Mollifier

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

Mollifiers were introduced by Kurt Otto Friedrichs in his paper (Friedrichs 1944, pp. 136–139), considered a watershed in the modern theory of partial differential equations. The name of this mathematical object had a curious genesis: Peter Lax tells the whole story of this genesis in his commentary (Friedrichs 1986, volume 1, p. 117). According to Lax, 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, 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 (1953, p. 196) himself acknowledged Sobolev's work on mollifiers stating that:-"These mollifiers were introduced by Sobolev and the author...".

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