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During my work, I encounter the function like $\frac{\sin(n \omega)}{n \sin \omega}$. I'm puzzled and knew nothing about this function before.

Given integer $n>1$, my question is how to find a simple function to bound $|\frac{\sin(n \omega)}{n \sin \omega}|$ as tight as possible for $0 \le \omega \le \pi/2$. That is to say, we need to find a polynomial $g(n,\omega)$ such that $|\frac{\sin(n \omega)}{n \sin \omega}| \le g(n,\omega)$ and $g(n,\omega)$ is as close to $|\frac{\sin(n \omega)}{n \sin \omega}|$ as possible. To the simpleness, I think polynomials with lower order meet the requirements.

My idea is to bound $|\frac{\sin(n \omega)}{n \sin \omega}|$ individually. For $0 \le \omega \le \frac{\pi}{2n}$, we bound $|\frac{\sin(n \omega)}{n \sin \omega}|$ by Taylor expansion at $\omega = 0$. However, when $\frac{\pi}{2n} \le \omega \le \frac{\pi}{2}$, I don't know how to find a appropriate polynomial to bound $|\frac{\sin(n \omega)}{n \sin \omega}|$.

For $\frac{\pi}{2n} \le \omega \le \frac{\pi}{2}$, the envelope of $|\frac{\sin(n \omega)}{n \sin \omega}|$ is $\frac{1}{n \sin \omega}$, i.e., $|\frac{\sin(n \omega)}{n \sin \omega}| \le \frac{1}{n \sin \omega}$. One of my ideas is to find a simple polynomial to bound $\frac{1}{n \sin \omega}$ as tight as possible. But how to find?

I do some experiments with matlab. Let $n=40$. Figure 1 is $\frac{\sin(n \omega)}{n \sin \omega}$. Figure 2 is $|\frac{\sin(n \omega)}{n \sin \omega}|$. In Figure 3, the red line is the Taylor expansion at $\omega=0$, the green line is $\frac{1}{n \sin \omega}$ with $\frac{\pi}{2n} \le |\omega| \le \frac{\pi}{2}$.

Thanks for any help!

Figure 1 Figure 2 Figure 3

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I've removed the irrelevant "triangulated-categories" tag. – Leonid Positselski Dec 25 '10 at 23:04
Your function can be expressed in terms of the Chebyshev polynomial of the second kind: $$\frac1{n}U_{n-1}(\cos\omega)$$ – J. M. Dec 26 '10 at 0:39
Now, why do you want to bound it by a "simpler" function (clearly, it is assumed to be "simpler" for some particular purpose but until we know what that purpose is, it is hard to tell what is helpful and what isn't.) – fedja Dec 26 '10 at 2:04
For instance, if you are integrating this function against another function it is better to use the oscillating nature than to bound it by absolute value. It looks very similar to the Dirichlet kernel. – timur Dec 26 '10 at 2:32

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