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傅里叶级数理论详讲&实例应用

过冷水

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久久不更，更则必是大招，本期推文有理论有案例，有兴趣的读者仔细阅读。

过冷水之前有和大家讲傅里叶级数，并给出以一个函数用傅里叶级数近似的案例。本期就进一步详讲傅里叶级数。傅里叶级数展开时基底函数取1,cosx、sinx,cos2x、sin2x.....cosnx、sinnx,傅里叶级数一般情况下表示为：

a₀、a_n、b_n是展开系数。假定一个周期为2π的函数f(x 2π)=f(x)现在计算其系数。这就需要一点灵活的数学思想来解决问题。我们对上式两边在[0,2π]范围内积分。可得：

我们继续采用两边积分的方法求系数a_n、b_n_，对泰勒级数两边同时乘以cos(mx),然后在[0,2π]范围积分，如下：

则当m=n时：同理推b_n：

许多实际问题中，函数f(x)是一个定义在有限区间0<x<X上的任意函数。对这样的函数如何展开傅里叶级数？大师告诉我们依旧可以用正弦函数、余弦函数展开：

现在求展开系数的表达式：

讲完傅里叶级数的理论，我们来看看实际傅里叶级数实际应用。过冷水在学习过程中遇到如下一幅图，需要对该函数进行积分求值：

数值图像求积分的问题过冷水的往期推文数值计算——MATLAB数值积分原理详讲、数值优化—三种复杂函数数值积分方法实例演示都有讲。过冷水本期的重点不是求数值积分的方法讨论，就想弄清楚如何很好的给出很好拟合该线段的函数表达式。这个函数图像算是比较复杂的，很少接触到所示的图像函数式，在此只能给出一个较为接近的拟合公式：

%原图像数据

x=[-0.0171386436557126,2.64214113473764,2.75334303194563,2.92014587775763,2.92134159708245,2.97813826501126,3.03453635983180,3.09133302776062,3.14852826879770,3.20592279638893,3.26311803742601,3.31991470535483,3.37691065983778,3.43350804121246,3.49030470914127,3.54490922497459,3.54371350564977,3.59811873492895,3.65232467765400,3.70633133382491,3.76053727654995,3.81514179238327,3.86974630821658,3.92435082404990,3.97935391299149,4.03415771537894,4.14416389326212,4.25436935769944,4.36457482213675,4.58578289722792,4.69598836166524,4.80738954542737,4.86318978058550,4.97419239123936,5.14019809083481,5.25080212838040,5.36160545248012,5.47280734968812,5.69441399788756,5.80581518164969,5.91721636541183,5.91781422507423,5.97401303334064,6.03001255505291,6.08581279021105,6.19581896809422,6.25082205703581,6.36122680802726,6.41642918352299,6.69343749377230,6.85944319336774,7.02584746607147,7.13625221706292,7.30185934355009,7.57866836724526,7.74527192650312,7.80127144821539,7.85707168337352,7.96727714781084,8.07748261224815,8.18768807668547,8.29829211423105,8.57530042448036,8.63110065963850,8.90810896988780,9.01931086709580,9.07531038880807,9.13130991052034,9.24251180772833,9.29831204288647,9.40851750732378,9.51872297176110,9.68433009824827,9.79533270890213,9.90593674644772,10.0719424460432,10.3489507562925,10.4601526535005,10.6819585882541,10.8475657147412,10.9583690388410,11.0695709360489,11.1805735467028,11.2907790111401,11.4013830486857,11.5677873213894,11.7341915940931,11.8997987205803,12.0660037067299,12.2326072659878,12.3436098766416,12.5100141493453,12.6214153331075,12.6774148548197,12.8432212678611,13.0088283943482,13.1750333804978,13.2302357559935,13.5626457282927,13.6736483389466,13.7286514278882,13.8392554654337,13.9496602164252,14.3374718507742,14.4486737479822,15.1134936925806,15.2804958249467,15.5575041351960,15.6135036569083,15.7247055541163,15.8909105402658,16.0013152912573,16.1117200422488,16.1667231311903,16.2773271687359,16.3325295442317,16.4987345303812,16.5539369058770,16.7755435540764,16.9419478267801,17.0523525777716,17.1629566153172,17.3845632635166,17.6065684848243,17.8283744195779,18.0499810677774,18.2167839135894,18.9370055202376,19.2134159708245,19.3796209569740,19.4906235676279,19.7122302158273,19.8226349668188];

y=[5.95683453237410,5.95683453237410,6.02158273381295,6.11870503597122,6.31294964028777,6.53956834532374,6.70143884892086,6.92805755395683,7.21942446043166,7.54316546762590,7.83453237410072,8.06115107913669,8.32014388489209,8.51438848920863,8.74100719424460,8.61151079136691,8.41726618705036,8.25539568345324,8.06115107913669,7.83453237410072,7.64028776978417,7.51079136690648,7.38129496402878,7.25179856115108,7.18705035971223,7.08992805755396,6.96043165467626,6.86330935251799,6.76618705035971,6.70143884892086,6.60431654676259,6.70143884892086,6.76618705035971,6.79856115107914,6.76618705035971,6.73381294964029,6.73381294964029,6.79856115107914,6.79856115107914,6.89568345323741,6.99280575539568,7.08992805755396,7.21942446043166,7.31654676258993,7.38129496402878,7.25179856115108,7.18705035971223,7.12230215827338,7.08992805755396,7.08992805755396,7.05755395683453,7.08992805755396,7.02517985611511,6.92805755395683,6.89568345323741,6.96043165467626,7.05755395683453,7.12230215827338,7.02517985611511,6.92805755395683,6.83093525179856,6.79856115107914,6.79856115107914,6.86330935251799,6.86330935251799,6.92805755395683,7.02517985611511,7.12230215827338,7.18705035971223,7.25179856115108,7.15467625899281,7.05755395683453,6.96043165467626,6.99280575539568,6.96043165467626,6.92805755395683,6.92805755395683,6.99280575539568,7.02517985611511,6.92805755395683,6.92805755395683,6.99280575539568,7.02517985611511,6.92805755395683,6.89568345323741,6.92805755395683,6.96043165467626,6.86330935251799,6.86330935251799,6.92805755395683,6.96043165467626,6.99280575539568,7.08992805755396,7.18705035971223,7.12230215827338,7.02517985611511,7.02517985611511,6.99280575539568,6.99280575539568,7.02517985611511,6.96043165467626,6.92805755395683,6.86330935251799,6.86330935251799,6.92805755395683,6.92805755395683,7.05755395683453,7.05755395683453,7.15467625899281,7.21942446043166,7.21942446043166,7.15467625899281,7.08992805755396,7.02517985611511,6.99280575539568,6.96043165467626,6.96043165467626,6.92805755395683,6.92805755395683,6.96043165467626,6.89568345323741,6.86330935251799,6.86330935251799,6.92805755395683,6.96043165467626,6.96043165467626,7.05755395683453,7.05755395683453,6.96043165467626,6.96043165467626,6.99280575539568,6.99280575539568,6.92805755395683];

%函数拟合

f=fit(x',y','smoothingspline');

x1=linspace(2,19,250)';

y1=f(x1);

%绘图

figure1 = figure;

axes1 = axes('Parent',figure1);

hold(axes1,'on');

plot(x,y,'LineWidth',2,'Parent',axes1);

plot(x1,y1,'LineWidth',2,'Parent',axes1);

legend('原函数图像','拟合函数图像');

xlabel('$x$','Interpreter','latex');