📝 题目
例 7 设 $f\left( x\right)$ 在 $\left( {-\infty ,\infty }\right)$ 连续,
$$ {\int }_{-\infty }^{+\infty }\left| {f\left( x\right) }\right| \mathrm{d}x < + \infty ,\;{\int }_{-\infty }^{+\infty }{\left| f\left( x\right) \right| }^{2}\mathrm{\;d}x < + \infty . $$
定义
$$ \psi \left( x\right) = {\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left( {\left| {x - \xi }\right| + \left| {x - \eta }\right| }\right) }\left| {f\left( \xi \right) }\right| \left| {f\left( \eta \right) }\right| \mathrm{d}\xi \mathrm{d}\eta . $$
求证:
$$ {\int }_{-\infty }^{+\infty }\psi \left( x\right) \mathrm{d}x \leq 4{\int }_{-\infty }^{+\infty }{\left| f\left( x\right) \right| }^{2}\mathrm{\;d}x. $$
💡 答案与解析
证法 1 利用 $\left| {f\left( \xi \right) }\right| \left| {f\left( \eta \right) }\right| \leq \frac{1}{2}\left\lbrack {{\left| f\left( \xi \right) \right| }^{2} + {\left| f\left( \eta \right) \right| }^{2}}\right\rbrack$ ,则
$$ \psi \left( x\right) \leq {\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left( {\left| {x - \xi }\right| + \left| {x - \eta }\right| }\right) }\frac{{\left| f\left( \xi \right) \right| }^{2} + {\left| f\left( \eta \right) \right| }^{2}}{2}\mathrm{\;d}\xi \mathrm{d}\eta $$
$$ = \frac{1}{2}{\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}{\mathrm{e}}^{-\left| {x - \xi }\right| }\mathrm{d}\xi {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \eta }\right| }\mathrm{d}\eta $$
$$ + \frac{1}{2}{\int }_{-\infty }^{+\infty }{\left| f\left( \eta \right) \right| }^{2}{\mathrm{e}}^{-\left| {x - \eta }\right| }\mathrm{d}\eta {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }\mathrm{d}\xi . \tag{7.22} $$
又
$$ {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \eta }\right| }\mathrm{d}\eta = {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }\mathrm{d}\xi = {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| u\right| }\mathrm{d}u $$
$$ = 2{\int }_{0}^{+\infty }{\mathrm{e}}^{-u}\mathrm{\;d}u = 2, \tag{7.23} $$
代入 (7.22) 式得到
$$ \psi \left( x\right) \leq 2{\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}{\mathrm{e}}^{-\left| {x - \xi }\right| }\mathrm{d}\xi . $$
再用 (7.23) 式推出
$$ {\int }_{-\infty }^{+\infty }\psi \left( x\right) \mathrm{d}x \leq 2{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }\mathrm{d}x{\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi = 4{\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi . $$
证法 2 利用柯西-施瓦兹不等式,
$$ \psi \left( x\right) = {\left( {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }f\left( \xi \right) \mathrm{d}\xi \right) }^{2} = {\left( {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\frac{\left| x - \xi \right| }{2}}{\mathrm{e}}^{-\frac{\left| x - \xi \right| }{2}}\left| f\left( \xi \right) \right| \mathrm{d}\xi \right) }^{2} $$
$$ \leq {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }\mathrm{d}\xi {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi $$
$$ = 2{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left| {x - \xi }\right| }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi . $$
下同证法 1.
证法 3 通过计算求得
$$ {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left( {\left| {x - \xi }\right| + \left| {x - \eta }\right| }\right) }\mathrm{d}x = {\mathrm{e}}^{-\left| {\xi - \eta }\right| }\left( {1 + \left| {\xi - \eta }\right| }\right) , $$
因此
$$ {\int }_{-\infty }^{+\infty }\psi \left( x\right) \mathrm{d}x = {\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\left| {f\left( \xi \right) }\right| \left| {f\left( \eta \right) }\right| \mathrm{d}\xi \mathrm{d}\eta {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\left( {\left| {x - \xi }\right| + \left| {x - \eta }\right| }\right) }\mathrm{d}x $$
$$ = {\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\left| {f\left( \xi \right) }\right| \left| {f\left( \eta \right) }\right| {\mathrm{e}}^{-\left| {\xi - \eta }\right| }\left( {1 + \left| {\xi - \eta }\right| }\right) \mathrm{d}\xi \mathrm{d}\eta $$
$$ \leq \frac{1}{2}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\left( {{\left| f\left( \xi \right) \right| }^{2} + {\left| f\left( \eta \right) \right| }^{2}}\right) $$
$$ \times {\mathrm{e}}^{-\left| {\xi - \eta }\right| }\left( {1 + \left| {\xi - \eta }\right| }\right) \mathrm{d}\xi \mathrm{d}\eta $$
$$ = {\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi {\int }_{-\infty }^{+\infty }\left( {1 + \left| {\xi - \eta }\right| }\right) {\mathrm{e}}^{-\left| {\xi - \eta }\right| }\mathrm{d}\eta $$
$$ = {\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi {\int }_{-\infty }^{+\infty }\left( {1 + \left| u\right| }\right) {\mathrm{e}}^{-u}\mathrm{\;d}u = 4{\int }_{-\infty }^{+\infty }{\left| f\left( \xi \right) \right| }^{2}\mathrm{\;d}\xi . $$