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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 Segal–Bargmann space ： ウィキペディア英語版 Segal–Bargmann space In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integrability condition: : where here ''dz'' denotes the 2''n''-dimensional Lebesgue measure on ''C''''n''. It is a Hilbert space with respect to the associated inner product: :$\backslash langle\; F\backslash mid\; G\backslash rangle\; =\; \backslash pi^\; \backslash int\_\; \backslash overlineG(z)\backslash exp(-|z|^2)\backslash ,dz.$ The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see and . Basic information about the material in this section may be found in and . Segal worked from the beginning in the infinite-dimensional setting; see and Section 10 of for more information on this aspect of the subject. == Properties == A basic property of this space is that ''pointwise evaluation is continuous'', meaning that for each ''a'' in ''C''''n'', there is a constant ''C'' such that : It then follows from the Riesz representation theorem that there exists a unique ''F''''a'' in the Segal–Bargmann space such that :$F(a)\; =\; \backslash langle\; F\_a\backslash mid\; F\backslash rangle.$ The function ''F''''a'' may be computed explicitly as :$F\_a(z)\; =\; \backslash exp(\backslash overline\backslash cdot\; z)$ where, explicitly, :$\backslash overline\backslash cdot\; z\; =\; \backslash sum\_^n\; \backslash overlinez\_j.$ The function ''F''''a'' is called the coherent state with parameter ''a'', and the function :$\backslash kappa(a,z)\; :=\; \backslash overline$ is known as the reproducing kernel for the Segal–Bargmann space. Note that : $F(a)\; =\; \backslash langle\; F\_a\backslash mid\; F\backslash rangle\; =\; \backslash pi^\; \backslash int\_\; \backslash kappa(a,z)F(z)\backslash exp(-|z|^2)\backslash ,dz,$ meaning that integration against the reproducing kernel simply gives back (i.e., reproduces) the function ''F'', provided, of course that ''F'' is holomorphic! Note that :$\backslash |F\_a\backslash |^2\; =\; \backslash langle\; F\_a\backslash mid\; F\_a\backslash rangle\; =\; F\_a(a)\; =\; \backslash exp(|a|^2).$ It follows from the Cauchy–Schwarz inequality that elements of the Segal–Bargmann space satisfy the pointwise bounds :$|F(a)|\; \backslash leq\; \backslash |F\_a\backslash |\; \backslash |F\backslash |\; =\; \backslash exp(|a|^2/2)\backslash |F\backslash |.$ 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「Segal–Bargmann space」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.