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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 theorem ： ウィキペディア英語版 Parseval's theorem In mathematics, Parseval's theorem 〔Parseval des Chênes, Marc-Antoine Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in ''Mémoires présentés à l’Institut des Sciences, ettres et Arts, par divers savans, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savans étrangers.)'', vol. 1, pages 638–648 (1806).〕 usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's Identity, after John William Strutt, Lord Rayleigh.〔Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature," ''Philosophical Magazine'', vol. 27, pages 460–469. Available on-line (here ).〕 Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics and engineering, the most general form of this property is more properly called the Plancherel theorem.〔Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," ''Rendiconti del Circolo Matematico di Palermo'', vol. 30, pages 298–335.〕 == Statement of Parseval's theorem == Suppose that ''A''(''x'') and ''B''(''x'') are two square integrable (with respect to the Lebesgue measure), complex-valued functions on R of period 2π with Fourier series :$A(x)=\backslash sum\_^\backslash infty\; a\_ne^$ and :$B(x)=\backslash sum\_^\backslash infty\; b\_ne^$ respectively. Then :$\backslash sum\_^\backslash infty\; a\_n\backslash overline\; =\; \backslash frac\; \backslash int\_^\backslash pi\; A(x)\backslash overline\; \backslash ,\; dx,$ where ''i'' is the imaginary unit and horizontal bars indicate complex conjugation. More generally, given an abelian topological group ''G'' with Pontryagin dual ''G^'', Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces ''L''2(''G'') and ''L''2(''G^'') (with integration being against the appropriately scaled Haar measures on the two groups.) When ''G'' is the unit circle T, ''G^'' is the integers and this is the case discussed above. When ''G'' is the real line R, ''G^'' is also R and the unitary transform is the Fourier transform on the real line. When ''G'' is the cyclic group Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts. 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 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.