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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.〔Respect to the topology of the given space of generalized functions.〕 They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them.〔See .〕
== Historical notes ==
Mollifiers were introduced by Kurt Otto Friedrichs in his paper , considered a watershed in the modern theory of partial differential equations.〔See the commentary of Peter Lax to the paper in .〕 The name of this mathematical object had a curious genesis: Peter Lax tells the whole story of this genesis in his commentary . 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 'to mollify', meaning 'to smooth over' in a figurative sense.〔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 remarked that thei could be named mollifier after himself; Friedrichs was delighted, as on other occasions, to carry this joke into print.''" 〕
Previously, Sergei Sobolev used mollifiers in his epoch making 1938 paper,〔See .〕 which contains the proof of the Sobolev embedding theorem: himself acknowledged Sobolev's work on mollifiers stating that:-"''These mollifiers were introduced by Sobolev and the author...''".
It must be pointed out that there is a little misunderstanding in the concept of mollifier: 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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