How to show a Fejér kernel is a good kernal??

Question

I can prove the other two properties,but I cant show that the integration of the modulus of Fejér kernel is bdd,that is

$\int$ |$K_n$|$\leq $ $M$ $for$ $all$ $n$ $\geq$$1$

Answer 1

Just note Fejer kernel can be expressed as $F_N(x)=\frac{\sin^2(\frac{Nx}{2})}{N\sin^2(\frac{x}{2})}$, which is positive, so it follows from property 1 with $M=1$.