关于$\bf{Riemann-Lebesgue引理}$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

$\bf命题：$