I want to compute the Fourier series for the following function
$$ g_n(\theta) = -2nK_{n}(\theta)\sin(n\theta)$$
where $K_n(\theta)$ is the Fejer Kernel.
I tried to compute the Fourier coefficients directly using this formula for $K_n(\theta)$
$$ K_n(\theta) = \frac{1}{n} \left(\frac{\sin^2(\frac{n\theta}{2})} {\sin^2(\frac{\theta}{2})}\right) $$
Then the coefficients are given by
\begin{align} c_k & = \frac{1}{2\pi} \int_{-\pi}^{\pi} g_n(\theta)e^{-ik\theta}d\theta \\ & = \frac{1}{2\pi} \int_{-\pi}^{\pi} \left(-2n \frac{1}{n} \left(\frac{\sin^2(\frac{n\theta}{2})} {\sin^2(\frac{\theta}{2})}\right)\sin(n\theta)\right)e^{-ik\theta}d\theta \\ & = -\frac{1}{\pi} \int_{-\pi}^{\pi} \left(\frac{\sin^2(\frac{n\theta}{2})} {\sin^2(\frac{\theta}{2})}\right)\sin(n\theta)e^{-ik\theta}d\theta \\ \end{align}
I'm not sure how to proceed after this. Can someone give me a hint as to how I should compute this Fourier series?
There is a problem with the significance of index $i$.
Don't you want to compute the $k$th coefficient, which is: \begin{align} c_k & = \frac{1}{2\pi} \int_{-\pi}^{\pi} g_n(\theta)e^{-ik\theta}d\theta \ ? \end{align}