---
tag: 3E740YTT9V
kind: definition
name: Fejér kernel
msc: ["42A24", "40G05"]
defines: DCCA2T9GKH
uses: [G12KXDHPPS, 0P0DJHBGBB]
sources:
  - url: https://en.wikipedia.org/wiki/Fejér_kernel
---

## Definition

For an integer $n \ge 0$, the Dirichlet kernel is the function $D_n(x) = \sum_{k=-n}^{n} e^{ikx}$ for real $x$.
For an integer $N \ge 1$, the *Fejér kernel* $F_N$ is the [[Cesàro mean|0P0DJHBGBB]] of $D_0, \dots, D_{N-1}$:
$$F_N(x) = \frac{1}{N} \sum_{n=0}^{N-1} D_n(x) = \sum_{k=-(N-1)}^{N-1} \left(1 - \frac{\abs{k}}{N}\right) e^{ikx}.$$

## 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|G12KXDHPPS]] partial sums $S_n f$, one has $\frac{1}{N} \sum_{n=0}^{N-1} S_n f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(x - t) f(t) \, dt$.