Dirichlet Kernel

Definition

$D_N(u)=\frac{1}{2}+{\sum }_{n=1}^{N}cos(nu)=\frac{\sin (Nu+\frac{1}{2}u)}{2\sin \frac{u}{2}}$

Facts

${\int }_0^{\pi }{D}_N(u)du=\frac{\pi }{2}$

$D_N(u)=\frac{\sin (\frac{u}{2}+N_u)}{2\sin (\frac{u}{2})}$ where $u\ne 2k\pi \forall k=0,\dots ,n$