[备课] 狄里赫利条件 Dirichlet conditions 汇集

Johann Peter Gustav Lejeune Dirichlet (1805-02-13 ~ 1859-05-05) 

https://mathshistory.st-andrews.ac.uk/Biographies/Dirichlet/

MacTutor History of Mathematics Archive, University of St Andrews, Scotland

一、狄里赫利条件

   (1)函数在任意有限区间内连续，或只有有限个第一类间断点（当t从左或右趋于这个间断点时，函数有有限的左极限和右极限）。

   (2)在一个周期内，函数有有限个极大值或极小值。

   (3)x(t)在单个周期内绝对可积，即

二、Dirichlet Fourier Series Conditions

   A piecewise regular function that

   1. Has a finite number of finite discontinuities and

   2. Has a finite number of extrema

can be expanded in a Fourier series which converges to the function at continuous points and the mean of the positive and negative limits at points of discontinuity. 

三、Convergence of Fourier Series

（1）The Weak Dirichlet Condition for the Fourier Series

   For the Fourier Series to exist, the Fourier coefficients must be finite. The Weak Dirichlet Condition guarantees this. It essentially says that the integral of the absolute value of the signal must be finite.

（2）The Strong Dirichlet Conditions for the Fourier Series

   For the Fourier Series to exist, the following two conditions must be satisfied (along with the Weak Dirichlet Condition):

   1. In one period, f(t) has only a finite number of minima and maxima.

   2. In one period, f(t) has only a finite number of discontinuities and each one is finite.

   These are what we refer to as the Strong Dirichlet Conditions. In theory we can think of signals that violate these conditions, sin(logt) for instance. However, it is not possible to create a signal that violates these conditions in a lab. Therefore, any real-world signal will have a Fourier representation.

   十多年前，根据狄里赫利条件，真傻得到了一个基本上属于“写入教材/专著”级别的从“1到2”的科技创新：已经有了他人的理论解释，但是别人的解释不太彻底（八成是“纯数学家们”干的！）；根据这些不彻底的解释，人类还没有得到消除哪些不预期“坏作用”的具体方法。真傻算是后期搞“封顶”的，尽管只封了一半左右的顶。主要是没有全部封顶需要的一些客观条件。真傻找到了一些近似的实用工程方法。编程序试过了，大约的确是可行的。代表中国人，真傻再宣布一个基本上属于“小小的写进教材/专著级别的优先权”吧！尽管还没有全部封顶。

   请您不要嫉妒俺：这个基本上有点属于“小小的写进教材/专著级别的优先权”，肯定不会拿诺贝尔奖的。请您放心吧！

   由于我在周二（2020-10-06）上午在本科生课堂上提到过这个创新点，只好在此写一下。讲本科生《电工学》的“非正弦周期”信号时，用到了狄里赫利条件，随便说了一下该创新点。

   狗熊掰棒子（一个民间谚语故事）：

   狗熊走进玉米地里，掰了个棒子夹到腋下，走了几步又掰了个棒子夹到腋下，但原先的棒子却掉了。狗熊在玉米地里忙活了半天，最终手上就只有一两个棒子。

参考资料：

[1] 狄里赫利条件 - 百度百科 

https://baike.baidu.com/item/%E7%8B%84%E9%87%8C%E8%B5%AB%E5%88%A9%E6%9D%A1%E4%BB%B6

[2] Weisstein, Eric W. "Dirichlet Fourier Series Conditions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletFourierSeriesConditions.html

[3] 6.6: Convergence of Fourier Series - Engineering LibreTexts

https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Book%3A_Signals_and_Systems_(Baraniuk_et_al.)/06%3A_Continuous_Time_Fourier_Series_(CTFS)/6.06%3A_Convergence_of_Fourier_Series

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