軟化子
↑ これは与えられた超函数の空間の位相に関する議論である。
↑ (Friedrichs 1944, pp. 136–139) を参照。
1 2 (Friedrichs 1986, volume 1, p. 117) 内の論文 (Friedrichs 1944) に対するピーター・ラックスの論評を参照。
↑ ラックス (Friedrichs 1986, volume 1, p. 117) では正確には次のように書かれている:-"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."
任意の超函数 T {\displaystyle T} に対し、実数 ε {\displaystyle \epsilon } を添え字とする次の畳み込みの族は、 T {\displaystyle T} に収束する。
lim ε → 0 T ε = lim ε → 0 T ∗ φ ε = T ∈ D ′ ( R n ) {\displaystyle \lim _{\epsilon \to 0}T_{\epsilon }=\lim _{\epsilon \to 0}T\ast \varphi _{\epsilon }=T\in D^{\prime }(\mathbb {R} ^{n})}