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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 Riesz–Fischer theorem ： ウィキペディア英語版 Riesz–Fischer theorem In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the ''L''''p'' spaces from Lebesgue integration theory are complete. == Modern forms of the theorem == The most common form of the theorem states that a measurable function on () is square integrable if and only if the corresponding Fourier series converges in the space ''L''2. This means that if the ''N''th partial sum of the Fourier series corresponding to a square-integrable function ''f'' is given by :$S\_N\; f(x)\; =\; \backslash sum\_^\; F\_n\; \backslash ,\; \backslash mathrm^,$ where ''F''''n'', the ''n''th Fourier coefficient, is given by :$F\_n\; =\backslash frac\backslash int\_^\backslash pi\; f(x)\backslash ,\; \backslash mathrm^\backslash ,\; \backslash mathrmx,$ then :$\backslash lim\_\; \backslash left\; \backslash Vert\; S\_N\; f\; -\; f\; \backslash right\; \backslash |\_2\; =\; 0,$ where $\backslash left\; \backslash Vert\; \backslash cdot\; \backslash right\; \backslash |\_2$ is the ''L''2-norm. Conversely, if $\backslash left\; \backslash \; \backslash ,$ is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that : then there exists a function ''f'' such that ''f'' is square-integrable and the values $a\_n$ are the Fourier coefficients of ''f''. This form of the Riesz–Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity for Fourier series. Other results are often called the Riesz–Fischer theorem . Among them is the theorem that, if ''A'' is an orthonormal set in a Hilbert space ''H'', and ''x'' ∈ ''H'', then :$\backslash langle\; x,\; y\backslash rangle\; =\; 0$ for all but countably many ''y'' ∈ ''A'', and :$\backslash sum\_\; |\backslash langle\; x,y\backslash rangle|^2\; \backslash le\; \backslash |x\backslash |^2.$ Furthermore, if ''A'' is an orthonormal basis for ''H'' and ''x'' an arbitrary vector, the series :$\backslash sum\_\; \backslash langle\; x,y\backslash rangle\; \backslash ,\; y$ converges ''commutatively'' (or ''unconditionally'') to ''x''. This is equivalent to saying that for every ''ε'' > 0, there exists a finite set ''B''0 in ''A'' such that : for every finite set ''B'' containing ''B''0. Moreover, the following conditions on the set ''A'' are equivalent: * the set ''A'' is an orthonormal basis of ''H'' * for every vector ''x'' ∈ ''H'', ::$\backslash |x\backslash |^2\; =\; \backslash sum\_\; |\backslash langle\; x,y\backslash rangle|^2.$ Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that ''L''2 (or more generally ''L''''p'', 0 < ''p'' ≤ ∞) is complete. 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「Riesz–Fischer 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.