Fejér kernel

Remarks

For $x$ not a multiple of $2\pi$, the closed form is $$F_N(x) = \frac{1}{N} \left( \frac{\sin(Nx/2)}{\sin(x/2)} \right)^2,$$ and $F_N(2\pi m) = N$ for integers $m$. So $F_N \ge 0$, $F_N$ is $2\pi$-periodic, and $\frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(x) \, dx = 1$, because only the term $k = 0$ has a nonzero integral. For $\delta \le \abs{x} \le \pi$ one has $F_N(x) \le \frac{1}{N \sin^2(\delta/2)}$, which tends to $0$.

Example: $F_1 = D_0 = 1$, and $F_2(x) = \frac12 (1 + (1 + 2\cos x)) = 1 + \cos x$, which agrees with $\frac12 \left(\frac{\sin x}{\sin(x/2)}\right)^2 = 2\cos^2(x/2)$.

Non-example: the Dirichlet kernel $D_1(x) = 1 + 2\cos x$ is not a Fejér kernel, and it takes the negative value $-1$ at $x = \pi$.

For a $2\pi$-periodic integrable function $f$ with Fourier series partial sums $S_n f$, one has $\frac{1}{N} \sum_{n=0}^{N-1} S_n f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(x - t) f(t) \, dt$.