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In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if ''f'':R → C is a continuous function with period 2π, then the sequence (σ''n'') of Cesàro means of the sequence (''s''''n'') of partial sums of the Fourier series of ''f'' converges uniformly to ''f'' on (). Explicitly, : where : and : with ''F''''n'' being the ''n''th order Fejér kernel. A more general form of the theorem applies to functions which are not necessarily continuous . Suppose that ''f'' is in ''L''1(-π,π). If the left and right limits ''f''(''x''0±0) of ''f''(''x'') exist at ''x''0, or if both limits are infinite of the same sign, then : Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of Marcel Riesz, Fejér's theorem holds precisely as stated if the (C, 1) mean σ''n'' is replaced with (C, α) mean of the Fourier series . ==References== * . 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「Fejér's theorem」の詳細全文を読む スポンサード リンク
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