Fejér kernel

File:Fejér kernels.svg

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).

Definition

The Fejér kernel has many equivalent definitions. We outline three such definitions below:

1) The traditional definition expresses the Fejér kernel $F_n(x)$ in terms of the Dirichlet kernel: $F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x)$

where

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

is the kth order Dirichlet kernel.

2) The Fejér kernel $F_n(x)$ may also be written in a closed form expression as follows{{cite book |last=Hoffman |first=Kenneth |title=Banach Spaces of Analytic Functions |publisher=Dover |year=1988 |isbn=0-486-45874-1 |page=17}}

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

This closed form expression may be derived from the definitions used above. The proof of this result goes as follows.

First, we use the fact that the Dirichlet kernel may be written as:{{Cite book |last=Konigsberger |first=Konrad |title=Analysis 1 |publisher=Springer |edition=6th |pages=322 |language=German}}

: $D_k(x)=\frac{
\sin((k
+\frac{1}{2})x)}{\sin\frac{x}{2}}$

Hence, using the definition of the Fejér kernel above we get:

: $F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x) = \frac{1}{n} \sum_{k=0}^{n-1} \frac{
\sin((k
+\frac{1}{2})x)}{\sin(\frac{x}{2})}
= \frac{1}{n} \frac{1}{\sin(\frac{x}{2})}\sum_{k=0}^{n-1}
\sin((k
+\frac{1}{2})x) = \frac{1}{n} \frac{1}{\sin^2(\frac{x}{2})}\sum_{k=0}^{n-1}
\big[\sin((k+\frac{1}{2})x) \cdot \sin(\frac{x}{2})\big]$

Using the trigonometric identity: $\sin(\alpha)\cdot\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))$

: $F_n(x) =\frac{1}{n} \frac{1}{\sin^2(\frac{x}{2})}\sum_{k=0}^{n-1}
[\sin((k
+\frac{1}{2})x) \cdot \sin(\frac{x}{2})] = \frac{1}{n} \frac{1}{2\sin^2(\frac{x}{2})}\sum_{k=0}^{n-1}
[\cos(kx)-\cos((k+1)x)]$

Hence it follows that:

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

3) 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 $F_n(x) \ge 0$ 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 convolution inequality,

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

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, $p\ge1$ 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 uniqueness of Fourier coefficients: If $f,g\in L^1$ with $\hat{f}=\hat{g}$ , then $f=g$ a.e. This follows from writing $f*F_n=\sum_{|j|\le n}\left(1-\frac{|j$

{n}\right)\hat{f}_je^{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.

=Applications=

The Fejér kernel is used in signal processing and Fourier analysis.

References

Category:Fourier series