关于黎曼—勒贝格引理的专题讨论

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

[mathop {lim }limits_{lambda   o { m{ + }}infty } int_a^b {fleft( x ight)sin lambda xdx}  = 0]

参考答案

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

参考答案

$f命题：$设$fleft( x ight),gleft( x ight) in Cleft( { - infty , + infty } ight)$，且对任意$x in left( { - infty , + infty } ight)$，有$gleft( {x + 1} ight) = gleft( x ight)$，则[mathop {lim }limits_{n o infty } int_0^1 {fleft( x ight)gleft( {nx} ight)dx}  = int_0^1 {fleft( x ight)dx} int_0^1 {gleft( x ight)dx} ]

方法一

$f命题：$