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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 kernel ： ウィキペディア英語版 Dirichlet kernel In mathematical analysis, the Dirichlet kernel is the collection of functions :$D\_n(x)=\backslash sum\_^n$ e^=1+2\sum_^n\cos(kx)=\frac. It is named after Peter Gustav Lejeune Dirichlet. The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of ''Dn''(''x'') with any function ''f'' of period 2π is the ''n''th-degree Fourier series approximation to ''f'', i.e., we have :$(D\_n$ *f)(x)=\frac\int_^\pi f(y)D_n(x-y)\,dy=\sum_^n \hat(k)e^, where :$\backslash hat(k)=\backslash frac\backslash int\_^\backslash pi\; f(x)e^\backslash ,dx$ is the ''k''th Fourier coefficient of ''f''. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel. Of particular importance is the fact that the ''L''1 norm of ''Dn'' diverges to infinity as ''n'' → ∞. One can estimate that :$\backslash |\; D\_n\; \backslash |\; \_\; =\; O(\backslash log\; n)$. By using a Riemann-sum argument to estimate the contribute in the largest neighbourhood of zero in which $D\_n$ is positive, and the Jensen's inequality for the remaining part, it is also possible to show that: :$\backslash |\; D\_n\; \backslash |\_\; \backslash geq\; 4\backslash operatorname(\backslash pi)+\backslash frac\backslash log\; n.$ This lack of uniform integrability is behind many divergence phenomena for the Fourier series. For example, together with the uniform boundedness principle, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion. See convergence of Fourier series for further details. ==Relation to the delta function== Take the periodic Dirac delta function, which is not really a function, in the sense of mapping one set into another, but is rather a "generalized function", also called a "distribution", and multiply by 2π. We get the identity element for convolution on functions of period 2π. In other words, we have :$f$ *(2\pi \delta)=f \, for every function ''f'' of period 2π. The Fourier series representation of this "function" is :$2\backslash pi\; \backslash delta(x)\backslash sim\backslash sum\_^\backslash infty\; e^=\backslash left(1\; +2\backslash sum\_^\backslash infty\backslash cos(kx)\backslash right).$ Therefore the Dirichlet kernel, which is just the sequence of partial sums of this series, can be thought of as an ''approximate identity''. Abstractly speaking it is not however an approximate identity of ''positive'' elements (hence the failures mentioned above). 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 kernel」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.