Hi, I need an estimation or an exact closed form expression for the following integral

$\int_{0}^{2\pi} K_N^4(s) ds $

where $K_N(s)= \frac{1}{N2\pi} (\frac{sin(Ns/2)}{sin(s/2)})^2$, the Fejer kernel.

I don't know how to obtain an estimation better than

$\int_{0}^{2\pi} K_N^4(s) ds < N^4$

Does anyone know a better estimation or some trigonometric tricks that can help me to improve my estimation?

Thanks in advance

Imma

integerpower of $K_N$ you can get an exact formula by expanding the exponential sum, though this gets tiresome past the first few cases. For any power, you get an easy upper bound from $K_N(s) < \min(N, 1/\sin(s/2)^2)$ [I assume that the factor of $1/2\pi$ should be applied to the integral, not to $K_N(s)$], and this should be within a small factor of the truth. – Noam D. Elkies Jul 29 '12 at 23:43