蒲和平非数习题解答 综合题 8* 第 42 题

设 $f\left( x\right)$ 是 $\left( {-\infty , + \infty }\right)$ 上以 ${2\pi }$ 为周期的具有二阶连续导数的函数. 记 ${b}_{n} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \sin {nx}\mathrm{\;d}x$ , ${b}_{n}^{\prime\prime} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }{f}^{\prime \prime }\left( x\right) \sin {nx}\mathrm{\;d}x$ . 证明若 $\mathop{\sum }\limits_{n = 1}^{\infty }{b}_{n}^{\prime\prime}$ 绝对收敛,则 $\mathop{\sum }\limits_{n = 1}^{\infty }\sqrt{\left| {b}_{n}\right| } < \frac{1}{2}\left( {2 + \mathop{\sum }\limits_{n = 1}^{\infty }\left| {b}_{n}^{\prime\prime}\right| }\right)$ .