= Fejér kernel
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{title2=$F_n$}
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Use $N\ge1$ terms in the arithmetic mean. The <Fejér kernel> is
$$
F_N(t)=\frac1N\sum_{j=0}^{N-1}D_j(t)
=\sum_{|k|<N}\left(1-\frac{|k|}{N}\right)e^{ikt}
=\frac1N\left|\sum_{j=0}^{N-1}e^{ijt}\right|^2.
$$
Here $D_j$ is the <Dirichlet kernel>. Counting pairs with a fixed index difference proves the middle coefficient formula. The finite <geometric series> gives
$$
F_N(t)=\frac1N\frac{\sin^2(Nt/2)}{\sin^2(t/2)},\qquad F_N(0)=N.
$$
The squared-modulus form proves nonnegativity, and the constant Fourier coefficient gives $(2\pi)^{-1}\int_{-\pi}^{\pi}F_N=1$. Thus $K_N=F_N/(2\pi)$ is the unit-mass convolution density. Another common half-kernel convention uses $H_N=F_N/2$ with convolution prefactor $1/\pi$; these normalizations must be distinguished.
For each fixed $\delta>0$, $F_N(t)\le[N\sin^2(\delta/2)]^{-1}$ when $\delta\le|t|\le\pi$, so the mass away from zero tends to zero. Positivity, total mass and this concentration make the kernels an <approximate identity>, explaining <Fejér's theorem>. In this convention the degree is $N-1$, while $N$ is the number of averaged partial sums.<245C, Notes 2: The Fourier transform by Terence Tao>[§2, Exercise 26, Cesàro averaging and the positive Fejér kernel]{ref}
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