# 狄利克雷定理 (傅里叶级数)

## 定理的叙述

${\displaystyle f}$ 为一个在${\displaystyle \mathbb {R} }$上的周期性局部可积函数，其周期为${\displaystyle 2\pi }$。给定 ${\displaystyle x_{0}\in \mathbb {R} }$，假设有以下条件成立：

1. 函数 ${\displaystyle f}$${\displaystyle x_{0}}$ 处有左极限和右极限，分别记为 ${\displaystyle f(x_{0}^{+})}$${\displaystyle f(x_{0}^{-})}$
2. 存在正实数：${\displaystyle \alpha >0}$，使得以下的两个积分收敛：
${\displaystyle \int _{0}^{\alpha }{\frac {|f(x_{0}+t)-f(x_{0}^{+})|}{t}}{\mathrm {d} }t,\qquad \int _{0}^{\alpha }{\frac {|f(x_{0}-t)-f(x_{0}^{-})|}{t}}{\mathrm {d} }t}$

${\displaystyle \lim \limits _{n}(S_{n}f(x_{0}))={\frac {1}{2}}(f(x_{0}^{+})+f(x_{0}^{-}))}$

## 证明

${\displaystyle D_{n}(x)=\sum _{k=-n}^{n}e^{ikx}={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}},}$
${\displaystyle S_{n}(f)(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)D_{n}(x-t)dt={\frac {1}{2\pi }}\int _{-\pi }^{\pi }D_{n}(t)f(x-t)dt}$

${\displaystyle S_{n}(f)(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin \left(\left(n+{\frac {1}{2}}\right)t\right){\frac {f(x-t)}{\sin {\frac {t}{2}}}}dt}$

${\displaystyle {\tilde {f}}(x)={\frac {f(x^{+})+f(x^{-})}{2}}}$

${\displaystyle S_{n}(f)(x)=\int _{0}^{\pi }\sin \left(\left(n+{\frac {1}{2}}\right)t\right){\frac {f(x+t)+f(x-t)}{\sin {\frac {t}{2}}}}dt}$

${\displaystyle S_{n}(f)(x)-{\tilde {f}}(x)={\frac {1}{2\pi }}\int _{0}^{\pi }\sin \left(\left(n+{\frac {1}{2}}\right)t\right){\frac {f(x+t)+f(x-t)}{\sin {\frac {t}{2}}}}dt-{\frac {1}{2}}\left(f(x^{+})+f(x^{-})\right)}$

${\displaystyle 1={\frac {1}{2\pi }}\int _{-\pi }^{\pi }D_{n}(t)dt={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{\frac {\sin \left(\left(n+{\frac {1}{2}}\right)t\right)}{\sin(t/2)}}dt=2\cdot {\frac {1}{2\pi }}\int _{0}^{\pi }{\frac {\sin \left(\left(n+{\frac {1}{2}}\right)t\right)}{\sin(t/2)}}dt}$
${\displaystyle {\frac {1}{2}}={\frac {1}{2\pi }}\int _{0}^{\pi }{\frac {\sin \left(\left(n+{\frac {1}{2}}\right)t\right)}{\sin(t/2)}}dt}$

{\displaystyle {\begin{aligned}S_{n}(f)(x)-{\tilde {f}}(x)&={\frac {1}{2\pi }}\int _{0}^{\pi }\sin \left(\left(n+{\frac {1}{2}}\right)t\right){\frac {f(x+t)+f(x-t)}{\sin {\frac {t}{2}}}}dt\\&-{\frac {1}{2\pi }}\int _{0}^{\pi }{\frac {\sin \left(\left(n+{\frac {1}{2}}\right)t\right)}{\sin {\frac {t}{2}}}}\left(f(x^{+})+f(x^{-})\right)dt\\&={\frac {1}{2\pi }}\int _{0}^{\pi }\sin \left(\left(n+{\frac {1}{2}}\right)t\right){\frac {f(x+t)+f(x-t)-f(x^{+})-f(x^{-})}{\sin {\frac {t}{2}}}}dt\end{aligned}}}

${\displaystyle \lim _{n\to \infty }S_{n}(f)(x)={\tilde {f}}(x)}$

## 注释与参考

1. ^ 狄利克雷, Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données, Journal de Crelle 4 (1829) p. 157-169
2. ^ 若尔当, Sur la série de Fourier, C. R. Acad. Sci. Paris, 92 p 228-230

## 参考书籍

• （英文）Allan Pinkus,Samy Zafrany. Fourier series and integral transforms. Cambridge University Press. 1997. ISBN 9780521597715.p.46-52.
• （法文）Jean-Pierre Kahane, Pierre-Gilles Lemarié-Rieusset. Séries de Fourier et ondelettes. Cassini. 1998. ISBN 284225001X.