Show that $\sup_{0<c<\delta<\frac{1}{2}}\left|\int_{c}^{\delta}\frac{\sin \pi(2n+1)t}{\sin\pi t}dt\right|<\infty$

Question

As I was trying to understand the proof of Jordan's theorem, I'm trying to show that $$\sup_{0<c<\delta<\frac{1}{2}}\left|\int_{c}^{\delta}\frac{\sin \pi(2n+1)t}{\sin\pi t}dt\right|<\infty$$

I found a similar question on here, but since the denominator has another $\sin \pi t$, changing the bounds would not do the trick. I tried to break it down to two different functions so I can use integration by parts, but that didn't really get me anywhere. I think I do have to use the fact that $\int_0^{\infty}\frac{\sin x}{x}dx=\frac{\pi}{2}$, but I'm not too sure.

Answer 1

Consider $$ \left|\int_{c}^{\delta}\frac{\sin \pi(2n+1)t}{\sin\pi t}dt-\int_{c}^{\delta}\frac{\sin \pi(2n+1)t}{\pi t}dt \right| $$ and note that for $0<x<\pi/2$, $$ \left|\frac{1}{\sin x}-\frac{1}{x}\right|\le cx $$ where $c$ is a constant.

Answer 2

Use a trigonometric identity to show

$$ \frac{\sin\pi(2n+1)t}{\sin\pi t}=1+2\sum_{m=1}^n\cos2m\pi t $$

This identity is one of the Lagrange's trigonometric identities and this function is called the Dirichlet kernel