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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 Bochner's theorem ： ウィキペディア英語版 Bochner's theorem In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. ==The theorem for locally compact abelian groups== Bochner's theorem for a locally compact abelian group ''G'', with dual group $\backslash widehat$, says the following: Theorem For any normalized continuous positive definite function ''f'' on ''G'' (normalization here means ''f'' is 1 at the unit of ''G''), there exists a unique probability measure on $\backslash widehat$ such that :$f(g)=\backslash int\_$. Conversely, the Fourier transform of a probability measure on $\backslash widehat$ is necessarily a normalized continuous positive definite function ''f'' on ''G''. This is in fact a one-to-one correspondence. The Gelfand-Fourier transform is an isomorphism between the group C *-algebra C *(''G'') and C0(''G''^). The theorem is essentially the dual statement for states of the two abelian C *-algebras. The proof of the theorem passes through vector states on strongly continuous unitary representations of ''G'' (the proof in fact shows every normalized continuous positive definite function must be of this form). Given a normalized continuous positive definite function ''f'' on ''G'', one can construct a strongly continuous unitary representation of ''G'' in a natural way: Let ''F''0(''G'') be the family of complex valued functions on ''G'' with finite support, i.e. ''h''(''g'') = 0 for all but finitely many ''g''. The positive definite kernel ''K''(''g''1, ''g''2) = ''f''(''g''1 - ''g''2) induces a (possibly degenerate) inner product on ''F''0(''G''). Quotiening out degeneracy and taking the completion gives a Hilbert space :$(\; \backslash mathcal,\; \backslash langle\; \backslash ;,\backslash ;\; \backslash rangle\_f\; )$ whose typical element is an equivalence class (). For a fixed ''g'' in ''G'', the "shift operator" ''Ug'' defined by (''Ug'')('' h '') (g') = ''h''(''g' - g''), for a representative of (), is unitary. So the map :$g\; \backslash ;\; \backslash mapsto\; \backslash ;\; U\_g$ is a unitary representations of ''G'' on $(\; \backslash mathcal,\; \backslash langle\; \backslash ;,\backslash ;\; \backslash rangle\_f\; )$. By continuity of ''f'', it is weakly continuous, therefore strongly continuous. By construction, we have :$\backslash langle\; U\_\; (),\; ()\; \backslash rangle\_f\; =\; f(g)$ where () is the class of the function that is 1 on the identity of ''G'' and zero elsewhere. But by Gelfand-Fourier isomorphism, the vector state $\backslash langle\; \backslash cdot\; (),\; ()\; \backslash rangle\_f$ on C *(''G'') is the pull-back of a state on $C\_0(\backslash widehat)$, which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives :$\backslash langle\; U\_\; (),\; ()\; \backslash rangle\_f\; =\; \backslash int\_$, the function :$f(g)\; =\; \backslash int\_)$. This extends uniquely to a representation of its multiplier algebra $C\_b(\backslash widehat)$ and therefore a strongly continuous unitary representation ''Ug''. As above we have ''f'' given by some vector state on ''Ug'' :$f(g)\; =\; \backslash langle\; U\_g\; v,\; v\; \backslash rangle,$ therefore positive-definite. The two constructions are mutual inverses. 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「Bochner'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.