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三角级数.
在数学中,三角级数是任何具有下述形式的级数:
[1]
当
和
具有以下形式时,该级数称为傅立叶级数:
![{\displaystyle A_{n}={\frac {1}{\pi ))\displaystyle \int _{0}^{2\pi }\!f(x)\cos {nx}\,dx\qquad (n=0,1,2,\dots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b507d57367433dced3b7e055ca40f683575314)
![{\displaystyle B_{n}={\frac {1}{\pi ))\displaystyle \int _{0}^{2\pi }\!f(x)\sin {nx}\,dx\qquad (n=1,2,3,\dots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81cb2d6854a0bc788f46a007336334b6c4d418dc)
其中
是可积函数。[1]
并不是所有三角级数都是傅立叶级数。一个有趣的问题是给定一个三角级数,当x取什么值时级数收敛。
格奥尔格·康托尔在1870年证明了这一定理。如果三角级数的和函数是零,那么,该三角级数的各项系数均为零。因此,如果两个三角级数的和函数相等,那么它们的各项系数也相等。
- A. Zygmund,1935, "Trigonmetric Series"
- ^ 1.0 1.1 Harry F. Davis, Fourier Series and Orthogonal Functions . 页89
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