle \varphi (x)=\mu (|x|)} for some infinitely differentiable function μ : R + → R {\displaystyle \mu :\mathbb {R} ^{+}\to \mathbb {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 (3) above was formulated by saying that the convolution of the function φ ϵ {\displaystyle \scriptstyle \varphi _{\epsilon }} with a given function belonging to a proper Hilbert or Banach space converges as ε → 0 to that function: this is exactly what Friedrichs did. This also clarifies why mollifiers are related to approximate identities. Note 2. As briefly pointed out in the "Historical notes" section of this entry, originally, the term "mollifier" identified the following convolution operator:
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