📝 题目
5.5.6 设 $f\left( x\right)$ 是周期为 ${2\pi }$ 的连续函数. ${a}_{n},{b}_{n}$ 为其傅氏系数, ${A}_{n},{B}_{n}$ 是卷积函数
$$ F\left( x\right) = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( t\right) f\left( {x - t}\right) \mathrm{d}t $$
的傅氏系数. 求证:
(1) ${A}_{0} = {a}_{0}^{2},{A}_{n} = {a}_{n}^{2} + {b}_{n}^{2},{B}_{n} = 0\left( {n = 1,2,\cdots }\right)$ ;
(2) $\frac{1}{\pi }{\int }_{-\pi }^{\pi }{f}^{2}\left( t\right) \mathrm{d}t = \frac{{a}_{0}^{2}}{2} + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{a}_{n}^{2} + {b}_{n}^{2}}\right)$ .