if$f,g\mathpunct{:}\mathbb{R}\to \mathbb{R}$satisfy$f,g,fg\in L^1$,and$\exists T(g(x+T)=g(x))$,Is the following equation valid? $$ \lim_{ n \to \infty} \int_{-\infty}^{+\infty} f(x)g(nx) \, \mathrm{d}x =\frac{1}{T}\int_{0}^{T}g(x) \, \mathrm{d}x \int_{-\infty}^{\infty} f(x) \, \mathrm{d}x $$ The teacher at school taught us the proof when g is bounded. I don't know how to handle general situations
Asked
Active
Viewed 72 times
0
-
Why not include the proof of the bounded case in your question? – Lorago Jun 06 '23 at 12:12
-
The bounded case us here. For unbounded, this holds for $L_p$, $p>1$. See here – Mittens Jun 06 '23 at 13:16
-
The assumption on $g$ is not quite correct Since it $g$ is $T$-periodic ($T>0$), $g\in L_1( \mathbb{R})$ would yield $g=0$ a.s. The assumption should be $g\in L_1[0,T]$ instead. – Mittens Jun 06 '23 at 13:56