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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 Parseval's identity ： ウィキペディア英語版 Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is the Pythagorean theorem for inner-product spaces. Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function, :$\backslash sum\_^\backslash infty\; |c\_n|^2\; =\; \backslash frac\backslash int\_^\backslash pi\; |f(x)|^2\; \backslash ,\; dx,$ where the Fourier coefficients ''c''''n'' of ''ƒ'' are given by :$c\_n\; =\; \backslash frac\backslash int\_^\; f(x)\; \backslash mathrm^\; \backslash ,\; dx.$ More formally, the result holds as stated provided ''ƒ'' is square-integrable or, more generally, in ''L''2(). A similar result is the Plancherel theorem, which asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself. In one-dimension, for , :$\backslash int\_^\backslash infty\; |\backslash hat(\backslash xi)|^2\backslash ,d\backslash xi\; =\; \backslash int\_^\backslash infty\; |f(x)|^2\backslash ,\; dx.$ == Generalization of the Pythagorean theorem == The identity is related to the Pythagorean theorem in the more general setting of a separable Hilbert space as follows. Suppose that ''H'' is a Hilbert space with inner product 〈•,•〉. Let (''e''''n'') be an orthonormal basis of ''H''; i.e., the linear span of the ''e''''n'' is dense in ''H'', and the ''e''''n'' are mutually orthonormal: : 0&\mbox\ m \not= n.\end Then Parseval's identity asserts that for every ''x'' ∈ ''H'', :$\backslash sum\_n\; |\backslash langle\; x,\; e\_n\backslash rangle|^2\; =\; \backslash |x\backslash |^2.$ This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval's identity by letting ''H'' be the Hilbert space ''L''2(), and setting ''e''''n'' = e−i''nx'' for More generally, Parseval's identity holds in any inner-product space, not just separable Hilbert spaces. Thus suppose that ''H'' is an inner-product space. Let ''B'' be an orthonormal basis of ''H''; i.e., an orthonormal set which is ''total'' in the sense that the linear span of ''B'' is dense in ''H''. Then :$\backslash |x\backslash |^2=\backslash langle\; x,x\backslash rangle=\backslash sum\_\backslash left|\backslash langle\; x,v\backslash rangle\backslash right|^2.$ The assumption that ''B'' is total is necessary for the validity of the identity. If ''B'' is not total, then the equality in Parseval's identity must be replaced yielding Bessel's inequality. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem. 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「Parseval's identity」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.