关于黎曼—勒贝格引理的专题讨论

$\bf命题：（Riemann-Lebesgue引理）$设函数$f\left( x \right)$在$\left[ {a,b} \right]$上可积，则

\[\mathop {\lim }\limits_{\lambda  \to {\rm{ + }}\infty } \int_a^b {f\left( x \right)\sin \lambda xdx}  = 0\]

参考答案

$\bf命题：（Riemann-Lebesgue引理的推广）$ 设函数$f\left( x \right),g\left( x \right)$均在$\left[ {a,b} \right]$上可积，且$g\left( x \right)$以正数$T$为周期，则\[\mathop {\lim }\limits_{\lambda  \to {\rm{ + }}\infty } \int_a^b {f\left( x \right)g\left( {\lambda x} \right)dx}  = \frac{1}{T}\int_0^T {g\left( x \right)dx} \int_a^b {f\left( x \right)dx} \]

参考答案

$\bf命题：$设$f\left( x \right),g\left( x \right) \in C\left( { - \infty , + \infty } \right)$，且对任意$x \in \left( { - \infty , + \infty } \right)$，有$g\left( {x + 1} \right) = g\left( x \right)$，则\[\mathop {\lim }\limits_{n \to \infty } \int_0^1 {f\left( x \right)g\left( {nx} \right)dx}  = \int_0^1 {f\left( x \right)dx} \int_0^1 {g\left( x \right)dx} \]

方法一

$\bf命题：$