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

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

where this expression is defined.

The Fejér kernel can also be expressed as

$F_{n}(x)=\sum _{|k|\leq n-1}\left(1-{\frac {|k|}{n}}\right)e^{ikx}.$

Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is $F_{n}(x)\geq 0$ with average value of $1$.

Convolution

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

$0\leq (f*F_{n})(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)F_{n}(x-y)\,dy.$

Since $f*D_{n}=S_{n}(f)=\sum _{|j|\leq n}{\widehat {f}}_{j}e^{ijx}$, 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 convolution inequality,

$\|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}{\text{ for every }}1\leq p\leq \infty {\text{ for }}f\in L^{p}.$

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, $p\geq 1$ 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 uniqueness of Fourier coefficients: If $f,g\in L^{1}$ with ${\hat {f}}={\hat {g}}$, then $f=g$ a.e. This follows from writing $f*F_{n}=\sum _{|j|\leq n}\left(1-{\frac {|j|}{n}}\right){\hat {f}}_{j}e^{ijt}$, 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