Fejér kernel

Definition

The Fejér kernel has many equivalent definitions. Three such definitions are outlined 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

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. A proof of this result goes as follows.

Using the fact that the Dirichlet kernel may be written as: :D_k(x)=\frac{ \sin((k+\frac{1}{2})x)}{\sin\frac{x}{2}}, one obtains from the definition of the Fejér kernel above: :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]

By the trigonometric identity: \sin(\alpha)\cdot\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta)), one has :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)], which allows evaluation of F_n(x) as a telescoping sum: :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 :fD_n=S_n(f)=\sum_{|j|\le n}\widehat{f}_je^{ijx} , we have :fF_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

References

1. Hoffman, Kenneth. (1988). "Banach Spaces of Analytic Functions". Dover.
2. Königsberger, Konrad. "Analysis 1". Springer.