柔化函数

在数学中，柔化函数（英語：mollifier）是某种特殊的光滑函数。在分布理论中，柔化函数和某个不光滑的目标函数（可以是广义的函数）的卷积将是光滑的，因此通过取一系列的柔化函数，我们可以以卷积的方式来“逼近”目标函数。直觉上，给定某个不光滑的函数，它和柔化函数卷积之后变得“柔滑”了。比如说一个有“棱角”的函数，和柔化函数的卷积将会使得“棱角”被“磨圆”，但这个卷积函数的形状仍然和原来的（广义）函数“大致”一样。最早提出柔化函数概念的数学家是Kurt Otto Friedrichs。

一个一维的柔化函数（上方）。它可以将一个函数的尖角（红色部分）“柔化”为光滑曲线段（蓝色部分）。

参考与注释

补充来源

• 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, （原始内容存档于2021-03-08）  引文格式1维护：日期与年 (link)。这篇论文引入了柔滑函数。

• 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, （原始内容存档于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. , 编, 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 （俄语） . The paper where Sergei Sobolev proved his embedding theorem, introducing and using integral operators very similar to mollifiers, without naming them.