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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 Dirichlet conditions ： ウィキペディア英語版 Dirichlet conditions In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function ''f''(''x'') to be equal to the sum of its Fourier series at each point where ''f'' is continuous. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). These conditions are named after Peter Gustav Lejeune Dirichlet. The conditions are: *''f''(''x'') must be absolutely integrable over a period. *''f''(''x'') must have a finite number of extrema in any given bounded interval, i.e. there must be a finite number of maxima and minima in the interval. *''f''(''x'') must have a finite number of discontinuities in any given bounded interval, however the discontinuity cannot be infinite. *''f''(''x'') must be bounded The last three conditions are satisfied if ''f'' is a function of bounded variation over a period. ==Dirichlet's theorem for 1-dimensional Fourier series== We state Dirichlet's theorem assuming ''f'' is a periodic function of period 2π with Fourier series expansion where :$a\_n\; =\; \backslash frac\; \backslash int\_^\backslash pi\; f(x)\; e^\backslash ,\; dx.$ The analogous statement holds irrespective of what the period of ''f'' is, or which version of the Fourier expansion is chosen (see Fourier series). :Dirichlet's theorem: If ''f'' satisfies Dirichlet conditions, then for all ''x'', we have that the series obtained by plugging ''x'' into the Fourier series is convergent, and is given by ::$\backslash sum\_^\backslash infty\; a\_n\; e^\; =\; \backslash frac(f(x+)\; +\; f(x-)),$ :where the notation ::$f(x+)\; =\; \backslash lim\_\; f(y)$ ::$f(x-)\; =\; \backslash lim\_\; f(y)$ :denotes the right/left limits of ''f''. A function satisfying Dirichlet's conditions must have right and left limits at each point of discontinuity, or else the function would need to oscillate at that point, violating the condition on maxima/minima. Note that at any point where ''f'' is continuous, :$\backslash frac(f(x+)\; +\; f(x-))\; =\; f(x).$ Thus Dirichlet's theorem says in particular that under the Dirichlet conditions the Fourier series for ''f'' converges and is equal to ''f'' wherever ''f'' is continuous. 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「Dirichlet conditions」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.