富里叶级数的收敛速度

对正弦和余弦富立叶级数,通过合并相邻同号项,使其重排成交错级数.讨论了重排形成的交错级数的敛散性.指出根据自变量x的不同取值,该交错级数可能是单调递减或周期递减的级数.按照莱布尼茨判定法提出了不同精度要求的级数项数的计算公式.选取一到三阶收敛的富立叶级数计算了不同比值精度及差值精度要求的级数项数.计算表明,在x的取值为2π的等分点时,富立叶级数的部分和随项数的增加单调地逼近其收敛值.在x的取值为其它点时,富立叶级数的部分和随项数的增加围绕收敛值上下变动,周期地逼近其收敛值.低收敛阶富立叶级数的收敛速度较慢.要达到0.01%的精度,一收敛阶富立叶级数需要数万项,二收敛阶富立叶级数也需要数百项.在不同计算点处,要达到相同的计算精度,需要的级数项数差别较大.

On the Speed of Convergence of Fourier Series

Both sine and cosine Fourier series were rearranged as alternating series by combining consecutive terms with the same sign in the series.Convergence and divergence of the rearrangement-formed alternating series was discussed.It was indicated that terms of the series could be monotonous decreasing or periodical decreasing with respect to different value of independent variable x.Based on Leibniz criterion,formulae were put forward to compute the number of terms needed for certain accuracy of calculation.The speed of convergence was investigated upon three series,namely sine series of first order,cosine series of second order as well as sine series of third order of convergence.The number of terms was obtained for two kinds of accuracies,ratio of remainder over convergence value and difference between partial sum and convergence value.Results show that the partial sum of the series monotonously approaches its convergence value as x is an equipartition of 2π;otherwise,the partial sum waves around and approximates its limit as the number of terms added increases.Fourier series with low order of convergence have low speed of convergence.To reach accuracy of 0.01%,ten thousands of terms are needed for the first order series while hundreds of terms are needed for the second order series.Terms needed for certain accuracy varies largely at different value of x.