蒲和平非数习题解答 综合题 8* 第 44 题

设 $f\left( x\right)$ 是以 ${2\pi }$ 为周期的可积函数,其傅里叶系数为 ${a}_{0},{a}_{n},{b}_{n}$ ,记 ${S}_{0}\left( x\right) = \frac{{a}_{0}}{2}$ , ${S}_{n}\left( x\right) = \frac{{a}_{0}}{2} + \mathop{\sum }\limits_{k = 1}^{n}\left( {{a}_{k}\cos {kx} + {b}_{k}\sin {kx}}\right) ,{\sigma }_{n}\left( x\right) = \frac{1}{n}\mathop{\sum }\limits_{k = 0}^{n - 1}{S}_{k}\left( x\right)$ . 证明:
(1) ${S}_{n}\left( x\right) = \frac{1}{2\pi }{\int }_{-\pi }^{\pi }f\left( {x + t}\right) \frac{\sin \left( {n + 1/2}\right) t}{\sin \left( {t/2}\right) }\mathrm{d}t$
(2) ${\sigma }_{n}\left( x\right) = \frac{1}{2n\pi }{\int }_{-\pi }^{\pi }f\left( {x + t}\right) {\left[ \frac{\sin \left( {{nt}/2}\right) }{\sin \left( {t/2}\right) }\right] }^{2}\mathrm{\;d}t$ .