柔化函数

在数学中，柔化函数（英语：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^[1]。

参考与注释

[1]
参见(Friedrichs 1944，第136–139页)。

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）。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.

[1] [1]
参见(Friedrichs 1944，第136–139页)。

[1]