Fejer kernel

The Fejer kernel $F_{n}$ of order $n$ is defined as

F_{n}(t)=\frac{1}{n}\sum_{k=0}^{n-1}D_{k}(t),

where $D_{n}$ is the Dirichlet kernel of order $n$ . The Fejer kernel can be written as

F_{n}(t)=\frac{1}{n}\left(\frac{\sin\frac{nt}{2}}{\sin\frac{t}{2}}\right)^{2}.

(1)

Proof: Since

D_{n}(t)=\frac{\sin\left(\left(n+\frac{1}{2}\right)t\right)}{\sin\frac{t}{2}}

we have

\sin\frac{t}{2}D_{n}(t)=\sin\left(\left(n+\frac{1}{2}\right)t\right).

Therefore

	$\displaystyle n\sin^{2}\frac{t}{2}F_{n}(t)$	$\displaystyle=\sum_{k=0}^{n-1}\sin\left(\left(k+\frac{1}{2}\right)t\right)\sin% \frac{t}{2}$
		$\displaystyle=\frac{1}{2}\sum_{k=0}^{n-1}(\cos kt-\cos((k+1)t)$
		$\displaystyle=\frac{1}{2}(1-\cos nt)$
		$\displaystyle=\sin^{2}\frac{nt}{2}.$

From this follows equation (1).