柔化函數

一個一維的柔化函數（上方）。它可以將一個函數的尖角（紅色部分）「柔化」為光滑曲線段（藍色部分）。

在數學中，柔化函數（英語：mollifier）是某種特殊的光滑函數。在分布理論中，柔化函數和某個不光滑的目標函數（可以是廣義的函數）的卷積將是光滑的，因此通過取一系列的柔化函數，我們可以以卷積的方式來「逼近」目標函數。直覺上，給定某個不光滑的函數，它和柔化函數卷積之後變得「柔滑」了。比如說一個有「稜角」的函數，和柔化函數的卷積將會使得「稜角」被「磨圓」，但這個卷積函數的形狀仍然和原來的（廣義）函數「大致」一樣。最早提出柔化函數概念的數學家是Kurt Otto Friedrichs。

參考與注釋

補充來源

• Friedrichs, Kurt Otto, The identity of weak and strong extensions of differential operators, Transactions of the American Mathematical Society, January 1944, 55 (1): 132–151 [2012-07-14], MR 0009701, Zbl 0061.26201, doi:10.1090/S0002-9947-1944-0009701-0, （原始內容存檔於2021-03-08）  引文格式1維護：日期與年 (link)。這篇論文引入了柔滑函數。

• Friedrichs, Kurt Otto, On the differentiability of the solutions of linear elliptic differential equations, Communications on Pure and Applied Mathematics, 1953, VI (3): 299–326 [2012-07-14], MR 0058828, Zbl 0051.32703, doi:10.1002/cpa.3160060301, （原始內容存檔於2013-01-05） . A paper where the differentiability of solutions of elliptic partial differential equations is investigated by using mollifiers.

• Friedrichs, Kurt Otto, Morawetz, Cathleen S. , 編, Selecta, Contemporary Mathematicians, Boston-Basel-Stuttgart: Birkhäuser Verlag: 427 (Vol. 1); pp. 608 (Vol. 2), 1986, ISBN 0-8176-3270-0, Zbl 0613.01020 . A selection from Friedrichs' works with a biography and commentaries of David Isaacson, Fritz John, Tosio Kato, Peter Lax, Louis Nirenberg, Wolfgag Wasow, Harold Weitzner.

• Giusti, Enrico, Minimal surfaces and functions of bounded variations, Monographs in Mathematics 80, Basel-Boston-Stuttgart: Birkhäuser Verlag: xii+240, 1984, ISBN 0-8176-3153-4, MR 0775682, Zbl 0545.49018, ISBN 3-7643-3153-4 .

• Hörmander, Lars, The analysis of linear partial differential operators I, Grundlehren der Mathematischen Wissenschaft 256 2nd, Berlin-Heidelberg-New York: Springer-Verlag, 1990, ISBN 0-387-52343-X, MR 1065136, Zbl 0712.35001, ISBN 3-540-52343-X .

• Sobolev, Sergei L., Sur un théorème d'analyse fonctionnelle, Recueil Mathématique (Matematicheskii Sbornik), 1938, 4(46) (3): 471–497, Zbl 0022.14803 （俄語） . The paper where Sergei Sobolev proved his embedding theorem, introducing and using integral operators very similar to mollifiers, without naming them.