Fejér kernel

In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

Plot of several Fejér kernels

Definition

The Fejér kernel is defined as

$F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x),$

where

$D_k(x)=\sum_{s=-k}^k {\rm e}^{isx}$

is the kth order Dirichlet kernel. It can also be written in a closed form as

$F_n(x) = \frac{1}{n} \left(\frac{\sin \frac{n x}{2}}{\sin \frac{x}{2}}\right)^2 = \frac{1}{n} \left(\frac{1 - \cos(nx)}{1 - \cos x}\right)$,

where this expression is defined.^[1]

The Fejér kernel can also be expressed as

$F_n(x)=\sum_{|j|\le n}\left(1-\frac{|j|}{n}\right)e^{ijx}$.

Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is $F_n(x) \ge 0$ with average value of $1$.

Convolution

The convolution F_n is positive: for $f \ge 0$ of period $2 \pi$ it satisfies

$0 \le (f*F_n)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y) F_n(x-y)\,dy.$

Since $f*D_n=S_n(f)=\sum_{|j|\le n}\widehat{f}_je^{ijx}$, we have $f*F_n=\frac{1}{n}\sum_{k=0}^{n-1}S_k(f)$, which is Cesàro summation of Fourier series.

By Young's inequality,

$\|F_n*f \|_{L^p([-\pi, \pi])} \le \|f\|_{L^p([-\pi, \pi])}$ for every $1 \le p \le \infty$

for $f\in L^p$.

Additionally, if $f\in L^1([-\pi,\pi])$, then

$f*F_n \rightarrow f$ a.e.

Since $[-\pi,\pi]$ is finite, $L^1([-\pi,\pi])\supset L^2([-\pi,\pi])\supset\cdots\supset L^\infty([-\pi,\pi])$, so the result holds for other $L^p$ spaces, $p\ge1$ as well.

If $f$ is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.

• One consequence of the pointwise a.e. convergence is the uniquess of Fourier coefficients: If $f,g\in L^1$ with $\hat{f}=\hat{g}$, then $f=g$ a.e. This follows from writing $f*F_n=\sum_{|j|\le n}\left(1-\frac{|j|}{n}\right)e^{ijt}$, which depends only on the Fourier coefficients.
• A second consequence is that if $\lim_{n\to\infty}S_n(f)$ exists a.e., then $\lim_{n\to\infty}F_n(f)=f$ a.e., since Cesàro means $F_n*f$ converge to the original sequence limit if it exists.

See also

• Fejér's theorem
• Dirichlet kernel
• Gibbs phenomenon
• Charles Jean de la Vallée-Poussin

References

1. Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.