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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 Wold's decomposition ： ウィキペディア英語版 Wold's decomposition In operator theory, a discipline within mathematics, the Wold decomposition, named after Herman Wold, or Wold–von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. == Details == Let ''H'' be a Hilbert space, ''L''(''H'') be the bounded operators on ''H'', and ''V'' ∈ ''L''(''H'') be an isometry. The Wold decomposition states that every isometry ''V'' takes the form :$V\; =\; (\backslash oplus\_\; S)\; \backslash oplus\; U$ for some index set ''A'', where ''S'' in the unilateral shift on a Hilbert space ''Hα'', and ''U'' is a unitary operator (possible vacuous). The family consists of isomorphic Hilbert spaces. A proof can be sketched as follows. Successive applications of ''V'' give a descending sequences of copies of ''H'' isomorphically embedded in itself: :$H\; =\; H\; \backslash supset\; V(H)\; \backslash supset\; V^2\; (H)\; \backslash supset\; \backslash cdots\; =\; H\_0\; \backslash supset\; H\_1\; \backslash supset\; H\_2\; \backslash supset\; \backslash cdots,$ where ''V''(''H'') denotes the range of ''V''. The above defined $H\_i\; =\; V^i(H)$. If one defines :$M\_i\; =\; H\_i\; \backslash ominus\; H\_\; =\; V^i\; (H\; \backslash ominus\; V(H))\; \backslash quad\; \backslash text\; \backslash quad\; i\; \backslash geq\; 0\; \backslash ;,$ then :$H\; =\; (\backslash oplus\_\; M\_i)\; \backslash oplus\; (\backslash cap\_\; H\_i)\; =\; K\_1\; \backslash oplus\; K\_2.$ It is clear that ''K''1 and ''K''2 are invariant subspaces of ''V''. So ''V''(''K''2) = ''K''2. In other words, ''V'' restricted to ''K''2 is a surjective isometry, i.e., a unitary operator ''U''. Furthermore, each ''Mi'' is isomorphic to another, with ''V'' being an isomorphism between ''Mi'' and ''M''''i''+1: ''V'' "shifts" ''Mi'' to ''M''''i''+1. Suppose the dimension of each ''Mi'' is some cardinal number ''α''. We see that ''K''1 can be written as a direct sum Hilbert spaces :$K\_1\; =\; \backslash oplus\; H\_$ where each ''Hα'' is an invariant subspaces of ''V'' and ''V'' restricted to each ''Hα'' is the unilateral shift ''S''. Therefore :$V\; =\; V\; \backslash vert\_\; \backslash oplus\; V\backslash vert\_\; =\; (\backslash oplus\_\; S)\; \backslash oplus\; U,$ which is a Wold decomposition of ''V''. 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「Wold's decomposition」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.