Fourier Series: Deriving Fejer's Kernel

PROBLEM

Derive Fejer's Kernel (we will use Dirichlet's Kernel), see the other post. 

SOLUTION

Note that $D_n(x)$ is the Dirichlet Kernel.

\begin{eqnarray*}
K_n(x)&=&(1/n)(D_0(x)+D_1(x)+\dots+D_{n-1}(x))\\
&=& (1/n)\sum_0^{j=n-1}\sum_{k=-j}^j e^{ikx}\\
&=& (1/n)\sum_{k=-n-1}^{n-1}(1-|k|)e^{ikx}\\
&=&  \sum_{k=-n-1}^{n-1}(1-|k|/n)e^{ikx}\\
&=& (1/n) \left(\displaystyle\frac{\sin(n+1/2)x}{\sin(x/2)}\right)^2
\end{eqnarray*}

As before the above holds for $0<|x|\leq \pi$. If $x=0$, then $K_n(x)=n$.