Fejér kernel

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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[edit]

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} \frac{1 - \cos(nx)}{1 - \cos x} 
,

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^{ijt}.

Properties[edit]

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[edit]

The convolution Fn 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_n(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}S_n(f)=f a.e., since Cesàro means F_n*f converge to the original sequence limit if it exists.

See also[edit]

References[edit]

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