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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 Convergence of Fourier series ： ウィキペディア英語版 Convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, ''L''''p'' spaces, summability methods and the Cesàro mean. ==Preliminaries== Consider ''ƒ'' an integrable function on the interval (). For such an ''ƒ'' the Fourier coefficients $\backslash widehat(n)$ are defined by the formula :$\backslash widehat(n)=\backslash frac\backslash int\_0^f(t)e^\backslash ,dt,\; \backslash quad\; n\; \backslash in\; \backslash mathbf.$ It is common to describe the connection between ''ƒ'' and its Fourier series by :$f\backslash sim\; \backslash sum\_n\; \backslash widehat(n)e^.$ The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined: :$S\_N(f;t)=\backslash sum\_^N\; \backslash widehat(n)e^.$ The question here is: Do the functions $S\_N(f)$ (which are functions of the variable ''t'' we omitted in the notation) converge to ''ƒ'' and in which sense? Are there conditions on ''ƒ'' ensuring this or that type of convergence? This is the main problem discussed in this article. Before continuing, the Dirichlet kernel must be introduced. Taking the formula for $\backslash widehat(n)$, inserting it into the formula for $S\_N$ and doing some algebra gives that :$S\_N(f)=f$ * D_N\, where ∗ stands for the periodic convolution and $D\_N$ is the Dirichlet kernel, which has an explicit formula, :$D\_n(t)=\backslash frac)t)\}.$ The Dirichlet kernel is ''not'' a positive kernel, and in fact, its norm diverges, namely :$\backslash int\; |D\_n(t)|\backslash ,dt\; \backslash to\; \backslash infty$ a fact that plays a crucial role in the discussion. The norm of ''D''''n'' in ''L''1(T) coincides with the norm of the convolution operator with ''D''''n'', acting on the space ''C''(T) of periodic continuous functions, or with the norm of the linear functional ''ƒ'' → (''S''''n''''ƒ'')(0) on ''C''(T). Hence, this family of linear functionals on ''C''(T) is unbounded, when ''n'' → ∞. 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「Convergence of 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.