[Google N-Gram shows](https://books.google.com/ngrams/graph?content=temperate%20distribution%2Ctempered%20distribution&year_start=1800&year_end=2019&corpus=26&smoothing=3&direct_url=t1%3B%2Ctemperate%20distribution%3B%2Cc0%3B.t1%3B%2Ctempered%20distribution%3B%2Cc0) that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term declines.


Schwartz himself seems to have used this term for the first time in his paper of 1951, *Analyse et synthèse harmoniques dans les espaces de distributions*,
Canad. J. Math. **3** (1951), 503–512, doi:[10.4153/CJM-1951-051-5](https://doi.org/10.4153/CJM-1951-051-5), and the French term was "distributions tempérées".


Google translates "tempérées" as "temperate".


I am not a native English speaker, but I understand "temperate" as a synonym of "moderate", which makes sense to me as a name of those distributions. For example, we say "temperate climate", not "tempered climate".
While "tempered" seems to be related to metallurgy, namely to a process making steel hard. What does tempering of steel have to do with distributions?


Why does the name "tempered" win in English?



Why are distributions "tempered"?

> 
> Can someone explain, why in English the name "tempered" wins?
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Presumably because that’s how the inventor himself translated it (French past participle to English past participle), on e.g. p. 188 of
*Schwartz, Laurent*, Mathematics for the physical sciences, Collection enseign. des sciences. ADIWES Internation Series in Mathematics. Paris: Hermann & Cie.; Reading, Mass. etc.: Addison-Wesley Publishing Company. 357 p. (1966). [ZBL0151.34001](https://zbmath.org/?q=an:0151.34001):



> 
> A distribution $\mathrm T$ (that is to say a continuous linear form on $\mathscr D$) is termed
> a *tempered distribution* if it may be extended to a continuous linear form on $\mathscr S$.
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The usage was apparently already well-established by 1956 when G. Mackey [reviewed](//mathscinet.ams.org/mathscinet-getitem?mr=84713) a French paper of F. Bruhat and wrote (first occurrence of the term in MathSciNet):



> 
> The representations concerned are assumed to yield “tempered representations” when restricted to the Abelian normal subgroup being studied. Here tempered means being not too badly unbounded in a precise sense suggested by Schwartz’s definition of tempered distribution.
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