I have a problem with the following exercise. I don't really have an idea where to start. I'm glad about every help. So here is the exercise:

Suppose $f\colon \Bbb{R} \to \Bbb{R}$ is a function in $L^1(\Bbb{R})$ (i.e integrable function). Show that $$\lim_{n \to \infty}\int f(x)\sin(nx)dx=0.$$

Thanks already for any help!

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    $\begingroup$ Orange you glad Arturo fixed your title? $\endgroup$ – Cam McLeman Dec 14 '11 at 20:19
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    $\begingroup$ The question does not make sense as written: $\int f(x)\sin(nx)\,dx$ is a family of functions; what does it mean for the family of functions to "go to zero"? $\endgroup$ – Arturo Magidin Dec 14 '11 at 20:19
  • $\begingroup$ Did you mean $\sin( \frac{x}{n})$? $\endgroup$ – Rudy the Reindeer Dec 14 '11 at 20:20
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    $\begingroup$ obviously $\int_a^b\sin(nx)dx\to0$. now approximate $f$ by simple functions (see planetmath.org/encyclopedia/RiemannLebesgueLemma.html) $\endgroup$ – yoyo Dec 14 '11 at 20:30

I'll assume you want to take definite integrals over $\Bbb R$ (see Arturo's comment).

Given $\epsilon>0$, there is an $M$ such that $\int_{[-M,M]^c} |f(x)\sin(nx)|<\epsilon$ for all positive integers $n$. From this, it follows that you need only show that $\lim\limits_{n\rightarrow\infty}\int_{-M}^M f(x)\sin(nx) =0 $ for any fixed $M$.

To do this, you can appeal to the theorem cited by yoyo in the comments; or, you can use the hints in this very similar post

  • $\begingroup$ I hope you can still answer questions here. How was the $M$ above chosen? $\endgroup$ – Kurome May 17 '16 at 15:32
  • $\begingroup$ @Kurome $f$ is in $L_1$ and the absolute value of the $\sin$ terms are uniformly bounded by $1$. So, you can just choose $M$ so that $\int_{[-M,M]^C }|f|<\epsilon$. $\endgroup$ – David Mitra May 17 '16 at 15:46
  • $\begingroup$ That's exactly what I want to know though, how is it possible that such an $M$ exists where $\int_{[-M,M]^C }|f|<\epsilon$. I originally thought that it was possible by the property that if $f$ is integrable in $\mathbb{R}$, then for every $\epsilon>0$, there is a $\delta>0$ s.t. if $A\subseteq R$ is measurable and $m(A)<\delta$ then $\int_{A} |f|<\epsilon$. But I can't seem to get $A=[-M,M]^C$ from that. $\endgroup$ – Kurome May 17 '16 at 16:21
  • $\begingroup$ @Kurome You could apply the Dominated Convergence Theorem to the sequence $(g_n)$ with $g_n =f\cdot\chi_{[-n,n]}$. $\endgroup$ – David Mitra May 17 '16 at 16:27

Here's an answer assuming that you meant to write $\sin \left( \frac{x}{n}\right)$:

You can use the Lebesgue dominated convergence theorem:

You have $|f(x) \sin(\frac{x}{n})| \leq |f(x)|$ which you know is integrable hence you can swap the limit and the integral:

$$ \lim_{n \to \infty} \int f(x) \sin (\frac{x}{n})\ dx = \int f(x) \lim_{n \to \infty} \sin (\frac{x}{n})\ dx = \int f(x) \cdot 0\ dx = 0.$$


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