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).