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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 Gibbs phenomena ： ウィキペディア英語版 Gibbs phenomenon In mathematics, the Gibbs phenomenon, discovered by 〔 Available on-line at: (National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. )〕 and rediscovered by , is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The ''n''th partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit.〔 〕 This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatuses.〔 Section 4.7.〕 These are one cause of ringing artifacts in signal processing. ==Description== The Gibbs phenomenon involves both the fact that Fourier sums overshoot at a jump discontinuity, and that this overshoot does not die out as the frequency increases. The three pictures on the right demonstrate the phenomenon for a square wave (of height $\backslash pi/4$) whose Fourier expansion is :$\backslash sin(x)+\backslash frac\backslash sin(3x)+\backslash frac\backslash sin(5x)+\backslash dotsb.$ More precisely, this is the function ''f'' which equals $\backslash pi/4$ between $2n\backslash pi$ and $(2n+1)\backslash pi$ and $-\backslash pi/4$ between $(2n+1)\backslash pi$ and $(2n+2)\backslash pi$ for every integer ''n''; thus this square wave has a jump discontinuity of height $\backslash pi/2$ at every integer multiple of $\backslash pi$. As can be seen, as the number of terms rises, the error of the approximation is reduced in width and energy, but converges to a fixed height. A calculation for the square wave (see Zygmund, chap. 8.5., or the computations at the end of this article) gives an explicit formula for the limit of the height of the error. It turns out that the Fourier series exceeds the height $\backslash pi/4$ of the square wave by :$\backslash frac\backslash int\_0^\backslash pi\; \backslash frac\backslash ,\; dt\; -\; \backslash frac\; =\; \backslash frac\backslash cdot\; (0.089490\backslash dots)$ or about 9 percent. More generally, at any jump point of a piecewise continuously differentiable function with a jump of ''a'', the ''n''th partial Fourier series will (for ''n'' very large) overshoot this jump by approximately $a\; \backslash cdot\; (0.089392\backslash dots)$ at one end and undershoot it by the same amount at the other end; thus the "jump" in the partial Fourier series will also be about 9% larger than the jump in the original function. At the location of the discontinuity itself, the partial Fourier series will converge to the midpoint of the jump (regardless of what the actual value of the original function is at this point). The quantity :$\backslash int\_0^\backslash pi\; \backslash frac\backslash \; dt\; =\; (1.851937052\backslash dots)\; =\; \backslash frac\; +\; \backslash pi\; \backslash cdot\; (0.089392\backslash dots)$ is sometimes known as the ''Wilbraham–Gibbs constant''. 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「Gibbs phenomenon」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.