柔化函數

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^[1]。

參考與注釋[編輯]

^ 參見（Friedrichs 1944，pp.136–139）。

補充來源[編輯]

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）。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.

[1] 參見（Friedrichs 1944，pp.136–139）。

[1]