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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 Paley–Wiener theorem ： ウィキペディア英語版 Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. ==Holomorphic Fourier transforms== The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform :$f(\backslash zeta)\; =\; \backslash int\_^\backslash infty\; F(x)e^\backslash ,dx$ and allow ''ζ'' to be a complex number in the upper half-plane. One may then expect to differentiate under the integral in order to verify that the Cauchy–Riemann equations hold, and thus that ''f'' defines an analytic function. However, this integral may not be well-defined, even for ''F'' in ''L''2(R) - indeed, since ''ζ'' is in the upper half plane, the modulus of ''e''''ixζ'' grows exponentially as x → -∞ - so differentiation under the integral sign is out of the question. One must impose further restrictions on ''F'' in order to ensure that this integral is well-defined. The first such restriction is that ''F'' be supported on R+: that is, ''F'' ∈ ''L''2(R+). The Paley–Wiener theorem now asserts the following:〔; ; 〕 The holomorphic Fourier transform of ''F'', defined by :$f(\backslash zeta)\; =\; \backslash int\_0^\backslash infty\; F(x)\; e^\backslash ,\; dx$ for ζ in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has :$\backslash int\_^\backslash infty\; \backslash left\; |f(\backslash xi+i\backslash eta)\; \backslash right|^2\backslash ,\; d\backslash xi\; \backslash le\; \backslash int\_0^\backslash infty\; |F(x)|^2\backslash ,\; dx$ and by dominated convergence, :$\backslash lim\_\backslash int\_^\backslash infty\; \backslash left|f(\backslash xi+i\backslash eta)-f(\backslash xi)\; \backslash right|^2\backslash ,d\backslash xi\; =\; 0.$ Conversely, if ''f'' is a holomorphic function in the upper half-plane satisfying : then there exists ''F'' in ''L''2(R+) such that ''f'' is the holomorphic Fourier transform of ''F''. In abstract terms, this version of the theorem explicitly describes the Hardy space ''H''2(R). The theorem states that :$\backslash mathcalH^2(\backslash mathbf)=L^2(\backslash mathbf).$ This is a very useful result as it enables one pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space ''L''2(R+) of square-integrable functions supported on the positive axis. By imposing the alternative restriction that ''F'' be compactly supported, one obtains another Paley–Wiener theorem.〔; 〕 Suppose that ''F'' is supported in (''A'' ), so that ''F'' ∈ ''L''2(−''A'',''A''). Then the holomorphic Fourier transform :$f(\backslash zeta)\; =\; \backslash int\_^A\; F(x)e^\backslash ,dx$ is an entire function of exponential type ''A'', meaning that there is a constant ''C'' such that :$|f(\backslash zeta)|\backslash le\; Ce^,$ and moreover, ''f'' is square-integrable over horizontal lines: : Conversely, any entire function of exponential type ''A'' which is square-integrable over horizontal lines is the holomorphic Fourier transform of an ''L''2 function supported in (''A'' ). 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「Paley–Wiener 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.