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If $h(x),g(x)$ continuous, and $h(x)$ is periodic function with a period $T$, then prove: $$\lim_{n\to \infty}\int_0^Tg(x)h(nx) dx=\cfrac{1}{T}\left(\int_0^Tg(x)dx\right)\left(\int_0^Th(x)dx\right)$$

I have no idea about this problem. I don’t know how to use the periodicity.

Larry
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Yan Peng
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    Hello and welcome to MSE! What are your thoughts on this problem? What have you tried so far? What related theorems do you know, like for example the Riemann-Lebesgue Lemma and how did you proof it? – F. Conrad Dec 19 '18 at 03:07
  • Do you mean $g(x)$ and $h(x)$, in which $h(x)$ is periodic, or your equation can not fit your assumption. – Nanayajitzuki Dec 19 '18 at 04:07
  • @Nanayajitzuki Oh, yes. Thank you for reminding me. I just found this mistake and I ’m correcting it now. – Yan Peng Dec 19 '18 at 08:29
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    @F.Conrad Thanks for your suggestions. I’ll come to know about the related theorems, like the Riemann-Lebesgue Lemma you mentioned, and rethink the problem. – Yan Peng Dec 19 '18 at 08:42

2 Answers2

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(edited for complement of some critical steps)

I think your assumption actually is If $g(x),h(x)$ are continuous, and $h(x)$ is periodic function with a period $T$.

I will give a hint under this assumption (some details may not be so strict), let

$$\widetilde h(x)=h(x)-\frac1{T} \int_0^{T}{h(t){\text d}t}$$

you will get

$$\int_0^{T}{\widetilde h(x){\text d}x}=\int_0^{T}{h(x){\text d}x}-\int_0^{T}{\left( \frac1{T}\int_0^{T}{h(t){\text d}t}\right){\text d}x}=0$$

so you just need to prove

$$\lim_{n\to \infty}\int_0^{T}{g(x)\widetilde h(nx){\text d}x}=\left(\frac1{T}\int_0^{T}\widetilde h(x){\text d}x\right)\left(\int_0^{T}{g(x){\text d}x}\right)=0$$

let $t=nx$ (here I assume $n$ is any positive real number), where $[n]$ means the integer part of $n$ and $a=T(n-[n])<T$

$$\begin{aligned} \int_0^{T}{g(x)\widetilde h(nx){\text d}x} & = \frac1{n} \int_0^{[n]T+a}{g(t/n)\widetilde h(t){\text d}t} \\ & = \frac1{n} \sum_{k=0}^{[n]} \int_{kT}^{(k+1)T}{g(t/n)\widetilde h(t){\text d}t} + \frac1{n} \int_{[n]T}^{[n]T+a} {g(t/n)\widetilde h(t){\text d}t} \end{aligned}$$

the continuity of $g(x)$ suggests that $g(t/n)$ in the $k$th period should almost performance as a constant when $n\to\infty$, thus for any $k$

$$\int_{kT}^{(k+1)T}{g(t/n)\widetilde h(t){\text d}t} \to g(t_{k}) \int_{kT}^{(k+1)T}{\widetilde h(t){\text d}t} = 0$$

as well notice that $g(x),h(x)$ are continuous, which means they are bounded in the period, assume $|g(x)|\le G$, $|\widetilde h(x)|\le H$

$$\frac1{n} \left| \int_{[n]T}^{[n]T+a} {g(t/n)\widetilde h(t){\text d}t} \right| \le \frac1{n} \int_{[n]T}^{[n]T+a} {|g(t/n)||\widetilde h(t)|{\text d}t} \le \frac{G}{n}\int_0^{a}{|\widetilde h(t)|{\text d}t} \le \frac{aGH}{n} \to 0$$

just for curiosity, in general:

If $h(x)$ is Lebesgue measurable and periodic in $\mathbb R$, $I$ is any interval, $g(x) \in \mathcal L(I)$, we have

$$\lim_{|\lambda| \to +\infty}\int_I {g(x)h(\lambda x){\text d}x}=\left(\frac1{T}\int_0^{T}h(x){\text d}x\right)\left(\int_I{g(x){\text d}x}\right)$$

under the meaning of Lebesgue integration.

Nanayajitzuki
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1

If the periodic one is $g$, the equality does not hold. For instance, take $g(x)=\sin^2x$, $h(x)=x$, $T=\pi$.

When $h$ has period $T$, use first the substitution $u=nx$ to get $$ \int_0^Tg(x)\,h(nx)\,dx=\frac1n\int_0^{nT} g(x/n)\,h(x)\,dx=\frac1n\sum_{k=0}^{n-1}\int_{kT}^{(k+1)T} g(x/n)\,h(x)\,dx. $$ Next substitute $v=x-kT$, to obtain \begin{align} \int_0^Tg(x)\,h(nx)\,dx&=\frac1n\sum_{k=0}^{n-1}\int_{0}^{T} g(\tfrac{x+kT}n)\,h(x+kT)\,dx=\frac1n\sum_{k=0}^{n-1}\int_{0}^{T} g(\tfrac{x+kT}n)\,h(x)\,dx\\ &=\frac1T\int_{0}^{T} \left(\sum_{k=0}^{n-1} g(\tfrac{x+kT}n)\,\frac Tn\right)\,h(x)\,dx. \end{align} Because $g$ is uniformly continuous on $[0,T]$, the values $g(\tfrac{x+kT}n)$ are very close to $g(\tfrac kTn)$. So $$ \lim_{n\to\infty}\sum_{k=0}^{n-1} g(\tfrac{x+kT}n)\,\frac Tn =\lim_{n\to\infty}\sum_{k=0}^{n-1} g(\tfrac{kT}n)\,\frac Tn=\int_0^Tg(x)\,dx. $$

Martin Argerami
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