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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 Generalized Fourier series ： ウィキペディア英語版 Generalized Fourier series In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory. ==Definition== Consider a set of square-integrable functions with values in $\backslash mathbb=\backslash mathbb\backslash mbox\backslash mathbb$, :$\backslash Phi\; =\; \backslash \_^\backslash infty,$ which are pairwise orthogonal for the inner product :$\backslash langle\; f,\; g\backslash rangle\_w\; =\; \backslash int\_a^b\; f(x)\backslash ,\backslash overline(x)\backslash ,w(x)\backslash ,dx$ where ''w''(''x'') is a weight function, and $\backslash overline\backslash cdot$ represents complex conjugation, i.e. $\backslash overline(x)=g(x)$ for $\backslash mathbb=\backslash mathbb$. The generalized Fourier series of a square-integrable function ''f'': (''b'' ) → $\backslash mathbb$, with respect to Φ, is then :$f(x)\; \backslash sim\; \backslash sum\_^\backslash infty\; c\_n\backslash varphi\_n(x),$ where the coefficients are given by :$c\_n\; =\; .$ If Φ is a complete set, i.e., an orthonormal basis of the space of all square-integrable functions on (''b'' ), as opposed to a smaller orthonormal set, the relation $\backslash sim\; \backslash ,$ becomes equality in the ''L²'' sense, more precisely modulo |·|''w'' (not necessarily pointwise, nor almost everywhere). 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「Generalized Fourier series」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.