Using Riemann-Lebesgue lemma

Question

I am reading a paper by VAALER. He is using Riemann-Lebesgue lemma and saying that below function tends to $0$ as $N \to \infty$

$$\int_{-2}^{2} \frac{\pi t}{\sin \pi t}(\cos \pi(2N+1)t)e^{2 \pi i t z} dt$$

Before this he has not written anything. I am not able to understand why we can use that theorem. If you can give any hint.

Thanks

Answer 1

Suppose $f$ is absolutely integrable on some finite interval $[a,b]$ of $\mathbb{R}$. Then one form of the Riemann-Lebesgue Lemma states that $$ \lim_{r\rightarrow\infty}\int_{a}^{b}f(t)\sin(rt+\phi)dt =0, \;\;\;\forall \phi\in\mathbb{R}. $$

You can use this version to get what you want. The above holds for $\sin$ if $\phi=0$ and it holds for $\cos$ if $\phi=-\pi/2$. Then $2\pi(N+1)$ becomes $r$.