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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fatou's theorem ： ウィキペディア英語版 Fatou's theorem In complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. ==Motivation and statement of theorem== If we have a holomorphic function $f$ defined on the open unit disk $D^=\backslash$, it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius $r$. This defines a new function on the circle $f\_:S^\backslash rightarrow\; \backslash mathbb$, defined by $f\_(e^)=f(re^)$, where $S^:=\backslash =\backslash$. Then it would be expected that the values of the extension of $f$ onto the circle should be the limit of these functions, and so the question reduces to determining when $f\_$ converges, and in what sense, as $r\backslash rightarrow\; 1$, and how well defined is this limit. In particular, if the $L^p$ norms of these $f\_$ are well behaved, we have an answer: :Theorem: Let $f:D^\backslash rightarrow\backslash mathbb$ be a holomorphic function such that :: Then $f\_$ converges to some function $f\_\backslash in\; L^(S^)$ pointwise almost everywhere and in $L^$. That is, :: $\backslash lVert\; f\_-f\_\backslash rVert\_)\}\backslash rightarrow\; 0$ :and :: $|f\_(e^)-f\_(e^)|\backslash rightarrow\; 0$ :for almost every $\backslash theta\backslash in\; ()$. Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that :$f(re^)\backslash rightarrow\; f\_(e^)$ for almost every $\backslash theta$. The natural question is, now with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve $\backslash gamma:$, $\backslash lim\_f(\backslash gamma(t))=f\_(e^)$ : for every nontangential limit $\backslash gamma$ converging to $e^$, where $f\_$ is defined as above. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fatou's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.